Maths Numeracy Intermediate WJEC GCSE - (FRACTION & PERCENTAGE)

2/7 =
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2/7 =
Two parts out of seven.
You must first divide by 7 and then multiply by 2.
To find a fraction of an amount:
Divide by the denominator
Multiply by the numerator
QUESTIONS
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What is 2/7 of 84?
Answer = 24

Divide by the denominator: 84 divided by 7 = 12
Multiply by the numerator: 12 x 2 = 24
QUESTIONS - Finding the original quantity
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2/5 of the people on a bus are children. If there are 4 people on the bus, how many are there in total?
Answer: = 10 people in total.


2/5 of the value = 4

Divide by 2 to find 1/5
1/5 of the value = 2

Multiply by the 5 to find 5/5 or 1 whole.
I eat 3/8 of a chocolate bar and am left with 20 squares, how many squares were there to begin with?
3/8 has been eaten, which leaves 5/8
5/8 = 20 squares (now divide by 5)
1/8 = 4 squares (now multiply by 8)

Answer: 1 whole = 32 squares
Louis' father increases his pocket money by 1/3. He now receives $26 a month. How much was he receiving originally?
Answer = $19.50

1 whole + 1/3 = 3/3 + 1/3 = 4/3
4/3 = $26 (now divide by 4)
1/3 = $6.50 (now multiply by 3)

1 whole = $19.50
PERCENTAGES OF AMOUNTS
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27/100 =
Represents a fraction given out of 100.27%
QUESTION - Calculator methodWhat is 23% of 460?460 divided by 100 = 4.6 4.6 x 23 = 105.8Non-calculator methodStart with finding 10% - found by dividing 100% by 10 From 10% we can multiply and divide to find other percentages and add parts together if needed.QUESTIONSWhat is 15% of 60?Divide by 10 to find 10% = 6 Divide 10% by 2 to find 5% = 3 Add together 10% and 5% to find 15% = 9 Answer = 9What is 28% of 50?Divide by 10 to find 10% = 5 Multiply 10% by 3 to find 30% = 15 Divide 10% by 10 to find 1% = 0.5 and then multiply by 2 to find 2% = 1 Subtract 2% from 30% to find 28% = 14 Answer = 14EXPRESSING NUMBERS AS FRACTIONS AND PERCENTAGESOut of a form group of 20 people, five people have chosen to take Art at GCSE. What fraction of the class will be taking the Art GCSE?Answer: = 1/4 There are 5 people out of 20 in total. We can write this as 5/20 This fraction can be simplified by dividing the numerator and denominator by the same value: 5/20 This can be simplified to: 5 divided by 5 = 1 & 20 divided by 5 = 4 —> 1/4QUESTIONIn June it rains for eight days. What fraction of the month did it rain?Answer: 4/15 There are 30 days in June, of which it rained on 8. This is written as 8/30 This can be simplified to: 8 divided by 2 = 4 & 30 divided by 2 = 15 —> 4/15KEYPOINTIT IS IMPORTANT TO ENSURE THAT BOTH QUANTITIES ARE IN THE SAME UNITS. For example, we cant compare a time in minutes with a number of hours. We must first convert the time in hours to minutes. We normally change to the smaller unit to avoid having to work decimals.QUESTIONJohnny is 20 minutes late for a film that is two hours long. What fraction of the film does he miss?Answer: 1/6 2 hours = 2 x 60 = 120mins 20/120 = 2/12 = 1/6Expressing one value as a percentage of another - Method 1Once you have expressed the quantity as a fraction of another number, this can be converted to a percentage:Method 1 - In a bunch of 32 grapes, 4 are mouldy. What percentage of the grapes is mouldy?Answer: = 12.5% Express as a fraction: 4/32 = 2/16 = 1/8 Divide the numerator by the denominator: 1 divided by 8 = 0.125 Multiply by 100 to convert to %: 0.125 x 100% = 12.5%Method 2 - Out of my $25 pocket money, I spend $7. What percentage is this?Answer: = 28% Express as a fraction: 7/25 Use equivalent fractions to get a denominator of 100: 7 x 4/25 x 4 = 28/100 Write as a percentage: 28%Finding a percentage increase: A fish tank is home to 15 fish, after a year the fish have bred and there are now 20 fish. By what percentage has the number of fish increased?Answer: = 33.3% increase Calculate the change: 20 - 15 = 5 Express the change as a percentage of the original value: 5/15 = 1/3 = 0.333 = 33.3%Finding a percentage decrease: On Monday there were 45 chocolates in a box. By Friday there were only 12. What is the percentage change?Answer: = 73.3% to one decimal point Calculate the change: 42 5 - 12 = 33 Express the change as a percentage of the original value: 33/45 = 0.73333333 = 73.3%PERCENTAGE CHANGE - Higher & IntermediateHow do you increase of decrease by a given percentage?First, calculate the percentage given Then, add or subtract the percentage as required.QUESTIONSA landlord raises his rental prices by 5%. How much will as two bedroom flat priced at $350 a month, cost after the increase?Answer: = $367.50 10% of $350 = $35 5% = $17..50 $350 + $17.50 = $367.50A shop reduces all their stock by 15%. How much would a jacket, costing $60, now cost in the sale?Answer: $51 10% of $60 = $6 5% = $3 15% = $9 $60 - $9 = $51FINDING THE ORIGINAL QUANTITY AFTER A PERCENTAGE INCREASEExample: A bottle of lemonade advertises that its now 50% bigger. If it is now 750ml, what was its original volume?Answer: = 500ml Solution: Start with the original volume as 100% The new volume is 50% higher than this: 100% + 50% = 150% 150% = 750ml. We can divide by 150 to fins 1% 1% = 5ml. We can now multiply by 100 to find 100%, which was the original volume. 100% = 500ml.QUESTIONSAfter a price increase of 9%, a TV costs $495. How much did it cost originally?Answer: = $454.13 109% = $495 (now divide by 9) 1% = 4.541284404 100% = $454.13 (rounded up)Finding the original quantity after a percentage decreaseExample: 20% of the apples in a harvest are mouldy and are thrown away, leaving 360 good apples. How many were harvested?Answer: = 450 apples. Solution: Start with the original harvest as 100% The apples have been reduced by 20% —> 100% - 20% = 80% 80% = 360 apples. We can divide by 80 to find 1% 1% = 4.5 apples. If we now multiply by 100 we will find 100%, which is the size of the original harvest. 100% = 450 apples.A snacks manufacturer has reduced the weight of his packets of crisps by 15% so that they now weigh 51g. How much did they weigh before?Answer: = 60g. 100% - 15% = 85% 85% = 51g (now divide by 85) 1% = 0.6g (now multiply by 100) 100% = 60g.USING MULTIPLIERS - Intermediate & HigherUsing multipliers is a more efficient method for calculating a percentage increase or decrease. It involves finding a number you can multiply by that represents the percentage change.What multiplier would represent an increase of 29%?An increase of 29% would result in a value of 100% + 29% = 129%. This is the same as 1.29. So multiplying by 1.29 is the same as increasing a value by 29%.QUESTIONSSuzie's investment of $200 increases by 3%. Using multipliers, what is her investment now worth?Answer: = $206. 100% + 3% = 103 103 divided by 100 = 1.03 (is the multiplier) $200 x 1.03 = $206.What multiplier would represent a decrease of 7.3%?Answer: = 0.927 A decrease of 7.3% would result in a value of 100% - 7.3% = 92.7% As a fraction this would be written as: 92.7/100 This is the same as 0.927 (Therefore, multiplying by 0.927 is the same as decreasing a value by 7.3%)QUESTIONAdele foes travelling with $1,500 and spends 65% of her savings. Using multipliers, calculate how much money she has at the end.Answer: = $525. 100% - 65% = 35% 35 divided by 100 = 0.35 —> this is the multiplier. $1,500 x 0.35 = $525.FURTHER EXAMPLES: Example 1 - Barry bought a brand new car in 2009. It cost him $16,000. It's value decreased by 8% every year and in 2014 he decided to sell it. How much was it worth then? Give your answer to the nearest dollar.Answer: = $10,545 (nearest dollar) Solution: You can work this out by using Powers, or the following - Decrease is 8%, so multipliers is 0.92 as 100% - 8% = 92% 210 value = 0.92 x $16,000 = $14,720 2011 value = 0.92 x $14,720 = $13,542.40 2012 value = 0.92 x $13,542.40 = $12,459.008 = $12,459.01 (nearest penny) 2013 value = 0.92 x $12,459.01 = $11,462.2892 = $11,462.29 (nearest penny) 2014 value = 0.92 x $11,462.29 = $10,545.3068 = $10,545.31 (nearest penny) and $10,545 (nearest pound)Example 2 - In a sale a tent was reduced each day by 1/4 until it was sold. It was originally priced $140. It was sold after 4 days. How much did it sell for? Give your answer to the nearest penny.Answer: = $44.30 (to nearest penny) Solution: At start price = $140 Day 1 price = 3/4 x $140 = $105 Day 2 price = 3/4 x $105 = $78.75 Day 3 price = 3/4 x $78.75 = $59.0625 = $59.06 (to nearest penny) Day 4 price = 3/4 x $59.06 = $44.295 = $44.30 (to nearest penny)