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1 Basic Math Review
Numbers
NATURAL NUMBERS
{1, 2, 3, 4, 5, …}
WHOLE NUMBERS
{0, 1, 2, 3, 4, …}
INTEGERS
{…, 3, 2, 1, 0, 1, 2, …}
RATIONAL NUMBERS
All numbers that can be written in the form , where a
and b are integers and .
IRRATIONAL NUMBERS
Real numbers that cannot be written as the quotient of two
integers but can be represented on the number line.
REAL NUMBERS
Include all numbers that can be represented on the number
line, that is, all rational and irrational numbers.
PRIME NUMBERS
A prime number is a number greater than 1 that has only
itself and 1 as factors.
Some examples:
2, 3, and 7 are prime numbers.
COMPOSITE NUMBERS
A composite number is a number that is not prime. For
example, 8 is a composite number since
8 = 2 . # 2 # 2 = 23
Rational Numbers
Real Numbers
23, 22.4, 21 , 0, 0.6, 1, etc. 4_ 2 5
25VN
Irrational
Numbers
Integers p 23, 22, 21, 0, 1, 2, 3, p
Whole Numbers 0, 1, 2, 3, p
Natural Numbers 1, 2, 3, p
3,
VN2, p, etc.
b Z 0
a>b
–5 –4 –3
Negative integers Positive integers
The Number Line
Zero
–2 –1 0 1 2 3 4 5
ISBN-13:
ISBN-10:
978-0-321-39476-7
0-321-39476-3
9 780321 394767
90000
Int Properties
PROPERTIES OF ADDITION
Identity Property of Zero:
Inverse Property:
Commutative Property:
Associative Property:
PROPERTIES OF MULTIPLICATION
Property of Zero:
Identity Property of One: , when .
Inverse Property: , when .
Commutative Property:
Associative Property:
PROPERTIES OF DIVISION
Property of Zero: , when .
Property of One: , when .
Identity Property of One:
Absolute Value
The absolute value of a number is always ≥ 0.
If , .
If , .
For example, and . In each case, the
answer is positive.
ƒ -5 ƒ = 5 ƒ 5 ƒ = 5
a 6 0 ƒ a ƒ = a
a 7 0 ƒ a ƒ = a
a
1 = a # 1
a Z 0 a
a = 1
a Z 0
0
a = 0
a # 1b # c2 = 1a # b2 # c
a # b = b # a
a # a Z 0
1
a = 1
a # 1 = a a Z 0
a # 0 = 0
a + 1b + c2 = 1a + b2 + c
a + b = b + a
a + 1-a2 = 0
a + 0 = a
Key Words
Text
8. Your Turn: (No Calculator first, then check.) a) 0.65 = b) 2.666 = c) 0.54 = Page 12 of 40 d) 3.14 = 9. Converting Fractions into Decimals Converting fractions into decimals is based on place value. For example, applying what you have understood about equivalent fractions, we can easily convert 2 5 into a decimal. First we need to convert to a denominator that has a 10 base. Let’s convert 2 5 into tenths → 2×2 5×2 = 4 10 ∴ we can say that two fifths is the same as four tenths: 0.4 Converting a fraction to decimal form is a simple procedure because we simply use the divide key on the calculator. Note: If you have a mixed number, convert it to an improper fraction before dividing it on your calculator. Example problems: 1. 2 3 = 2 ÷ 3 = 0.66666666666 … ≈ 0.67 2. 3 8 = 3 ÷ 8 = 0.375 3. 17 3 = 17 ÷ 3 = 5.6666666 … ≈ 5.67 4. 3 5 9 = (27 + 5) ÷ 9 = 3.555555556 … ≈ 3.56 9. Your Turn: (Round your answer to three decimal places where appropriate) a) 17 23 = b) 5 72 = c) 56 2 3 = d) 29 5 = Watch this short Khan Academy video for further explanation: “Converting fractions to decimals” (and vice versa) https://www.khanacademy.org/math/pre-algebra/decimals-pre-alg/decimal-to-fraction-pre-alg/v/converting-fractions-to-decimals 10. Fraction Addition and Subtraction Adding and subtracting fractions draws on the concept of equivalent fractions. The golden rule is that you can only add and subtract fractions if they have the same denominator, for example, 1 3 + 1 3 = 2 3 . However, if two fractions do not have the same denominator, we must use equivalent fractions to find a “common denominator” before they can be added together. For instance, we cannot simply add 1 4 + 1 2 because these fractions have different denominators (4 and 2). As such, arriving at an answer of 2 6 (two sixths) would be incorrect. Before these fractions can be added together, they must both have the same denominator. Page 13 of 40 From the image at right, we can see that we have three quarters of a whole cake. So to work this abstractly, we need to decide on a common denominator, 4, which is the lowest common denominator. Now use the equivalent fractions concept to change 1 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2 4 by multiplying both the numerator and denominator by two: 1 2 x 2 2 = 2 4 Now that the denominators are the same, the addition can be carried out: 1 4 + 2 4 = 3 4 Let’s try another: 1 3 + 1 2 We cannot simply add these fractions; again we need to find the lowest common denominator. The easiest way to do this is to multiply the denominators: 1 3 𝑎𝑎𝑎𝑎𝑎𝑎 1 2 (2 x 3 = 6). Therefore, both fractions can have a denominator of 6, yet we need to change the numerator. The next step is to convert both fractions into sixths as an equivalent form. How many sixths is one third? 1 3 = 2 6 � 1 3 × 2 2 = 2 6 � And how many sixths is one half? 1 2 = 3 6 � 1 2 × 3 3 = 3 6 � Therefore: 1 3 + 1 2 = 2 6 + 3 6 = 5 6 With practise, a pattern forms, as is illustrated in the next example: 1 3 + 2 5 = (1 × 5) + (2 × 3) (3 × 5) = 5 + 6 15 = 11 15 In the example above, the lowest common denominator is found by multiplying 3 and 5, and then the numerators are multiplied by 5 and 3 respectively. Use the following problems to reinforce the pattern. 10. Your Turn: a) 1 3 + 2 5 = b) 3 4 + 2 7 = Subtraction is the same procedure but with a negative symbol: 2 3 − 1 4 = (2 × 4) − (1 × 3) (3 × 4) = 8 − 3 12 = 5 12 10. (continued) Your Turn: e) 9 12 − 1 3 = f) 1 3 − 1 2 = Page 14 of 40 Watch this short Khan Academy video for further explanation: “Adding and subtraction fractions” https://www.khanacademy.org/math/arithmetic/fractions/fractions-unlike-denom/v/adding-and-subtracting-fractions 11. Fraction Multiplication and Division Compared to addition and subtraction, multiplication and division of fractions is easy to do, but sometimes a challenge to understand how and why the procedure works mathematically. For example, imagine I have 1 2 of a pie and I want to share it between 2 people. Each person gets a quarter of the pie. Mathematically, this example would be written as: 1 2 × 1 2 = 1 4 . Remember that fractions and division are related; in this way, multiplying by a half is the same as dividing by two. So 1 2 (two people to share) of 1 2 (the amount of pie) is 1 4 (the amount each person will get). But what if the question was more challenging: 2 3 × 7 16 =? This problem is not as easy as splitting pies. A mathematical strategy to use is: “Multiply the numerators then multiply the denominators” Therefore, 2 3 × 7 16 = (2×7) (3×16) = 14 48 = 7 24 However, we can also apply a cancel out method – which you may recall from school. The rule you may recall is, ‘What we do to one side, we must do to the other.’ Thus, in the above example, we could simplify first: 2 3 × 7 16 = ? The first thing we do is look to see if there are any common multiples. Here we can see that 2 is a multiple of 16, which means that we can divide top and bottom by 2: 2÷2 3 × 7 16÷2 = 1 3 × 7 8 = 1×7 3×8 = 7 24 Example Multiplication Problems: 1. 4 9 × 3 4 = (4×3) (9×4) = 12 36 = 1 3 Have a go at simplifying first and then perform the multiplication. 4÷4 9÷3 × 3÷3 4÷4 = (1×1) (3×1) = 1 3 2. 2 4 9 × 3 3 5 = 22 9 × 18 5 = (18×22) (9×5) = 396 45 = 396 ÷ 45 = 8.8 22 9÷9 × 18÷9 5 = 22×2 1×5 = 44 5 so 44 ÷ 5 = 8 4 5 Page 15 of 40 Watch this short Khan Academy video for further explanation: “Multiplying negative and positive fractions” https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-fractions-decimals/cc-7th-mult-div-frac/v/multiplyingnegative-and-positive-fractions Division of fractions seems odd, but it is a simple concept: You may recall the expression ‘invert and multiply’ which means we flip the fraction; we switch the numerator and the denominator. Hence, ÷ 1 2 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 × 2 1 If the sign is swapped to its opposite, the fraction is flipped upside down, this ‘flipped’ fraction is referred to as the reciprocal of the original fraction. Therefore, 2 3 ÷ 1 2 is the same as 2 3 × 2 1 = (2×2) (3×1) = 4 3 = 1 1 3 Note: dividing by half doubled the answer. Example Division Problems: 1. 2 3 ÷ 3 5 = 2 3 × 5 3 = (2×5) (3×3) = 10 9 = 1 1 9 2. 3 3 4 ÷ 2 2 3 = 15 4 ÷ 8 3 = 15 4 × 3 8 = (15×3) (4×8) = 45 32 = 1 13 32 Watch this short Khan Academy video for further explanation: “Dividing fractions example” https://www.khanacademy.org/math/arithmetic/fractions/div-fractions-fractions/v/another-dividing-fractions-example 11. Your Turn: a) Find the reciprocal of 2 2 5 b) 2 3 × 7 13 = c) 1 1 6 × 2 9 = d) 8 × 3 4 × 5 6 × 1 1 2 = e) 3 7 ÷ 2 5 = f) 2 2 5 ÷ 3 8 9 = g) (−25)÷(−5) 4−2×7 = h) −7 2 ÷ −4 9 = i) If we multiply 8 and the reciprocal of 2, what do we get? j) What is 40 mulitplied by 0.2? (use your knowledge of fractions to solve) Page 16 of 40 12. Percentage The concept of percentage is an extension of the material we have already covered about fractions. To allow comparisons between fractions we need to use the same denominator. As such, all percentages use 100 as the denominator. The word percent or “per cent” means per 100. Therefore, 27% is 27 100 . To use percentage in a calculation, the simple mathematical procedure is modelled below: For example, 25% of 40 is 25 100 × 40 = 10 Percentages are most commonly used to compare parts of an original. For instance, the phrase ‘30% off sale,’ indicates that whatever the original price, the new price is 30% less. However, percentages are not often as simple as, for example, 23% of 60. Percentages are commonly used as part of a more complex question. Often the questions might be, “How much is left?” or “How much was the original?” Example problems: A. An advertisement at the chicken shop states that on Tuesday everything is 22% off. If chicken breasts are normally $9.99 per kilo. What is the new per kilo price? Step 1: SIMPLE PERCENTAGE: 22 100 × 9.99 = 2.20 Step 2: DIFFERENCE: Since the price is cheaper by 22%, $2.20 is subtracted from the original: 9.99 – 2.20 = $7.79 B. For the new financial year you have been given an automatic 3.5% pay rise. If you were earning $17.60 per hour what would be your new rate? Step 1: SIMPLE PERCENTAGE 3.5 100 × 17.6 = 0.62 Step 2: DIFFERENCE: Since it is a 3.5% pay RISE, $0.62 is added to the original: 17.60 + 0.62 = $18.22 C. A new dress is now $237 reduced from $410. What is the percentage difference? As you can see, the problem is in reverse, so we approach it in reverse. Step 1: DIFFERENCE: Since it is a discount the difference between the two is the discount. Thus we need to subtract $237.00 from $410 to see what the discount was that we received. $410 – $237 = $173 Step 2: SIMPLE PERCENTAGE: now we need to calculate what percentage of $410 was $173, and so we can use this equation: 𝑥𝑥 100 × 410 = 173 We can rearrange the problem in steps: 𝑥𝑥 100 × 410÷410 = 173÷410 this step involved dividing 410 from both sides to get 𝑥𝑥 100 = 173 410 Next we work to get the 𝑥𝑥 on its own, so we multiply both sides by 100. Now we have 𝑥𝑥 = 173 410 × 100 1 Next we solve, so 0.42 multiplied by 100, ∴ 0.42 × 100 and we get 42. ∴ The percentage difference was 42%. Let’s check: 42% of $410 is $173, $410 - $173 = $237, the cost of the dress was $237.00 . 12. Your Turn: a) GST adds 10% to the price of most things. How much does a can of soft drink cost if it is 80c before GST? b) A Computer screen was $299 but is on special for $249. What is the percentage discount? c) Which of the following is the largest? 3 5 𝑜𝑜𝑜𝑜 16 25 𝑜𝑜𝑜𝑜 0.065 𝑜𝑜𝑜𝑜 63%? (Convert to percentages) Page 17 of 40 12 (continued). Activity: What do I need to get on the final exam??? Grade (%) Weight (%) Assessment 1 30.0% 10% Assessment 2 61.0% 15% Assessment 3 73.2% 30% Assessment 4 51.2% 5% Final Exam 40% Final Grade Overall Percentage Needed High Distinction 100 - 85% Distinction 84 - 75% Credit 74 - 65% Pass 64 - 50% Fail 49 - 0% 1. How much does each of my assessments contribute to my overall percentage, which determines my final grade? Calculation Overall % Assessment 1 This assessment contributes a maximum of 10% to my overall percentage and I scored 30.0% of those 10%, so 30.0 ÷ 100 x 10 = 3 𝟑𝟑𝟑𝟑 𝟏𝟏𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 𝟏𝟏 = 𝟑𝟑 3.0 % Assessment 2 This assessment contributes a maximum of 15% to my overall percentage and I scored 61.0% of those 15%, so Assessment 3 Assessment 4 Total Check: Your total should be 36.67% Page 18 of 40 2. What do I need to score on the final exam to get a P, C, or a D? Can I still get a HD? Calculation Required score P For a Pass, I need to get at least 50% overall. I already have 36.67%, so the final exam needs to contribute 50 - 36.67 = 13.33 The exam contributes a maximum of 40% to my overall grade, but I only need to get 13.33%, so how many percent of 40% is 13.33%? ? ÷ 100 x 40 = 13.33 ? = 13.33 ÷ 40% x 100 ? = 33.33 33.33% C For a Credit, I need to get at least 65% overall. I already have 36.67%, so
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5. Equivalent Fractions Equivalence is a concept that is easy to understand when a fraction wall is used. As you can see, each row has been split into different fractions: top row into 2 halves, bottom row 12 twelfths. An equivalent fraction splits the row at the same place. Therefore: 1 2 = 2 4 = 3 6 = 4 8 = 5 10 = 6 12 The more pieces I split the row into (denominator), the more pieces I will need (numerator). Mathematically, whatever I do to the numerator (multiply or divide), I must also do to the denominator and vice versa, whatever I do to the denominator I must do to the numerator. Take 2 3 as an example. If I multiply the numerator by 4, then I must multiply the denominator by 4 to create an equivalent fraction: 2 × 4 3 × 4 = 8 12 Example problems: Use what you have understood about equivalent fractions to find the missing values in these fraction pairs. 1. 3 5 = 20 Answer: The denominator was multiplied by 4. (20 ÷5 =4) So the numerator must by multiplied by 4. ∴ 3×4 5×4 = 12 20 2. 27 81 = 9 Answer: The numerator was divided by 3. (27 ÷9 =3) So the denominator must be divided by 3. ∴ 27÷3 81÷3 = 9 27 5. Your Turn: a) 2 3 = 9 b) 5 7 = 45 c) 9 10 = 30 d) 52 = 4 13 Return to Your Turn Activity 4. e) What fraction of the large square has dots? f) What fraction of the large square has horizontal lines? Watch this short Khan Academy video for further explanation: “Equivalent fractions” https://www.khanacademy.org/math/arithmetic/fractions/Equivalent_fractions/v/equivalent-fractions Page 9 of 40 6. Converting Mixed Numbers to Improper Fractions A mixed number is a way of expressing quantities greater than 1. A mixed number represents the number of wholes and remaining parts of a whole that you have, while an improper fraction represents how many parts you have. The diagram below illustrates the difference between a mixed number and an improper fraction, using a quantity of car oil as an example. On the left, we use a mixed number to represent 3 whole litres and 1 half litre. We write this mixed number as 3 ½. On the right, we use an improper fraction to represent 7 half litres. We write this improper fraction as 7 2 . Is the same as 3 1 2 = 7 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 You are more likely to encounter mixed numbers than improper fractions in everyday language. For example, you are more likely to say, ‘my car requires 3 ½ litres of oil,’ rather than, ‘my car requires 7 2 litres of oil.’ It is much easier to multiply or divide fractions when they are in improper form. As such, mixed numbers are usually converted to improper fractions before they are used in calculations. To convert from a mixed number to an improper fraction, multiply the whole number by the denominator then add the numerator. This total then becomes the new numerator which is placed over the original denominator. For example: Convert 3 1 2 into an improper fraction. working: 3(𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) × 2(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) + 1(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) = 7 Therefore,𝑡𝑡ℎ𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖𝑠𝑠 7 2 Example problems: 1. 2 2 3 = 8 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 3 + 2 = 8) 2. 2 3 7 = 17 7 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 7 + 3 = 17) 6. Your Turn: Convert these mixed numbers to improper fractions. a) 4 1 2 = c) 7 3 5 = b) 5 1 3 = d) 2 1 8 = Page 10 of 40 7. Converting Improper Fractions to Mixed Numbers While improper fractions are good for calculations, they are rarely used in everyday situations. For example, people do not wear a size 23 2 shoe; instead they wear a size 11 1 2 shoe. = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 To convert to an improper fraction we need to work out how many whole numbers we have. Here we reverse the procedure from the previous section. We can see that 6 of the halves combine to form 3 wholes; with a half left over. = 31 2 So to work this symbolically as a mathematical calculation we simply divide the numerator by the denominator. Whatever the remainder is becomes the new numerator. Using a worked example of the diagram above: Convert 7 2 7 ÷ 2 = 3 1 2 If I have three whole numbers, then I also have six halves and we have one half remaining. ∴ 7 2 = 3 1 2 That was an easy one. Another example: Convert 17 5 into a 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 . working: 17 ÷ 5 = the whole number is 3 with some remaining. If I have 3 whole numbers that is 15 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠. (3 × 5) I must now have 2 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠 remaining. (17 − 15) Therefore, I have 3 2 5 Example problems: Convert the improper fractions to mixed numbers: 1. 27 6 = 4 3 6 = 4 1 2 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (27 ÷ 6 = 4.5) 𝑠𝑠𝑠𝑠 (4 × 6 = 24) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (27 − 24 = 3) remaining. Don’t forget equivalent fractions. 2. 8 3 = 2 2 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (8 ÷ 3 = 2.67) 𝑠𝑠𝑠𝑠 (2 × 3 = 6) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (8 − 6 = 2) remaining. 7. Your Turn: Convert the following improper fractions to mixed numbers: a) 7 5 = c) 53 9 = b) 12 9 = d) 27 7 = Watch this short Khan Academy video for further explanation: “Mixed numbers and improper fractions” (converting both ways) https://www.khanacademy.org/math/cc-fourth-grade-math/imp-fractions-2/imp-mixed-numbers/v/changing-a-mixed-number-to-animproper-fraction Page 11 of 40 8. Converting Decimals into fractions Decimals are an almost universal method of displaying data, particularly given that it is easier to enter decimals, rather than fractions, into computers. But fractions can be more accurate. For example, 1 3 is not 0.33 it is 0.33̇ The method used to convert decimals into fractions is based on the notion of place value. The place value of the last digit in the decimal determines the denominator: tenths, hundredths, thousandths, and so on… Example problems: 1. 0.5 has 5 in the tenths column. Therefore, 0.5 is 5 10 = 1 2 (simplified to an equivalent fraction). 2. 0.375 has the 5 in the thousandth column. Therefore, 0.375 is 375 1000 = 3 8 3. 1.25 has 5 in the hundredths column and you have 1 25 100 = 1 1 4 The hardest part is converting to the lowest equivalent fraction. If you have a scientific calculator, you can use the fraction button. This button looks different on different calculators so read your manual if unsure. If we take 375 1000 from example 2 above: Enter 375 then followed by 1000 press = and answer shows as 3 8 . NOTE: The calculator does not work for rounded decimals; especially thirds. For example, 0.333 ≈ 1 3 The table below lists some commonly encountered fractions expressed in their decimal form: Decimal Fraction 0.125 1 8 0.25 1 4 0.33333 1 3 0.375 3 8
8. Your Turn: (No Calculator first, then check.) a) 0.65 = b) 2.666 = c) 0.54 = Page 12 of 40 d) 3.14 = 9. Converting Fractions into Decimals Converting fractions into decimals is based on place value. For example, applying what you have understood about equivalent fractions, we can easily convert 2 5 into a decimal. First we need to convert to a denominator that has a 10 base. Let’s convert 2 5 into tenths → 2×2 5×2 = 4 10 ∴ we can say that two fifths is the same as four tenths: 0.4 Converting a fraction to decimal form is a simple procedure because we simply use the divide key on the calculator. Note: If you have a mixed number, convert it to an improper fraction before dividing it on your calculator. Example problems: 1. 2 3 = 2 ÷ 3 = 0.66666666666 … ≈ 0.67 2. 3 8 = 3 ÷ 8 = 0.375 3. 17 3 = 17 ÷ 3 = 5.6666666 … ≈ 5.67 4. 3 5 9 = (27 + 5) ÷ 9 = 3.555555556 … ≈ 3.56 9. Your Turn: (Round your answer to three decimal places where appropriate) a) 17 23 = b) 5 72 = c) 56 2 3 = d) 29 5 = Watch this short Khan Academy video for further explanation: “Converting fractions to decimals” (and vice versa) https://www.khanacademy.org/math/pre-algebra/decimals-pre-alg/decimal-to-fraction-pre-alg/v/converting-fractions-to-decimals 10. Fraction Addition and Subtraction Adding and subtracting fractions draws on the concept of equivalent fractions. The golden rule is that you can only add and subtract fractions if they have the same denominator, for example, 1 3 + 1 3 = 2 3 . However, if two fractions do not have the same denominator, we must use equivalent fractions to find a “common denominator” before they can be added together. For instance, we cannot simply add 1 4 + 1 2 because these fractions have different denominators (4 and 2). As such, arriving at an answer of 2 6 (two sixths) would be incorrect. Before these fractions can be added together, they must both have the same denominator. Page 13 of 40 From the image at right, we can see that we have three quarters of a whole cake. So to work this abstractly, we need to decide on a common denominator, 4, which is the lowest common denominator. Now use the equivalent fractions concept to change 1 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2 4 by multiplying both the numerator and denominator by two: 1 2 x 2 2 = 2 4 Now that the denominators are the same, the addition can be carried out: 1 4 + 2 4 = 3 4 Let’s try another: 1 3 + 1 2 We cannot simply add these fractions; again we need to find the lowest common denominator. The easiest way to do this is to multiply the denominators: 1 3 𝑎𝑎𝑎𝑎𝑎𝑎 1 2 (2 x 3 = 6). Therefore, both fractions can have a denominator of 6, yet we need to change the numerator. The next step is to convert both fractions into sixths as an equivalent form. How many sixths is one third? 1 3 = 2 6 � 1 3 × 2 2 = 2 6 � And how many sixths is one half? 1 2 = 3 6 � 1 2 × 3 3 = 3 6 � Therefore: 1 3 + 1 2 = 2 6 + 3 6 = 5 6 With practise, a pattern forms, as is illustrated in the next example: 1 3 + 2 5 = (1 × 5) + (2 × 3) (3 × 5) = 5 + 6 15 = 11 15 In the example above, the lowest common denominator is found by multiplying 3 and 5, and then the numerators are multiplied by 5 and 3 respectively. Use the following problems to reinforce the pattern. 10. Your Turn: a) 1 3 + 2 5 = b) 3 4 + 2 7 = Subtraction is the same procedure but with a negative symbol: 2 3 − 1 4 = (2 × 4) − (1 × 3) (3 × 4) = 8 − 3 12 = 5 12 10. (continued) Your Turn: e) 9 12 − 1 3 = f) 1 3 − 1 2 = Page 14 of 40 Watch this short Khan Academy video for further explanation: “Adding and subtraction fractions” https://www.khanacademy.org/math/arithmetic/fractions/fractions-unlike-denom/v/adding-and-subtracting-fractions 11. Fraction Multiplication and Division Compared to addition and subtraction, multiplication and division of fractions is easy to do, but sometimes a challenge to understand how and why the procedure works mathematically. For example, imagine I have 1 2 of a pie and I want to share it between 2 people. Each person gets a quarter of the pie. Mathematically, this example would be written as: 1 2 × 1 2 = 1 4 . Remember that fractions and division are related; in this way, multiplying by a half is the same as dividing by two. So 1 2 (two people to share) of 1 2 (the amount of pie) is 1 4 (the amount each person will get). But what if the question was more challenging: 2 3 × 7 16 =? This problem is not as easy as splitting pies. A mathematical strategy to use is: “Multiply the numerators then multiply the denominators” Therefore, 2 3 × 7 16 = (2×7) (3×16) = 14 48 = 7 24 However, we can also apply a cancel out method – which you may recall from school. The rule you may recall is, ‘What we do to one side, we must do to the other.’ Thus, in the above example, we could simplify first: 2 3 × 7 16 = ? The first thing we do is look to see if there are any common multiples. Here we can see that 2 is a multiple of 16, which means that we can divide top and bottom by 2: 2÷2 3 × 7 16÷2 = 1 3 × 7 8 = 1×7 3×8 = 7 24 Example Multiplication Problems: 1. 4 9 × 3 4 = (4×3) (9×4) = 12 36 = 1 3 Have a go at simplifying first and then perform the multiplication. 4÷4 9÷3 × 3÷3 4÷4 = (1×1) (3×1) = 1 3 2. 2 4 9 × 3 3 5 = 22 9 × 18 5 = (18×22) (9×5) = 396 45 = 396 ÷ 45 = 8.8 22 9÷9 × 18÷9 5 = 22×2 1×5 = 44 5 so 44 ÷ 5 = 8 4 5 Page 15 of 40 Watch this short Khan Academy video for further explanation: “Multiplying negative and positive fractions” https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-fractions-decimals/cc-7th-mult-div-frac/v/multiplyingnegative-and-positive-fractions Division of fractions seems odd, but it is a simple concept: You may recall the expression ‘invert and multiply’ which means we flip the fraction; we switch the numerator and the denominator. Hence, ÷ 1 2 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 × 2 1 If the sign is swapped to its opposite, the fraction is flipped upside down, this ‘flipped’ fraction is referred to as the reciprocal of the original fraction. Therefore, 2 3 ÷ 1 2 is the same as 2 3 × 2 1 = (2×2) (3×1) = 4 3 = 1 1 3 Note: dividing by half doubled the answer. Example Division Problems: 1. 2 3 ÷ 3 5 = 2 3 × 5 3 = (2×5) (3×3) = 10 9 = 1 1 9 2. 3 3 4 ÷ 2 2 3 = 15 4 ÷ 8 3 = 15 4 × 3 8 = (15×3) (4×8) = 45 32 = 1 13 32 Watch this short Khan Academy video for further explanation: “Dividing fractions example” https://www.khanacademy.org/math/arithmetic/fractions/div-fractions-fractions/v/another-dividing-fractions-example 11. Your Turn: a) Find the reciprocal of 2 2 5 b) 2 3 × 7 13 = c) 1 1 6 × 2 9 = d) 8 × 3 4 × 5 6 × 1 1 2 = e) 3 7 ÷ 2 5 = f) 2 2 5 ÷ 3 8 9 = g) (−25)÷(−5) 4−2×7 = h) −7 2 ÷ −4 9 = i) If we multiply 8 and the reciprocal of 2, what do we get? j) What is 40 mulitplied by 0.2? (use your knowledge of fractions to solve) Page 16 of 40 12. Percentage The concept of percentage is an extension of the material we have already covered about fractions. To allow comparisons between fractions we need to use the same denominator. As such, all percentages use 100 as the denominator. The word percent or “per cent” means per 100. Therefore, 27% is 27 100 . To use percentage in a calculation, the simple mathematical procedure is modelled below: For example, 25% of 40 is 25 100 × 40 = 10 Percentages are most commonly used to compare parts of an original. For instance, the phrase ‘30% off sale,’ indicates that whatever the original price, the new price is 30% less. However, percentages are not often as simple as, for example, 23% of 60. Percentages are commonly used as part of a more complex question. Often the questions might be, “How much is left?” or “How much was the original?” Example problems: A. An advertisement at the chicken shop states that on Tuesday everything is 22% off. If chicken breasts are normally $9.99 per kilo. What is the new per kilo price? Step 1: SIMPLE PERCENTAGE: 22 100 × 9.99 = 2.20 Step 2: DIFFERENCE: Since the price is cheaper by 22%, $2.20 is subtracted from the original: 9.99 – 2.20 = $7.79 B. For the new financial year you have been given an automatic 3.5% pay rise. If you were earning $17.60 per hour what would be your new rate? Step 1: SIMPLE PERCENTAGE 3.5 100 × 17.6 = 0.62 Step 2: DIFFERENCE: Since it is a 3.5% pay RISE, $0.62 is added to the original: 17.60 + 0.62 = $18.22 C. A new dress is now $237 reduced from $410. What is the percentage difference? As you can see, the problem is in reverse, so we approach it in reverse. Step 1: DIFFERENCE: Since it is a discount the difference between the two is the discount. Thus we need to subtract $237.00 from $410 to see what the discount was that we received. $410 – $237 = $173 Step 2: SIMPLE PERCENTAGE: now we need to calculate what percentage of $410 was $173, and so we can use this equation: 𝑥𝑥 100 × 410 = 173 We can rearrange the problem in steps: 𝑥𝑥 100 × 410÷410 = 173÷410 this step involved dividing 410 from both sides to get 𝑥𝑥 100 = 173 410 Next we work to get the 𝑥𝑥 on its own, so we multiply both sides by 100. Now we have 𝑥𝑥 = 173 410 × 100 1 Next we solve, so 0.42 multiplied by 100, ∴ 0.42 × 100 and we get 42. ∴ The percentage difference was 42%. Let’s check: 42% of $410 is $173, $410 - $173 = $237, the cost of the dress was $237.00 . 12. Your Turn: a) GST adds 10% to the price of most things. How much does a can of soft drink cost if it is 80c before GST? b) A Computer screen was $299 but is on special for $249. What is the percentage discount? c) Which of the following is the largest? 3 5 𝑜𝑜𝑜𝑜 16 25 𝑜𝑜𝑜𝑜 0.065 𝑜𝑜𝑜𝑜 63%? (Convert to percentages) Page 17 of 40 12 (continued). Activity: What do I need to get on the final exam??? Grade (%) Weight (%) Assessment 1 30.0% 10% Assessment 2 61.0% 15% Assessment 3 73.2% 30% Assessment 4 51.2% 5% Final Exam 40% Final Grade Overall Percentage Needed High Distinction 100 - 85% Distinction 84 - 75% Credit 74 - 65% Pass 64 - 50% Fail 49 - 0% 1. How much does each of my assessments contribute to my overall percentage, which determines my final grade? Calculation Overall % Assessment 1 This assessment contributes a maximum of 10% to my overall percentage and I scored 30.0% of those 10%, so 30.0 ÷ 100 x 10 = 3 𝟑𝟑𝟑𝟑 𝟏𝟏𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 𝟏𝟏 = 𝟑𝟑 3.0 % Assessment 2 This assessment contributes a maximum of 15% to my overall percentage and I scored 61.0% of those 15%, so Assessment 3 Assessment 4 Total Check: Your total should be 36.67% Page 18 of 40 2. What do I need to score on the final exam to get a P, C, or a D? Can I still get a HD? Calculation Required score P For a Pass, I need to get at least 50% overall. I already have 36.67%, so the final exam needs to contribute 50 - 36.67 = 13.33 The exam contributes a maximum of 40% to my overall grade, but I only need to get 13.33%, so how many percent of 40% is 13.33%? ? ÷ 100 x 40 = 13.33 ? = 13.33 ÷ 40% x 100 ? = 33.33 33.33% C For a Credit, I need to get at least 65% overall. I already have 36.67%, so5. Equivalent Fractions Equivalence is a concept that is easy to understand when a fraction wall is used. As you can see, each row has been split into different fractions: top row into 2 halves, bottom row 12 twelfths. An equivalent fraction splits the row at the same place. Therefore: 1 2 = 2 4 = 3 6 = 4 8 = 5 10 = 6 12 The more pieces I split the row into (denominator), the more pieces I will need (numerator). Mathematically, whatever I do to the numerator (multiply or divide), I must also do to the denominator and vice versa, whatever I do to the denominator I must do to the numerator. Take 2 3 as an example. If I multiply the numerator by 4, then I must multiply the denominator by 4 to create an equivalent fraction: 2 × 4 3 × 4 = 8 12 Example problems: Use what you have understood about equivalent fractions to find the missing values in these fraction pairs. 1. 3 5 = 20 Answer: The denominator was multiplied by 4. (20 ÷5 =4) So the numerator must by multiplied by 4. ∴ 3×4 5×4 = 12 20 2. 27 81 = 9 Answer: The numerator was divided by 3. (27 ÷9 =3) So the denominator must be divided by 3. ∴ 27÷3 81÷3 = 9 27 5. Your Turn: a) 2 3 = 9 b) 5 7 = 45 c) 9 10 = 30 d) 52 = 4 13 Return to Your Turn Activity 4. e) What fraction of the large square has dots? f) What fraction of the large square has horizontal lines? Watch this short Khan Academy video for further explanation: “Equivalent fractions” https://www.khanacademy.org/math/arithmetic/fractions/Equivalent_fractions/v/equivalent-fractions Page 9 of 40 6. Converting Mixed Numbers to Improper Fractions A mixed number is a way of expressing quantities greater than 1. A mixed number represents the number of wholes and remaining parts of a whole that you have, while an improper fraction represents how many parts you have. The diagram below illustrates the difference between a mixed number and an improper fraction, using a quantity of car oil as an example. On the left, we use a mixed number to represent 3 whole litres and 1 half litre. We write this mixed number as 3 ½. On the right, we use an improper fraction to represent 7 half litres. We write this improper fraction as 7 2 . Is the same as 3 1 2 = 7 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 You are more likely to encounter mixed numbers than improper fractions in everyday language. For example, you are more likely to say, ‘my car requires 3 ½ litres of oil,’ rather than, ‘my car requires 7 2 litres of oil.’ It is much easier to multiply or divide fractions when they are in improper form. As such, mixed numbers are usually converted to improper fractions before they are used in calculations. To convert from a mixed number to an improper fraction, multiply the whole number by the denominator then add the numerator. This total then becomes the new numerator which is placed over the original denominator. For example: Convert 3 1 2 into an improper fraction. working: 3(𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) × 2(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) + 1(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) = 7 Therefore,𝑡𝑡ℎ𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖𝑠𝑠 7 2 Example problems: 1. 2 2 3 = 8 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 3 + 2 = 8) 2. 2 3 7 = 17 7 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 7 + 3 = 17) 6. Your Turn: Convert these mixed numbers to improper fractions. a) 4 1 2 = c) 7 3 5 = b) 5 1 3 = d) 2 1 8 = Page 10 of 40 7. Converting Improper Fractions to Mixed Numbers While improper fractions are good for calculations, they are rarely used in everyday situations. For example, people do not wear a size 23 2 shoe; instead they wear a size 11 1 2 shoe. = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 To convert to an improper fraction we need to work out how many whole numbers we have. Here we reverse the procedure from the previous section. We can see that 6 of the halves combine to form 3 wholes; with a half left over. = 31 2 So to work this symbolically as a mathematical calculation we simply divide the numerator by the denominator. Whatever the remainder is becomes the new numerator. Using a worked example of the diagram above: Convert 7 2 7 ÷ 2 = 3 1 2 If I have three whole numbers, then I also have six halves and we have one half remaining. ∴ 7 2 = 3 1 2 That was an easy one. Another example: Convert 17 5 into a 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 . working: 17 ÷ 5 = the whole number is 3 with some remaining. If I have 3 whole numbers that is 15 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠. (3 × 5) I must now have 2 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠 remaining. (17 − 15) Therefore, I have 3 2 5 Example problems: Convert the improper fractions to mixed numbers: 1. 27 6 = 4 3 6 = 4 1 2 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (27 ÷ 6 = 4.5) 𝑠𝑠𝑠𝑠 (4 × 6 = 24) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (27 − 24 = 3) remaining. Don’t forget equivalent fractions. 2. 8 3 = 2 2 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (8 ÷ 3 = 2.67) 𝑠𝑠𝑠𝑠 (2 × 3 = 6) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (8 − 6 = 2) remaining. 7. Your Turn: Convert the following improper fractions to mixed numbers: a) 7 5 = c) 53 9 = b) 12 9 = d) 27 7 = Watch this short Khan Academy video for further explanation: “Mixed numbers and improper fractions” (converting both ways) https://www.khanacademy.org/math/cc-fourth-grade-math/imp-fractions-2/imp-mixed-numbers/v/changing-a-mixed-number-to-animproper-fraction Page 11 of 40 8. Converting Decimals into fractions Decimals are an almost universal method of displaying data, particularly given that it is easier to enter decimals, rather than fractions, into computers. But fractions can be more accurate. For example, 1 3 is not 0.33 it is 0.33̇ The method used to convert decimals into fractions is based on the notion of place value. The place value of the last digit in the decimal determines the denominator: tenths, hundredths, thousandths, and so on… Example problems: 1. 0.5 has 5 in the tenths column. Therefore, 0.5 is 5 10 = 1 2 (simplified to an equivalent fraction). 2. 0.375 has the 5 in the thousandth column. Therefore, 0.375 is 375 1000 = 3 8 3. 1.25 has 5 in the hundredths column and you have 1 25 100 = 1 1 4 The hardest part is converting to the lowest equivalent fraction. If you have a scientific calculator, you can use the fraction button. This button looks different on different calculators so read your manual if unsure. If we take 375 1000 from example 2 above: Enter 375 then followed by 1000 press = and answer shows as 3 8 . NOTE: The calculator does not work for rounded decimals; especially thirds. For example, 0.333 ≈ 1 3 The table below lists some commonly encountered fractions expressed in their decimal form: Decimal Fraction 0.125 1 8 0.25 1 4 0.33333 1 3 0.375 3 8
8. Your Turn: (No Calculator first, then check.) a) 0.65 = b) 2.666 = c) 0.54 = Page 12 of 40 d) 3.14 = 9. Converting Fractions into Decimals Converting fractions into decimals is based on place value. For example, applying what you have understood about equivalent fractions, we can easily convert 2 5 into a decimal. First we need to convert to a denominator that has a 10 base. Let’s convert 2 5 into tenths → 2×2 5×2 = 4 10 ∴ we can say that two fifths is the same as four tenths: 0.4 Converting a fraction to decimal form is a simple procedure because we simply use the divide key on the calculator. Note: If you have a mixed number, convert it to an improper fraction before dividing it on your calculator. Example problems: 1. 2 3 = 2 ÷ 3 = 0.66666666666 … ≈ 0.67 2. 3 8 = 3 ÷ 8 = 0.375 3. 17 3 = 17 ÷ 3 = 5.6666666 … ≈ 5.67 4. 3 5 9 = (27 + 5) ÷ 9 = 3.555555556 … ≈ 3.56 9. Your Turn: (Round your answer to three decimal places where appropriate) a) 17 23 = b) 5 72 = c) 56 2 3 = d) 29 5 = Watch this short Khan Academy video for further explanation: “Converting fractions to decimals” (and vice versa) https://www.khanacademy.org/math/pre-algebra/decimals-pre-alg/decimal-to-fraction-pre-alg/v/converting-fractions-to-decimals 10. Fraction Addition and Subtraction Adding and subtracting fractions draws on the concept of equivalent fractions. The golden rule is that you can only add and subtract fractions if they have the same denominator, for example, 1 3 + 1 3 = 2 3 . However, if two fractions do not have the same denominator, we must use equivalent fractions to find a “common denominator” before they can be added together. For instance, we cannot simply add 1 4 + 1 2 because these fractions have different denominators (4 and 2). As such, arriving at an answer of 2 6 (two sixths) would be incorrect. Before these fractions can be added together, they must both have the same denominator. Page 13 of 40 From the image at right, we can see that we have three quarters of a whole cake. So to work this abstractly, we need to decide on a common denominator, 4, which is the lowest common denominator. Now use the equivalent fractions concept to change 1 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2 4 by multiplying both the numerator and denominator by two: 1 2 x 2 2 = 2 4 Now that the denominators are the same, the addition can be carried out: 1 4 + 2 4 = 3 4 Let’s try another: 1 3 + 1 2 We cannot simply add these fractions; again we need to find the lowest common denominator. The easiest way to do this is to multiply the denominators: 1 3 𝑎𝑎𝑎𝑎𝑎𝑎 1 2 (2 x 3 = 6). Therefore, both fractions can have a denominator of 6, yet we need to change the numerator. The next step is to convert both fractions into sixths as an equivalent form. How many sixths is one third? 1 3 = 2 6 � 1 3 × 2 2 = 2 6 � And how many sixths is one half? 1 2 = 3 6 � 1 2 × 3 3 = 3 6 � Therefore: 1 3 + 1 2 = 2 6 + 3 6 = 5 6 With practise, a pattern forms, as is illustrated in the next example: 1 3 + 2 5 = (1 × 5) + (2 × 3) (3 × 5) = 5 + 6 15 = 11 15 In the example above, the lowest common denominator is found by multiplying 3 and 5, and then the numerators are multiplied by 5 and 3 respectively. Use the following problems to reinforce the pattern. 10. Your Turn: a) 1 3 + 2 5 = b) 3 4 + 2 7 = Subtraction is the same procedure but with a negative symbol: 2 3 − 1 4 = (2 × 4) − (1 × 3) (3 × 4) = 8 − 3 12 = 5 12 10. (continued) Your Turn: e) 9 12 − 1 3 = f) 1 3 − 1 2 = Page 14 of 40 Watch this short Khan Academy video for further explanation: “Adding and subtraction fractions” https://www.khanacademy.org/math/arithmetic/fractions/fractions-unlike-denom/v/adding-and-subtracting-fractions 11. Fraction Multiplication and Division Compared to addition and subtraction, multiplication and division of fractions is easy to do, but sometimes a challenge to understand how and why the procedure works mathematically. For example, imagine I have 1 2 of a pie and I want to share it between 2 people. Each person gets a quarter of the pie. Mathematically, this example would be written as: 1 2 × 1 2 = 1 4 . Remember that fractions and division are related; in this way, multiplying by a half is the same as dividing by two. So 1 2 (two people to share) of 1 2 (the amount of pie) is 1 4 (the amount each person will get). But what if the question was more challenging: 2 3 × 7 16 =? This problem is not as easy as splitting pies. A mathematical strategy to use is: “Multiply the numerators then multiply the denominators” Therefore, 2 3 × 7 16 = (2×7) (3×16) = 14 48 = 7 24 However, we can also apply a cancel out method – which you may recall from school. The rule you may recall is, ‘What we do to one side, we must do to the other.’ Thus, in the above example, we could simplify first: 2 3 × 7 16 = ? The first thing we do is look to see if there are any common multiples. Here we can see that 2 is a multiple of 16, which means that we can divide top and bottom by 2: 2÷2 3 × 7 16÷2 = 1 3 × 7 8 = 1×7 3×8 = 7 24 Example Multiplication Problems: 1. 4 9 × 3 4 = (4×3) (9×4) = 12 36 = 1 3 Have a go at simplifying first and then perform the multiplication. 4÷4 9÷3 × 3÷3 4÷4 = (1×1) (3×1) = 1 3 2. 2 4 9 × 3 3 5 = 22 9 × 18 5 = (18×22) (9×5) = 396 45 = 396 ÷ 45 = 8.8 22 9÷9 × 18÷9 5 = 22×2 1×5 = 44 5 so 44 ÷ 5 = 8 4 5 Page 15 of 40 Watch this short Khan Academy video for further explanation: “Multiplying negative and positive fractions” https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-fractions-decimals/cc-7th-mult-div-frac/v/multiplyingnegative-and-positive-fractions Division of fractions seems odd, but it is a simple concept: You may recall the expression ‘invert and multiply’ which means we flip the fraction; we switch the numerator and the denominator. Hence, ÷ 1 2 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 × 2 1 If the sign is swapped to its opposite, the fraction is flipped upside down, this ‘flipped’ fraction is referred to as the reciprocal of the original fraction. Therefore, 2 3 ÷ 1 2 is the same as 2 3 × 2 1 = (2×2) (3×1) = 4 3 = 1 1 3 Note: dividing by half doubled the answer. Example Division Problems: 1. 2 3 ÷ 3 5 = 2 3 × 5 3 = (2×5) (3×3) = 10 9 = 1 1 9 2. 3 3 4 ÷ 2 2 3 = 15 4 ÷ 8 3 = 15 4 × 3 8 = (15×3) (4×8) = 45 32 = 1 13 32 Watch this short Khan Academy video for further explanation: “Dividing fractions example” https://www.khanacademy.org/math/arithmetic/fractions/div-fractions-fractions/v/another-dividing-fractions-example 11. Your Turn: a) Find the reciprocal of 2 2 5 b) 2 3 × 7 13 = c) 1 1 6 × 2 9 = d) 8 × 3 4 × 5 6 × 1 1 2 = e) 3 7 ÷ 2 5 = f) 2 2 5 ÷ 3 8 9 = g) (−25)÷(−5) 4−2×7 = h) −7 2 ÷ −4 9 = i) If we multiply 8 and the reciprocal of 2, what do we get? j) What is 40 mulitplied by 0.2? (use your knowledge of fractions to solve) Page 16 of 40 12. Percentage The concept of percentage is an extension of the material we have already covered about fractions. To allow comparisons between fractions we need to use the same denominator. As such, all percentages use 100 as the denominator. The word percent or “per cent” means per 100. Therefore, 27% is 27 100 . To use percentage in a calculation, the simple mathematical procedure is modelled below: For example, 25% of 40 is 25 100 × 40 = 10 Percentages are most commonly used to compare parts of an original. For instance, the phrase ‘30% off sale,’ indicates that whatever the original price, the new price is 30% less. However, percentages are not often as simple as, for example, 23% of 60. Percentages are commonly used as part of a more complex question. Often the questions might be, “How much is left?” or “How much was the original?” Example problems: A. An advertisement at the chicken shop states that on Tuesday everything is 22% off. If chicken breasts are normally $9.99 per kilo. What is the new per kilo price? Step 1: SIMPLE PERCENTAGE: 22 100 × 9.99 = 2.20 Step 2: DIFFERENCE: Since the price is cheaper by 22%, $2.20 is subtracted from the original: 9.99 – 2.20 = $7.79 B. For the new financial year you have been given an automatic 3.5% pay rise. If you were earning $17.60 per hour what would be your new rate? Step 1: SIMPLE PERCENTAGE 3.5 100 × 17.6 = 0.62 Step 2: DIFFERENCE: Since it is a 3.5% pay RISE, $0.62 is added to the original: 17.60 + 0.62 = $18.22 C. A new dress is now $237 reduced from $410. What is the percentage difference? As you can see, the problem is in reverse, so we approach it in reverse. Step 1: DIFFERENCE: Since it is a discount the difference between the two is the discount. Thus we need to subtract $237.00 from $410 to see what the discount was that we received. $410 – $237 = $173 Step 2: SIMPLE PERCENTAGE: now we need to calculate what percentage of $410 was $173, and so we can use this equation: 𝑥𝑥 100 × 410 = 173 We can rearrange the problem in steps: 𝑥𝑥 100 × 410÷410 = 173÷410 this step involved dividing 410 from both sides to get 𝑥𝑥 100 = 173 410 Next we work to get the 𝑥𝑥 on its own, so we multiply both sides by 100. Now we have 𝑥𝑥 = 173 410 × 100 1 Next we solve, so 0.42 multiplied by 100, ∴ 0.42 × 100 and we get 42. ∴ The percentage difference was 42%. Let’s check: 42% of $410 is $173, $410 - $173 = $237, the cost of the dress was $237.00 . 12. Your Turn: a) GST adds 10% to the price of most things. How much does a can of soft drink cost if it is 80c before GST? b) A Computer screen was $299 but is on special for $249. What is the percentage discount? c) Which of the following is the largest? 3 5 𝑜𝑜𝑜𝑜 16 25 𝑜𝑜𝑜𝑜 0.065 𝑜𝑜𝑜𝑜 63%? (Convert to percentages) Page 17 of 40 12 (continued). Activity: What do I need to get on the final exam??? Grade (%) Weight (%) Assessment 1 30.0% 10% Assessment 2 61.0% 15% Assessment 3 73.2% 30% Assessment 4 51.2% 5% Final Exam 40% Final Grade Overall Percentage Needed High Distinction 100 - 85% Distinction 84 - 75% Credit 74 - 65% Pass 64 - 50% Fail 49 - 0% 1. How much does each of my assessments contribute to my overall percentage, which determines my final grade? Calculation Overall % Assessment 1 This assessment contributes a maximum of 10% to my overall percentage and I scored 30.0% of those 10%, so 30.0 ÷ 100 x 10 = 3 𝟑𝟑𝟑𝟑 𝟏𝟏𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 𝟏𝟏 = 𝟑𝟑 3.0 % Assessment 2 This assessment contributes a maximum of 15% to my overall percentage and I scored 61.0% of those 15%, so Assessment 3 Assessment 4 Total Check: Your total should be 36.67% Page 18 of 40 2. What do I need to score on the final exam to get a P, C, or a D? Can I still get a HD? Calculation Required score P For a Pass, I need to get at least 50% overall. I already have 36.67%, so the final exam needs to contribute 50 - 36.67 = 13.33 The exam contributes a maximum of 40% to my overall grade, but I only need to get 13.33%, so how many percent of 40% is 13.33%? ? ÷ 100 x 40 = 13.33 ? = 13.33 ÷ 40% x 100 ? = 33.33 33.33% C For a Credit, I need to get at least 65% overall. I already have 36.67%, so
5. Equivalent Fractions Equivalence is a concept that is easy to understand when a fraction wall is used. As you can see, each row has been split into different fractions: top row into 2 halves, bottom row 12 twelfths. An equivalent fraction splits the row at the same place. Therefore: 1 2 = 2 4 = 3 6 = 4 8 = 5 10 = 6 12 The more pieces I split the row into (denominator), the more pieces I will need (numerator). Mathematically, whatever I do to the numerator (multiply or divide), I must also do to the denominator and vice versa, whatever I do to the denominator I must do to the numerator. Take 2 3 as an example. If I multiply the numerator by 4, then I must multiply the denominator by 4 to create an equivalent fraction: 2 × 4 3 × 4 = 8 12 Example problems: Use what you have understood about equivalent fractions to find the missing values in these fraction pairs. 1. 3 5 = 20 Answer: The denominator was multiplied by 4. (20 ÷5 =4) So the numerator must by multiplied by 4. ∴ 3×4 5×4 = 12 20 2. 27 81 = 9 Answer: The numerator was divided by 3. (27 ÷9 =3) So the denominator must be divided by 3. ∴ 27÷3 81÷3 = 9 27 5. Your Turn: a) 2 3 = 9 b) 5 7 = 45 c) 9 10 = 30 d) 52 = 4 13 Return to Your Turn Activity 4. e) What fraction of the large square has dots? f) What fraction of the large square has horizontal lines? Watch this short Khan Academy video for further explanation: “Equivalent fractions” https://www.khanacademy.org/math/arithmetic/fractions/Equivalent_fractions/v/equivalent-fractions Page 9 of 40 6. Converting Mixed Numbers to Improper Fractions A mixed number is a way of expressing quantities greater than 1. A mixed number represents the number of wholes and remaining parts of a whole that you have, while an improper fraction represents how many parts you have. The diagram below illustrates the difference between a mixed number and an improper fraction, using a quantity of car oil as an example. On the left, we use a mixed number to represent 3 whole litres and 1 half litre. We write this mixed number as 3 ½. On the right, we use an improper fraction to represent 7 half litres. We write this improper fraction as 7 2 . Is the same as 3 1 2 = 7 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 You are more likely to encounter mixed numbers than improper fractions in everyday language. For example, you are more likely to say, ‘my car requires 3 ½ litres of oil,’ rather than, ‘my car requires 7 2 litres of oil.’ It is much easier to multiply or divide fractions when they are in improper form. As such, mixed numbers are usually converted to improper fractions before they are used in calculations. To convert from a mixed number to an improper fraction, multiply the whole number by the denominator then add the numerator. This total then becomes the new numerator which is placed over the original denominator. For example: Convert 3 1 2 into an improper fraction. working: 3(𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) × 2(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) + 1(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) = 7 Therefore,𝑡𝑡ℎ𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖𝑠𝑠 7 2 Example problems: 1. 2 2 3 = 8 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 3 + 2 = 8) 2. 2 3 7 = 17 7 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 7 + 3 = 17) 6. Your Turn: Convert these mixed numbers to improper fractions. a) 4 1 2 = c) 7 3 5 = b) 5 1 3 = d) 2 1 8 = Page 10 of 40 7. Converting Improper Fractions to Mixed Numbers While improper fractions are good for calculations, they are rarely used in everyday situations. For example, people do not wear a size 23 2 shoe; instead they wear a size 11 1 2 shoe. = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 To convert to an improper fraction we need to work out how many whole numbers we have. Here we reverse the procedure from the previous section. We can see that 6 of the halves combine to form 3 wholes; with a half left over. = 31 2 So to work this symbolically as a mathematical calculation we simply divide the numerator by the denominator. Whatever the remainder is becomes the new numerator. Using a worked example of the diagram above: Convert 7 2 7 ÷ 2 = 3 1 2 If I have three whole numbers, then I also have six halves and we have one half remaining. ∴ 7 2 = 3 1 2 That was an easy one. Another example: Convert 17 5 into a 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 . working: 17 ÷ 5 = the whole number is 3 with some remaining. If I have 3 whole numbers that is 15 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠. (3 × 5) I must now have 2 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠 remaining. (17 − 15) Therefore, I have 3 2 5 Example problems: Convert the improper fractions to mixed numbers: 1. 27 6 = 4 3 6 = 4 1 2 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (27 ÷ 6 = 4.5) 𝑠𝑠𝑠𝑠 (4 × 6 = 24) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (27 − 24 = 3) remaining. Don’t forget equivalent fractions. 2. 8 3 = 2 2 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (8 ÷ 3 = 2.67) 𝑠𝑠𝑠𝑠 (2 × 3 = 6) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (8 − 6 = 2) remaining. 7. Your Turn: Convert the following improper fractions to mixed numbers: a) 7 5 = c) 53 9 = b) 12 9 = d) 27 7 = Watch this short Khan Academy video for further explanation: “Mixed numbers and improper fractions” (converting both ways) https://www.khanacademy.org/math/cc-fourth-grade-math/imp-fractions-2/imp-mixed-numbers/v/changing-a-mixed-number-to-animproper-fraction Page 11 of 40 8. Converting Decimals into fractions Decimals are an almost universal method of displaying data, particularly given that it is easier to enter decimals, rather than fractions, into computers. But fractions can be more accurate. For example, 1 3 is not 0.33 it is 0.33̇ The method used to convert decimals into fractions is based on the notion of place value. The place value of the last digit in the decimal determines the denominator: tenths, hundredths, thousandths, and so on… Example problems: 1. 0.5 has 5 in the tenths column. Therefore, 0.5 is 5 10 = 1 2 (simplified to an equivalent fraction). 2. 0.375 has the 5 in the thousandth column. Therefore, 0.375 is 375 1000 = 3 8 3. 1.25 has 5 in the hundredths column and you have 1 25 100 = 1 1 4 The hardest part is converting to the lowest equivalent fraction. If you have a scientific calculator, you can use the fraction button. This button looks different on different calculators so read your manual if unsure. If we take 375 1000 from example 2 above: Enter 375 then followed by 1000 press = and answer shows as 3 8 . NOTE: The calculator does not work for rounded decimals; especially thirds. For example, 0.333 ≈ 1 3 The table below lists some commonly encountered fractions expressed in their decimal form: Decimal Fraction 0.125 1 8 0.25 1 4 0.33333 1 3 0.375 3 8
8. Your Turn: (No Calculator first, then check.) a) 0.65 = b) 2.666 = c) 0.54 = Page 12 of 40 d) 3.14 = 9. Converting Fractions into Decimals Converting fractions into decimals is based on place value. For example, applying what you have understood about equivalent fractions, we can easily convert 2 5 into a decimal. First we need to convert to a denominator that has a 10 base. Let’s convert 2 5 into tenths → 2×2 5×2 = 4 10 ∴ we can say that two fifths is the same as four tenths: 0.4 Converting a fraction to decimal form is a simple procedure because we simply use the divide key on the calculator. Note: If you have a mixed number, convert it to an improper fraction before dividing it on your calculator. Example problems: 1. 2 3 = 2 ÷ 3 = 0.66666666666 … ≈ 0.67 2. 3 8 = 3 ÷ 8 = 0.375 3. 17 3 = 17 ÷ 3 = 5.6666666 … ≈ 5.67 4. 3 5 9 = (27 + 5) ÷ 9 = 3.555555556 … ≈ 3.56 9. Your Turn: (Round your answer to three decimal places where appropriate) a) 17 23 = b) 5 72 = c) 56 2 3 = d) 29 5 = Watch this short Khan Academy video for further explanation: “Converting fractions to decimals” (and vice versa) https://www.khanacademy.org/math/pre-algebra/decimals-pre-alg/decimal-to-fraction-pre-alg/v/converting-fractions-to-decimals 10. Fraction Addition and Subtraction Adding and subtracting fractions draws on the concept of equivalent fractions. The golden rule is that you can only add and subtract fractions if they have the same denominator, for example, 1 3 + 1 3 = 2 3 . However, if two fractions do not have the same denominator, we must use equivalent fractions to find a “common denominator” before they can be added together. For instance, we cannot simply add 1 4 + 1 2 because these fractions have different denominators (4 and 2). As such, arriving at an answer of 2 6 (two sixths) would be incorrect. Before these fractions can be added together, they must both have the same denominator. Page 13 of 40 From the image at right, we can see that we have three quarters of a whole cake. So to work this abstractly, we need to decide on a common denominator, 4, which is the lowest common denominator. Now use the equivalent fractions concept to change 1 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2 4 by multiplying both the numerator and denominator by two: 1 2 x 2 2 = 2 4 Now that the denominators are the same, the addition can be carried out: 1 4 + 2 4 = 3 4 Let’s try another: 1 3 + 1 2 We cannot simply add these fractions; again we need to find the lowest common denominator. The easiest way to do this is to multiply the denominators: 1 3 𝑎𝑎𝑎𝑎𝑎𝑎 1 2 (2 x 3 = 6). Therefore, both fractions can have a denominator of 6, yet we need to change the numerator. The next step is to convert both fractions into sixths as an equivalent form. How many sixths is one third? 1 3 = 2 6 � 1 3 × 2 2 = 2 6 � And how many sixths is one half? 1 2 = 3 6 � 1 2 × 3 3 = 3 6 � Therefore: 1 3 + 1 2 = 2 6 + 3 6 = 5 6 With practise, a pattern forms, as is illustrated in the next example: 1 3 + 2 5 = (1 × 5) + (2 × 3) (3 × 5) = 5 + 6 15 = 11 15 In the example above, the lowest common denominator is found by multiplying 3 and 5, and then the numerators are multiplied by 5 and 3 respectively. Use the following problems to reinforce the pattern. 10. Your Turn: a) 1 3 + 2 5 = b) 3 4 + 2 7 = Subtraction is the same procedure but with a negative symbol: 2 3 − 1 4 = (2 × 4) − (1 × 3) (3 × 4) = 8 − 3 12 = 5 12 10. (continued) Your Turn: e) 9 12 − 1 3 = f) 1 3 − 1 2 = Page 14 of 40 Watch this short Khan Academy video for further explanation: “Adding and subtraction fractions” https://www.khanacademy.org/math/arithmetic/fractions/fractions-unlike-denom/v/adding-and-subtracting-fractions 11. Fraction Multiplication and Division Compared to addition and subtraction, multiplication and division of fractions is easy to do, but sometimes a challenge to understand how and why the procedure works mathematically. For example, imagine I have 1 2 of a pie and I want to share it between 2 people. Each person gets a quarter of the pie. Mathematically, this example would be written as: 1 2 × 1 2 = 1 4 . Remember that fractions and division are related; in this way, multiplying by a half is the same as dividing by two. So 1 2 (two people to share) of 1 2 (the amount of pie) is 1 4 (the amount each person will get). But what if the question was more challenging: 2 3 × 7 16 =? This problem is not as easy as splitting pies. A mathematical strategy to use is: “Multiply the numerators then multiply the denominators” Therefore, 2 3 × 7 16 = (2×7) (3×16) = 14 48 = 7 24 However, we can also apply a cancel out method – which you may recall from school. The rule you may recall is, ‘What we do to one side, we must do to the other.’ Thus, in the above example, we could simplify first: 2 3 × 7 16 = ? The first thing we do is look to see if there are any common multiples. Here we can see that 2 is a multiple of 16, which means that we can divide top and bottom by 2: 2÷2 3 × 7 16÷2 = 1 3 × 7 8 = 1×7 3×8 = 7 24 Example Multiplication Problems: 1. 4 9 × 3 4 = (4×3) (9×4) = 12 36 = 1 3 Have a go at simplifying first and then perform the multiplication. 4÷4 9÷3 × 3÷3 4÷4 = (1×1) (3×1) = 1 3 2. 2 4 9 × 3 3 5 = 22 9 × 18 5 = (18×22) (9×5) = 396 45 = 396 ÷ 45 = 8.8 22 9÷9 × 18÷9 5 = 22×2 1×5 = 44 5 so 44 ÷ 5 = 8 4 5 Page 15 of 40 Watch this short Khan Academy video for further explanation: “Multiplying negative and positive fractions” https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-fractions-decimals/cc-7th-mult-div-frac/v/multiplyingnegative-and-positive-fractions Division of fractions seems odd, but it is a simple concept: You may recall the expression ‘invert and multiply’ which means we flip the fraction; we switch the numerator and the denominator. Hence, ÷ 1 2 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 × 2 1 If the sign is swapped to its opposite, the fraction is flipped upside down, this ‘flipped’ fraction is referred to as the reciprocal of the original fraction. Therefore, 2 3 ÷ 1 2 is the same as 2 3 × 2 1 = (2×2) (3×1) = 4 3 = 1 1 3 Note: dividing by half doubled the answer. Example Division Problems: 1. 2 3 ÷ 3 5 = 2 3 × 5 3 = (2×5) (3×3) = 10 9 = 1 1 9 2. 3 3 4 ÷ 2 2 3 = 15 4 ÷ 8 3 = 15 4 × 3 8 = (15×3) (4×8) = 45 32 = 1 13 32 Watch this short Khan Academy video for further explanation: “Dividing fractions example” https://www.khanacademy.org/math/arithmetic/fractions/div-fractions-fractions/v/another-dividing-fractions-example 11. Your Turn: a) Find the reciprocal of 2 2 5 b) 2 3 × 7 13 = c) 1 1 6 × 2 9 = d) 8 × 3 4 × 5 6 × 1 1 2 = e) 3 7 ÷ 2 5 = f) 2 2 5 ÷ 3 8 9 = g) (−25)÷(−5) 4−2×7 = h) −7 2 ÷ −4 9 = i) If we multiply 8 and the reciprocal of 2, what do we get? j) What is 40 mulitplied by 0.2? (use your knowledge of fractions to solve) Page 16 of 40 12. Percentage The concept of percentage is an extension of the material we have already covered about fractions. To allow comparisons between fractions we need to use the same denominator. As such, all percentages use 100 as the denominator. The word percent or “per cent” means per 100. Therefore, 27% is 27 100 . To use percentage in a calculation, the simple mathematical procedure is modelled below: For example, 25% of 40 is 25 100 × 40 = 10 Percentages are most commonly used to compare parts of an original. For instance, the phrase ‘30% off sale,’ indicates that whatever the original price, the new price is 30% less. However, percentages are not often as simple as, for example, 23% of 60. Percentages are commonly used as part of a more complex question. Often the questions might be, “How much is left?” or “How much was the original?” Example problems: A. An advertisement at the chicken shop states that on Tuesday everything is 22% off. If chicken breasts are normally $9.99 per kilo. What is the new per kilo price? Step 1: SIMPLE PERCENTAGE: 22 100 × 9.99 = 2.20 Step 2: DIFFERENCE: Since the price is cheaper by 22%, $2.20 is subtracted from the original: 9.99 – 2.20 = $7.79 B. For the new financial year you have been given an automatic 3.5% pay rise. If you were earning $17.60 per hour what would be your new rate? Step 1: SIMPLE PERCENTAGE 3.5 100 × 17.6 = 0.62 Step 2: DIFFERENCE: Since it is a 3.5% pay RISE, $0.62 is added to the original: 17.60 + 0.62 = $18.22 C. A new dress is now $237 reduced from $410. What is the percentage difference? As you can see, the problem is in reverse, so we approach it in reverse. Step 1: DIFFERENCE: Since it is a discount the difference between the two is the discount. Thus we need to subtract $237.00 from $410 to see what the discount was that we received. $410 – $237 = $173 Step 2: SIMPLE PERCENTAGE: now we need to calculate what percentage of $410 was $173, and so we can use this equation: 𝑥𝑥 100 × 410 = 173 We can rearrange the problem in steps: 𝑥𝑥 100 × 410÷410 = 173÷410 this step involved dividing 410 from both sides to get 𝑥𝑥 100 = 173 410 Next we work to get the 𝑥𝑥 on its own, so we multiply both sides by 100. Now we have 𝑥𝑥 = 173 410 × 100 1 Next we solve, so 0.42 multiplied by 100, ∴ 0.42 × 100 and we get 42. ∴ The percentage difference was 42%. Let’s check: 42% of $410 is $173, $410 - $173 = $237, the cost of the dress was $237.00 . 12. Your Turn: a) GST adds 10% to the price of most things. How much does a can of soft drink cost if it is 80c before GST? b) A Computer screen was $299 but is on special for $249. What is the percentage discount? c) Which of the following is the largest? 3 5 𝑜𝑜𝑜𝑜 16 25 𝑜𝑜𝑜𝑜 0.065 𝑜𝑜𝑜𝑜 63%? (Convert to percentages) Page 17 of 40 12 (continued). Activity: What do I need to get on the final exam??? Grade (%) Weight (%) Assessment 1 30.0% 10% Assessment 2 61.0% 15% Assessment 3 73.2% 30% Assessment 4 51.2% 5% Final Exam 40% Final Grade Overall Percentage Needed High Distinction 100 - 85% Distinction 84 - 75% Credit 74 - 65% Pass 64 - 50% Fail 49 - 0% 1. How much does each of my assessments contribute to my overall percentage, which determines my final grade? Calculation Overall % Assessment 1 This assessment contributes a maximum of 10% to my overall percentage and I scored 30.0% of those 10%, so 30.0 ÷ 100 x 10 = 3 𝟑𝟑𝟑𝟑 𝟏𝟏𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 𝟏𝟏 = 𝟑𝟑 3.0 % Assessment 2 This assessment contributes a maximum of 15% to my overall percentage and I scored 61.0% of those 15%, so Assessment 3 Assessment 4 Total Check: Your total should be 36.67% Page 18 of 40 2. What do I need to score on the final exam to get a P, C, or a D? Can I still get a HD? Calculation Required score P For a Pass, I need to get at least 50% overall. I already have 36.67%, so the final exam needs to contribute 50 - 36.67 = 13.33 The exam contributes a maximum of 40% to my overall grade, but I only need to get 13.33%, so how many percent of 40% is 13.33%? ? ÷ 100 x 40 = 13.33 ? = 13.33 ÷ 40% x 100 ? = 33.33 33.33% C For a Credit, I need to get at least 65% overall. I already have 36.67%, so5. Equivalent Fractions Equivalence is a concept that is easy to understand when a fraction wall is used. As you can see, each row has been split into different fractions: top row into 2 halves, bottom row 12 twelfths. An equivalent fraction splits the row at the same place. Therefore: 1 2 = 2 4 = 3 6 = 4 8 = 5 10 = 6 12 The more pieces I split the row into (denominator), the more pieces I will need (numerator). Mathematically, whatever I do to the numerator (multiply or divide), I must also do to the denominator and vice versa, whatever I do to the denominator I must do to the numerator. Take 2 3 as an example. If I multiply the numerator by 4, then I must multiply the denominator by 4 to create an equivalent fraction: 2 × 4 3 × 4 = 8 12 Example problems: Use what you have understood about equivalent fractions to find the missing values in these fraction pairs. 1. 3 5 = 20 Answer: The denominator was multiplied by 4. (20 ÷5 =4) So the numerator must by multiplied by 4. ∴ 3×4 5×4 = 12 20 2. 27 81 = 9 Answer: The numerator was divided by 3. (27 ÷9 =3) So the denominator must be divided by 3. ∴ 27÷3 81÷3 = 9 27 5. Your Turn: a) 2 3 = 9 b) 5 7 = 45 c) 9 10 = 30 d) 52 = 4 13 Return to Your Turn Activity 4. e) What fraction of the large square has dots? f) What fraction of the large square has horizontal lines? Watch this short Khan Academy video for further explanation: “Equivalent fractions” https://www.khanacademy.org/math/arithmetic/fractions/Equivalent_fractions/v/equivalent-fractions Page 9 of 40 6. Converting Mixed Numbers to Improper Fractions A mixed number is a way of expressing quantities greater than 1. A mixed number represents the number of wholes and remaining parts of a whole that you have, while an improper fraction represents how many parts you have. The diagram below illustrates the difference between a mixed number and an improper fraction, using a quantity of car oil as an example. On the left, we use a mixed number to represent 3 whole litres and 1 half litre. We write this mixed number as 3 ½. On the right, we use an improper fraction to represent 7 half litres. We write this improper fraction as 7 2 . Is the same as 3 1 2 = 7 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 You are more likely to encounter mixed numbers than improper fractions in everyday language. For example, you are more likely to say, ‘my car requires 3 ½ litres of oil,’ rather than, ‘my car requires 7 2 litres of oil.’ It is much easier to multiply or divide fractions when they are in improper form. As such, mixed numbers are usually converted to improper fractions before they are used in calculations. To convert from a mixed number to an improper fraction, multiply the whole number by the denominator then add the numerator. This total then becomes the new numerator which is placed over the original denominator. For example: Convert 3 1 2 into an improper fraction. working: 3(𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) × 2(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) + 1(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) = 7 Therefore,𝑡𝑡ℎ𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖𝑠𝑠 7 2 Example problems: 1. 2 2 3 = 8 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 3 + 2 = 8) 2. 2 3 7 = 17 7 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 7 + 3 = 17) 6. Your Turn: Convert these mixed numbers to improper fractions. a) 4 1 2 = c) 7 3 5 = b) 5 1 3 = d) 2 1 8 = Page 10 of 40 7. Converting Improper Fractions to Mixed Numbers While improper fractions are good for calculations, they are rarely used in everyday situations. For example, people do not wear a size 23 2 shoe; instead they wear a size 11 1 2 shoe. = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 To convert to an improper fraction we need to work out how many whole numbers we have. Here we reverse the procedure from the previous section. We can see that 6 of the halves combine to form 3 wholes; with a half left over. = 31 2 So to work this symbolically as a mathematical calculation we simply divide the numerator by the denominator. Whatever the remainder is becomes the new numerator. Using a worked example of the diagram above: Convert 7 2 7 ÷ 2 = 3 1 2 If I have three whole numbers, then I also have six halves and we have one half remaining. ∴ 7 2 = 3 1 2 That was an easy one. Another example: Convert 17 5 into a 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 . working: 17 ÷ 5 = the whole number is 3 with some remaining. If I have 3 whole numbers that is 15 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠. (3 × 5) I must now have 2 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠 remaining. (17 − 15) Therefore, I have 3 2 5 Example problems: Convert the improper fractions to mixed numbers: 1. 27 6 = 4 3 6 = 4 1 2 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (27 ÷ 6 = 4.5) 𝑠𝑠𝑠𝑠 (4 × 6 = 24) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (27 − 24 = 3) remaining. Don’t forget equivalent fractions. 2. 8 3 = 2 2 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (8 ÷ 3 = 2.67) 𝑠𝑠𝑠𝑠 (2 × 3 = 6) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (8 − 6 = 2) remaining. 7. Your Turn: Convert the following improper fractions to mixed numbers: a) 7 5 = c) 53 9 = b) 12 9 = d) 27 7 = Watch this short Khan Academy video for further explanation: “Mixed numbers and improper fractions” (converting both ways) https://www.khanacademy.org/math/cc-fourth-grade-math/imp-fractions-2/imp-mixed-numbers/v/changing-a-mixed-number-to-animproper-fraction Page 11 of 40 8. Converting Decimals into fractions Decimals are an almost universal method of displaying data, particularly given that it is easier to enter decimals, rather than fractions, into computers. But fractions can be more accurate. For example, 1 3 is not 0.33 it is 0.33̇ The method used to convert decimals into fractions is based on the notion of place value. The place value of the last digit in the decimal determines the denominator: tenths, hundredths, thousandths, and so on… Example problems: 1. 0.5 has 5 in the tenths column. Therefore, 0.5 is 5 10 = 1 2 (simplified to an equivalent fraction). 2. 0.375 has the 5 in the thousandth column. Therefore, 0.375 is 375 1000 = 3 8 3. 1.25 has 5 in the hundredths column and you have 1 25 100 = 1 1 4 The hardest part is converting to the lowest equivalent fraction. If you have a scientific calculator, you can use the fraction button. This button looks different on different calculators so read your manual if unsure. If we take 375 1000 from example 2 above: Enter 375 then followed by 1000 press = and answer shows as 3 8 . NOTE: The calculator does not work for rounded decimals; especially thirds. For example, 0.333 ≈ 1 3 The table below lists some commonly encountered fractions expressed in their decimal form: Decimal Fraction 0.125 1 8 0.25 1 4 0.33333 1 3 0.375 3 8
8. Your Turn: (No Calculator first, then check.) a) 0.65 = b) 2.666 = c) 0.54 = Page 12 of 40 d) 3.14 = 9. Converting Fractions into Decimals Converting fractions into decimals is based on place value. For example, applying what you have understood about equivalent fractions, we can easily convert 2 5 into a decimal. First we need to convert to a denominator that has a 10 base. Let’s convert 2 5 into tenths → 2×2 5×2 = 4 10 ∴ we can say that two fifths is the same as four tenths: 0.4 Converting a fraction to decimal form is a simple procedure because we simply use the divide key on the calculator. Note: If you have a mixed number, convert it to an improper fraction before dividing it on your calculator. Example problems: 1. 2 3 = 2 ÷ 3 = 0.66666666666 … ≈ 0.67 2. 3 8 = 3 ÷ 8 = 0.375 3. 17 3 = 17 ÷ 3 = 5.6666666 … ≈ 5.67 4. 3 5 9 = (27 + 5) ÷ 9 = 3.555555556 … ≈ 3.56 9. Your Turn: (Round your answer to three decimal places where appropriate) a) 17 23 = b) 5 72 = c) 56 2 3 = d) 29 5 = Watch this short Khan Academy video for further explanation: “Converting fractions to decimals” (and vice versa) https://www.khanacademy.org/math/pre-algebra/decimals-pre-alg/decimal-to-fraction-pre-alg/v/converting-fractions-to-decimals 10. Fraction Addition and Subtraction Adding and subtracting fractions draws on the concept of equivalent fractions. The golden rule is that you can only add and subtract fractions if they have the same denominator, for example, 1 3 + 1 3 = 2 3 . However, if two fractions do not have the same denominator, we must use equivalent fractions to find a “common denominator” before they can be added together. For instance, we cannot simply add 1 4 + 1 2 because these fractions have different denominators (4 and 2). As such, arriving at an answer of 2 6 (two sixths) would be incorrect. Before these fractions can be added together, they must both have the same denominator. Page 13 of 40 From the image at right, we can see that we have three quarters of a whole cake. So to work this abstractly, we need to decide on a common denominator, 4, which is the lowest common denominator. Now use the equivalent fractions concept to change 1 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2 4 by multiplying both the numerator and denominator by two: 1 2 x 2 2 = 2 4 Now that the denominators are the same, the addition can be carried out: 1 4 + 2 4 = 3 4 Let’s try another: 1 3 + 1 2 We cannot simply add these fractions; again we need to find the lowest common denominator. The easiest way to do this is to multiply the denominators: 1 3 𝑎𝑎𝑎𝑎𝑎𝑎 1 2 (2 x 3 = 6). Therefore, both fractions can have a denominator of 6, yet we need to change the numerator. The next step is to convert both fractions into sixths as an equivalent form. How many sixths is one third? 1 3 = 2 6 � 1 3 × 2 2 = 2 6 � And how many sixths is one half? 1 2 = 3 6 � 1 2 × 3 3 = 3 6 � Therefore: 1 3 + 1 2 = 2 6 + 3 6 = 5 6 With practise, a pattern forms, as is illustrated in the next example: 1 3 + 2 5 = (1 × 5) + (2 × 3) (3 × 5) = 5 + 6 15 = 11 15 In the example above, the lowest common denominator is found by multiplying 3 and 5, and then the numerators are multiplied by 5 and 3 respectively. Use the following problems to reinforce the pattern. 10. Your Turn: a) 1 3 + 2 5 = b) 3 4 + 2 7 = Subtraction is the same procedure but with a negative symbol: 2 3 − 1 4 = (2 × 4) − (1 × 3) (3 × 4) = 8 − 3 12 = 5 12 10. (continued) Your Turn: e) 9 12 − 1 3 = f) 1 3 − 1 2 = Page 14 of 40 Watch this short Khan Academy video for further explanation: “Adding and subtraction fractions” https://www.khanacademy.org/math/arithmetic/fractions/fractions-unlike-denom/v/adding-and-subtracting-fractions 11. Fraction Multiplication and Division Compared to addition and subtraction, multiplication and division of fractions is easy to do, but sometimes a challenge to understand how and why the procedure works mathematically. For example, imagine I have 1 2 of a pie and I want to share it between 2 people. Each person gets a quarter of the pie. Mathematically, this example would be written as: 1 2 × 1 2 = 1 4 . Remember that fractions and division are related; in this way, multiplying by a half is the same as dividing by two. So 1 2 (two people to share) of 1 2 (the amount of pie) is 1 4 (the amount each person will get). But what if the question was more challenging: 2 3 × 7 16 =? This problem is not as easy as splitting pies. A mathematical strategy to use is: “Multiply the numerators then multiply the denominators” Therefore, 2 3 × 7 16 = (2×7) (3×16) = 14 48 = 7 24 However, we can also apply a cancel out method – which you may recall from school. The rule you may recall is, ‘What we do to one side, we must do to the other.’ Thus, in the above example, we could simplify first: 2 3 × 7 16 = ? The first thing we do is look to see if there are any common multiples. Here we can see that 2 is a multiple of 16, which means that we can divide top and bottom by 2: 2÷2 3 × 7 16÷2 = 1 3 × 7 8 = 1×7 3×8 = 7 24 Example Multiplication Problems: 1. 4 9 × 3 4 = (4×3) (9×4) = 12 36 = 1 3 Have a go at simplifying first and then perform the multiplication. 4÷4 9÷3 × 3÷3 4÷4 = (1×1) (3×1) = 1 3 2. 2 4 9 × 3 3 5 = 22 9 × 18 5 = (18×22) (9×5) = 396 45 = 396 ÷ 45 = 8.8 22 9÷9 × 18÷9 5 = 22×2 1×5 = 44 5 so 44 ÷ 5 = 8 4 5 Page 15 of 40 Watch this short Khan Academy video for further explanation: “Multiplying negative and positive fractions” https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-fractions-decimals/cc-7th-mult-div-frac/v/multiplyingnegative-and-positive-fractions Division of fractions seems odd, but it is a simple concept: You may recall the expression ‘invert and multiply’ which means we flip the fraction; we switch the numerator and the denominator. Hence, ÷ 1 2 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 × 2 1 If the sign is swapped to its opposite, the fraction is flipped upside down, this ‘flipped’ fraction is referred to as the reciprocal of the original fraction. Therefore, 2 3 ÷ 1 2 is the same as 2 3 × 2 1 = (2×2) (3×1) = 4 3 = 1 1 3 Note: dividing by half doubled the answer. Example Division Problems: 1. 2 3 ÷ 3 5 = 2 3 × 5 3 = (2×5) (3×3) = 10 9 = 1 1 9 2. 3 3 4 ÷ 2 2 3 = 15 4 ÷ 8 3 = 15 4 × 3 8 = (15×3) (4×8) = 45 32 = 1 13 32 Watch this short Khan Academy video for further explanation: “Dividing fractions example” https://www.khanacademy.org/math/arithmetic/fractions/div-fractions-fractions/v/another-dividing-fractions-example 11. Your Turn: a) Find the reciprocal of 2 2 5 b) 2 3 × 7 13 = c) 1 1 6 × 2 9 = d) 8 × 3 4 × 5 6 × 1 1 2 = e) 3 7 ÷ 2 5 = f) 2 2 5 ÷ 3 8 9 = g) (−25)÷(−5) 4−2×7 = h) −7 2 ÷ −4 9 = i) If we multiply 8 and the reciprocal of 2, what do we get? j) What is 40 mulitplied by 0.2? (use your knowledge of fractions to solve) Page 16 of 40 12. Percentage The concept of percentage is an extension of the material we have already covered about fractions. To allow comparisons between fractions we need to use the same denominator. As such, all percentages use 100 as the denominator. The word percent or “per cent” means per 100. Therefore, 27% is 27 100 . To use percentage in a calculation, the simple mathematical procedure is modelled below: For example, 25% of 40 is 25 100 × 40 = 10 Percentages are most commonly used to compare parts of an original. For instance, the phrase ‘30% off sale,’ indicates that whatever the original price, the new price is 30% less. However, percentages are not often as simple as, for example, 23% of 60. Percentages are commonly used as part of a more complex question. Often the questions might be, “How much is left?” or “How much was the original?” Example problems: A. An advertisement at the chicken shop states that on Tuesday everything is 22% off. If chicken breasts are normally $9.99 per kilo. What is the new per kilo price? Step 1: SIMPLE PERCENTAGE: 22 100 × 9.99 = 2.20 Step 2: DIFFERENCE: Since the price is cheaper by 22%, $2.20 is subtracted from the original: 9.99 – 2.20 = $7.79 B. For the new financial year you have been given an automatic 3.5% pay rise. If you were earning $17.60 per hour what would be your new rate? Step 1: SIMPLE PERCENTAGE 3.5 100 × 17.6 = 0.62 Step 2: DIFFERENCE: Since it is a 3.5% pay RISE, $0.62 is added to the original: 17.60 + 0.62 = $18.22 C. A new dress is now $237 reduced from $410. What is the percentage difference? As you can see, the problem is in reverse, so we approach it in reverse. Step 1: DIFFERENCE: Since it is a discount the difference between the two is the discount. Thus we need to subtract $237.00 from $410 to see what the discount was that we received. $410 – $237 = $173 Step 2: SIMPLE PERCENTAGE: now we need to calculate what percentage of $410 was $173, and so we can use this equation: 𝑥𝑥 100 × 410 = 173 We can rearrange the problem in steps: 𝑥𝑥 100 × 410÷410 = 173÷410 this step involved dividing 410 from both sides to get 𝑥𝑥 100 = 173 410 Next we work to get the 𝑥𝑥 on its own, so we multiply both sides by 100. Now we have 𝑥𝑥 = 173 410 × 100 1 Next we solve, so 0.42 multiplied by 100, ∴ 0.42 × 100 and we get 42. ∴ The percentage difference was 42%. Let’s check: 42% of $410 is $173, $410 - $173 = $237, the cost of the dress was $237.00 . 12. Your Turn: a) GST adds 10% to the price of most things. How much does a can of soft drink cost if it is 80c before GST? b) A Computer screen was $299 but is on special for $249. What is the percentage discount? c) Which of the following is the largest? 3 5 𝑜𝑜𝑜𝑜 16 25 𝑜𝑜𝑜𝑜 0.065 𝑜𝑜𝑜𝑜 63%? (Convert to percentages) Page 17 of 40 12 (continued). Activity: What do I need to get on the final exam??? Grade (%) Weight (%) Assessment 1 30.0% 10% Assessment 2 61.0% 15% Assessment 3 73.2% 30% Assessment 4 51.2% 5% Final Exam 40% Final Grade Overall Percentage Needed High Distinction 100 - 85% Distinction 84 - 75% Credit 74 - 65% Pass 64 - 50% Fail 49 - 0% 1. How much does each of my assessments contribute to my overall percentage, which determines my final grade? Calculation Overall % Assessment 1 This assessment contributes a maximum of 10% to my overall percentage and I scored 30.0% of those 10%, so 30.0 ÷ 100 x 10 = 3 𝟑𝟑𝟑𝟑 𝟏𝟏𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 𝟏𝟏 = 𝟑𝟑 3.0 % Assessment 2 This assessment contributes a maximum of 15% to my overall percentage and I scored 61.0% of those 15%, so Assessment 3 Assessment 4 Total Check: Your total should be 36.67% Page 18 of 40 2. What do I need to score on the final exam to get a P, C, or a D? Can I still get a HD? Calculation Required score P For a Pass, I need to get at least 50% overall. I already have 36.67%, so the final exam needs to contribute 50 - 36.67 = 13.33 The exam contributes a maximum of 40% to my overall grade, but I only need to get 13.33%, so how many percent of 40% is 13.33%? ? ÷ 100 x 40 = 13.33 ? = 13.33 ÷ 40% x 100 ? = 33.33 33.33% C For a Credit, I need to get at least 65% overall. I already have 36.67%, so
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