Proportion Learning Outcomes

Transcription

1 1 Proportion Learning Outcomes Express quantities as proportions of each other using fractions, decimals, ratios, and percentages. Divide quantities into a given proportion. Express errors as a proportion. Calculate tolerances using absolute values and proportions.

2 2 Express Quantities as Proportions Proportions describe how the size of one number compares to a whole as a factor. e.g. my niece is half as tall as I am. e.g. Cork is four times as far from Dublin as Wicklow is. e.g. my savings account offers 1.5% AER. e.g. a circle s circumference is times as long as its diameter. e.g. the cake was divided in the ratio 3:5.

3 3 Express Quantities as Proportions Proportions are usually calculated as a fraction using: Formula: proportion = part whole e.g. a shirt is on sale for 16. Its normal price is 20. We can express this proportion as a simplified fraction: 16 as a decimal: 4 5 = = 4 5 as a percentage: = 80% as a ratio: 4: 5

4 4 Express Quantities as Proportions Express each of the following as a proportion, simplified fraction, decimal, percentage, and ratio: a) Two cheese & onion from a multi-pack of six crisp packets. b) Consonants in the word consonants. c) Winning from a jackpot. d) 120 spaces free in a 600 space car park. e) 16 m 2 of a 56 m 2 room taken up with furniture. f) One side of a piece of bread toasted.

5 5 Express Quantities as Proportions If a proportion is known, either the part or whole can be calculated using ratios. e.g. if Annie has saved 45% as much as Charles, and Annie has 900, how much has Charles saved? If Annie is 45%, then Charles is 100%: 45% : : 45 1% : : % : 2 000

6 6 Express Quantities as Proportions e.g. Kate grew her collection of beanie babies by 25% last year. If she currently has 15 beanie babies, how many did she have last year? Kate now has 125% of last year s collection: we need 100%. 125% : : 125 1% : : % : 12

7 7 Express Quantities as Proportions a) Robyn worked 18 hours last week. This week, she worked 2 as many hours. How many hours did she work 3 this week? b) Lyn got 80% of questions correct on a test. If she answered 16 questions correctly, how many questions were on the test? c) Scott buys a shirt costing including VAT at 13.5%. What does the shirt cost before VAT? d) Yaru buys a couch wholesale for 300. She sells it in her shop for , which includes VAT at 13.5% as well as a mark-up (percentage profit). Calculate her mark-up to one decimal place.

8 FL Q5 Express Quantities as Proportions Rena wants to buy a new car. She intends to trade in her old car. She has found three different options, A, B, and C. Calculate what the new car will cost for each of the options A, B, and C. a) Option A: The new car will cost In exchange for her old car, she will get a discount of 1 on this price. 3 b) Option B: The new car will cost In exchange for her old car, she will get a 40% discount on this price. c) Option C: She trades in her old car. The new car will cost per month for 4 years. d) Which option should Rena choose if she wants to spend the least amount of money overall?

9 OL P1 Q OL P1 Q1 Express Quantities as Proportions a) Express 40 metres as a fraction of 1 kilometre. Give your answer in its simplest form. a) Express 35 cm as a fraction of 1 m. Give your answer in its simplest form.

10 OLP1 Q1 Express Quantities as Proportions A retailer buys an item for 73. She wants to apply a mark-up of 40% of the cost price of the item. She must then add VAT at 21% to this amount to find the price that she would need to charge the customer. a) Find this price to the nearest cent. The retailer adjusts the price charged to the customer so that it is 1 cent less than a multiple of 10, while keeping the mark-up as close as possible to 40%. b) Using this adjusted price, calculate the actual percentage mark-up achieved, correct to the nearest percent.

11 OL P1 Q OL P1 Q1 Express Quantities as Proportions 320 is 4 9 of a prize fund. Find the total prize fund. At the start of the year 2000, the population of a particular town was P. During the year 2000, the population of the town increased by 10%. a) Express, in terms of P, the population of the town at the end of the year b) During the year 2001, the population of the town increased by 4%. During the year 2002, the population of the town increased by 2%. Find the total percentage increase in population of the town over the three years. c) The actual increase in the population was Find the value of P.

12 12 Divide Quantities into a Given Proportion Amounts are often split into ratios. Ratios are more versatile than proportions since they can reference multiple amounts at once, while proportions can only refer to a single part of a whole. e.g. An inheritance is split between two children in the ratio 1:2:2. Using proportions, we would get 1, 2, and 2. Ratios allow us to express more information in a compact way.

13 13 Divide Quantities into a Given Proportion e.g. An inheritance of is split between Becky, Millie, and Cian in the ratio 1:2:2. How much did each of them get? Figure out what 1 means in the ratio: how many parts are there? = = Becky gets Millie and Cian each get = 4 000

14 14 Divide Quantities into a Given Proportion a) is split in the ratio 2: 3. Find each amount. b) Kate and Grace split a pack of fruit pastilles between them in the ratio 2: 5. If Kate gets 4 fruit pastilles, how many does Grace get? c) In a box of 48 oranges, the ratio of rotten oranges to ripe oranges is 1: 5. How many ripe oranges are there? d) The ratio of geese to ducks on a lake is 4: 9. When 10 extra geese land on the lake (and there are no duck changes), the ratio is 2: 3. How many of each were there initially?

15 OL P1 Q OL P1 Q1 Divide Quantities into a Given Proportion Aoife and Brian share a prize fund in the ratio 4: 3. Aoife gets 56. a) Find the total prize fund. b) How much does Brian get? Conor and Alice share 50 apples in the ratio 3: 7. a) How many apples does Connor get? b) How many apples does Alice get? 2005 OL P1 Q1 a) Express the ratio 1 2 : 1 3 : 1 4 as a ratio of natural numbers. b) Divide 325 into the ratio 1 2 : 1 3 : 1 4

16 16 Express Errors as a Proportion Errors often occur due to estimation or inaccurate measurement. Formula: Error as proportion = error true value In particular, we often deal with percentage error: Formula: Percentage error = error true value 100 e.g. a physics experiment determines the speed of sound to be 330 m/s, while it was actually 340 m/s. Calculate the percentage error to one decimal place. % error = error true value 100 = = 2.9%

17 17 Express Errors as a Proportion a) Elva miscalculates as 6. What is her percentage error? b) Millie mistakenly writes x + y 3 = x 3 + y 3. What is her percentage error if x = 3 and y = 1? c) Daniel mistakenly writes x+3 = x+3 = 3 = 1. What is his x+6 x percentage error if x = 6? d) Find the percentage difference in using the estimates 3.14 and 22 instead of the actual π button on your 7 calculator.

18 OL P1 Q OL P1 Q1 Express Errors as a Proportion When using a calculator to add 1.7 and 2.2, a student strikes the multiplication key instead of the addition key. Calculate the percentage error in the result, correct to one decimal place. The approximation was used for the calculation Find the percentage error, correct to one decimal place.

19 19 Calculate Tolerances Industrial processes are not perfect. A manufactured item may be too big or too small, a bag of sweets may be too heavy or too light, etc. The acceptable range of value is called the tolerance and is usually given as a target value ± tolerance. e.g. to fit properly, a plastic bottle cap needs to have an internal diameter of 28mm ± 1 mm. Calculate the widest bottle cap. Calculate the narrowest bottle cap. Determine the tolerance as a percentage of the internal diameter to one decimal place.

20 20 Calculate Tolerances Calculate the extrema of acceptable values for each of the following and express the tolerances as a percentage of the target value: a) A large packet of Doritos may be 225 g ± 10 g. b) A watch gear may be 10 mm ± 0.5 mm. c) A concrete footpath may expand or contract in the heat. It should be 5 m ± 5 cm. d) A plain 6 meatball marinara on Italian herbs & cheese bread with Monterey cheddar is 550calories ± 50 calories.