Fractions, decimals and percentages


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1 Fractions, decimals and percentages Some notes for the lesson. Extra practice questions available. A. Quick quiz on units Some of the exam questions will have units in them, and you may have to convert 1. How many metres in a kilometre? 2. How many milligrams in a gram? 3. How many millimetres in a centimetre? 4. How many millilitres in a litre? 5. 7 Km = metres 6. 42mm = centimetres Km = m = cm B. What can you remember about fractions and percentages? Try the card sorting game as well What is 10% of 70? 2. Can you find 15% of 70 without a calculator? 3. What is 5% of 6.50? 4. Convert 17% to a decimal number 5. Convert 140% to a decimal number (hint: think pounds and pence) 6. Freda estimates that the winter gas bill is going to be 170. She puts 12 a week away for the bill. How many weeks before Freda has enough to pay the bill? 7. What is the largest number that divides 12 and 18 without leaving a remainder? C. Finding a fraction of an amount using a calculator Divide by the number on the bottom Multiply by the number on the top Example: find 3 5 of 40. You do 40 5 then multiply the answer by 3. Gives 24 A drawing can help sometimes. What is each box worth if all five of them are worth 40? How much are 3 boxes worth? Try these (some give whole numbers, some give decimals) 1) 2 7 of 63 2) 4 9 of 36 3) 7 10 of ) Four out of ten cats prefer fish. You have 75 cats. How many prefer fish?
2 D. Writing one quantity as a fraction of another Write both quantities in the same units (watch out for this) Check if the question tells you the 'whole' or if you have to find it first Write the 'part' over the 'whole' Cancel the fraction down Example: write 5p as a fraction of Step 1, change the 0.75 to 75 pence. 5 Step 2, write 75. Step 3, cancel the fraction down 1 15 Example 2: sometimes you have to add up to find the whole There are 24 people on the train and 36 empty seats. What fraction of the train is 36 empty? The 'whole' is = 60. Fraction is 60. This cancels down to 3 5. Try these 1) Nigel has three red fish and twelve gold fish. What fraction of his fish are red? 2) Evadne said I have 4 apples and 7 oranges in my fruit bowl, so the fraction of my 4 fruit bowl that is apples is. Is Evadne right? Explain your answer. 7 3) There are 12 people in a room and four are male. What fraction of the people is male? 4) What is 2 as a fraction of 10? 5) What is 35p as a fraction of 1.50? 6) What is 400m as a fraction of 1.2Km? 7) A computer monitor has a resolution of pixels horizontally and 1080 pixels vertically. Write the vertical size of the monitor as a fraction of the horizontal size. 8) The train runs every day including Weekends. It is late 12 days one September. What fraction of the days is the train late? E. Finding the value of a percentage Per Cent just means 'out of a hundred' Find 1% of a quantity by dividing by 100 Find any other percentage by multiplying by the percent figure Its just the same as Section C but with 100 as the denominator of the fraction 35% off ipad Was 499 How much is the ipad in the sale? How could you work this out without a calculator?
3 F. Writing one quantity as a percentage of another Percentages are often used to compare different organisations or areas. E.G. crime rates in different wards in Birmingham, or success rates for a specific operation between different hospitals. Both of these have been in the news recently. The method is very similar to Section D but with fractions out of 100. Decide which is the 'whole' and the 'part' You might have to calculate the part or the whole Convert units if you need to: must be in same units Divide the part by the whole, don't round off your decimal too much Multiply the answer by 100 Round off to a sensible number of figures (one decimal place of percentages is usually OK if they don't ask for something specific) Example 1: Write 25 as a percentage of 75. Start at Step 3: Your calculator will give so round it to 33.3% Example 2: Nathan was 84Kg and is now 70Kg. What percentage of his weight has he lost? Part is = 14 and the whole is 84Kg = so round that to 16.7% Example 3: A 3Kg bag of carrots usually costs 1.20 but has been reduced by 24p. What is the percentage reduction? Part is 20p but we have to write that as Whole is Percentage is = 20% Try these 1) What is 15p as a percentage of 1.80? 2) Write 200g as a percentage of 1.25 Kg 3) You have to take 150ml of a reagent and add enough water to make 1 litre of solution. What percentage of the solution is reagent? 4) Tom scores 15 out of 20, Hari scores 35 out of 45 and Faith scores 18 out of 24. Work out their percentage scores. Who scored highest? 5) Lionel weighed 106Kg at the start of his diet. He now weighs 89Kg. What percentage of his starting weight has he lost? Give your answer to one decimal place. 6) Billy the bull has lost 340Kg over the last three months, and he know weighs 730 Kg. What percentage of the starting weight has the bull lost? 7) A supermarket buys in 250 bags of salad for 40p each and sells 207 of them for 95p each. The rest were not sold as they were out of date. What was the percentage profit on the salad bags? 8) A local mathematics club has the following members Men Women Girls Boys a) What percentage of the members are boys? b) What percentage of the members are female?
4 G. Percentage Increase and Decrease You can find the value of the percentage increase or decrease and then add it on or take it away from the whole. That was how the questions in the last section worked. Or you can use a 'multiplying factor'. Suppose Fred earns 9 per hour and then has a 10% increase. You can work out his new hourly rate by multiplying by 1.1. The 1.1 came from 100% + 10% = 110% divided by 100%. Example: it costs 3.60 to park a car for three hours in a car park. Prices go up by 5%. What is the new cost to park?. 100% + 5% = = 1.05 times as much. So = 3.78 Example: A laptop cost 300. It is reduced by 15% in a sale. Work out the new cost of the laptop = 85% 100 = So the new cost is = 255. Use either method to work out the following questions 1) Increase by 15% 2) Decrease by 8%. Round to nearest penny 3) A barrel of whisky loses 8% of its capacity to evaporation when it is maturing, this is known as the 'angels share'. If the barrel contained 240 quarts to begin with, how many are left when mature? 4) Aaron starts a new job on per year but with a 3% rise after three months. Hartinder starts a new job on per year and receives a pay rise of 5% after three months. Who has the greater annual salary after 6 months? 5) Which would you rather have, a pay rise of 10% each year or a pay rise of 5% every 6 months. Can you explain your answer? H. Index Numbers The retail price index tells you how much the cost of a 'standard' bag of shopping has changed over the years. One year is adopted as 100% and other years are written as percentages of that 'base year'. Below are the January RPI figures for 5 year intervals Year Index Food Suppose a typical weekly food shop for a couple cost in What would it cost in 2011? = Suppose a typical wage for a 18 year old in 1977 was 35 per week. What would that be worth in today's money? = Try the review exercise for more searching questions and past paper exam questions.
5 Answers I've put some methods with later questions so I hope they can act as extra examples. Section A 1) ) ) 10 4) ) ) 4.2 7) 400m, cm Section B 1) 7 (divide by 10 to find 10%) 2) Find 10%, then find half of that to get 5%, so 15% is = ) 33p 4) ) ) = 14 rem 2 or so 15 weeks 7) 6 This is called the Highest Common Factor Section C 1) 63 7 = 9 2 = 18 2) 36 9 = 4 4 = 16 3) = = ) = = 30 cats Section D 1) Whole = = 15 fish, so 3 15 = 1 5 2) Evadne is wrong, because the denominator of the fraction is = 11 pieces of fruit 3) 4 12 =1 3 6) Change units Section E 4) = =1 5 5) Change units 7) = = ) = reduction, so = ) = 2 5 Non calculator: find 10% of 499, Add this three times, then add half as much. That should give 24.95, and the total reduction comes to Or you could argue that if 35% off, then the cost is 65%. Then you could find half of 499, 10% and 5% and then add those three amounts. Section F 1) = 8.3% 2) = 16% (change unit) 3) = 15% (whole is 1000ml 'water to make') 4) = 75% = 77.8% = 75% Hari has most 5) = 17Kg is part = 16.0% (rounded from 16.03) 6) You need to find original weight of the bull! Tricky! Ended up at 730, lost 340, so original weight was = 1070Kg. Then = 31.8% loss. This is a Grade C high scoring question. 7) Costs = 96 Sales = Profit Percentage: = 104.8% 8) 12.5% and 56.25%
6 Section G 1) = 115 so new amount is = ) = 92% so new amount is = ) = 92% so new capacity is = quarts rounding to 221 4) Aaron: = Hartinder: = Aaron gets more. Watch out for = 103 so 1.03 multiplying factor, not 1.3. Missing the zero out of the decimal is a common mistake 5) Make up a wage, say % rise is = % every 6 months, you get 5% extra in 6 months, then that figure is the basis for the next pay rise of 5% at the end of the year. You can just multiply by 1.05 twice = So you are 50 better off... That 50 is the 5% on the extra 5% you earned in the first 6 months... Compound interest works like this. Section H Look at the Index questions from the textbook and in the review exercise
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