This starts off with with reasonably easy stuff about everyday ratios and proportions and then builds into more demanding stuff about using equations to solve problems involving inverse proportion. The more demanding topics are Grade A, and I think that’s fully justified.
As usual, the best way for me to firmly fix this subject in my mind, is to set a question, explain the approach and knowledge required and then provide a detailed answer.
Ratio and Proportion Questions
1) In John’s class of 30 pupils there are 20 boys. What is the ratio of boys to girls?
2) In Mary’s class the ratio of girls to boys is 3:1. Mary says that 1/3 of the class are boys. Is Mary correct? Explain your answer.
3) The ratio of boys to girls in Luke’s class is 5:4. Write this in the form n:1.
4) Arsenal’s squad has a mixture of British, European (mainland) and South American players in the ratio 5:12:7 The equivalent ratios for Chelsea’s squad are 6:15:9. Which squad has the higher proportion of British players?
5) Mr. Wenger has a total bonus pool of £1m for his main strikers. He wishes to award the bonus based on goals scored last season.
Goals Scored Last Season
How much bonus should be paid to each striker?
6) Gervinho complains to his Mr. Wenger that his brother, Servinho, who plays for Barcelona is paid considerably more than him. Servinho is paid €16,000 per week. Gervinho is paid £13,500 per week. The current exchange rate is £1 = €1.20. How should Mr Wenger reply to Grevinho’s complaint?
7) Gervinho asks Mr. Wenger for some financial advice. His favourite hair conditioner comes in 2 sizes, 750ml for £1.50 or one litre for £1.95. Which size should he buy?
8) It takes 3 men 6 hours to creosote a 100 metre fence. How long would it take 10 men to creosote the same fence?
9) The number of goals scored by a striker is directly proportional to the number of games played. Van Perise scored 32 goals (g) in 56 games played ℗. Write a statement of proportionality to connect goals (g) to games played ℗. Next season he hopes to play 65 games, based on last season, how many goals might he expect to score?
10) Look at the following sets of data. For one set of data y is inversely proportional to x, for another set of data y is in directly proportional to x² and in the final set of data y is inversely related to x². For each set of data state in which way y is proportional to x.
11) For all sets of data in question 10 above, the constant of proportionality is 4. For each set of data work out an equation connecting x and y.
Ratio and Proportion Questions– Approach
1) You use notation “:” to indicate ratio, as in 1:2. In this question you can just write 30:20 and then simplify to 3:2.
2) To convert ratios to fractions add both numbers to give the bottom half (denominator) of the fraction.
3) To convert a ratio to the form n:1, decide which number you wish to convert to be 1 and then divide both sides of the ratio by that number. For example if you has 200:100 you would divide both 200 and 100 by 100 to give you 2:1.
4) Convert the ratios to fractions and then use knowledge of fractions to find which is the larger fraction.
5) Add the numbers to get the total number of parts, work out the value of one part and then calculate the amount for each player.
6) In this example the exchange rate had been expressed as a ratio (£1 = €1.20). So to convert £‘s to €‘s you need to multiply by 1.20.
7) This requires you to calculate the cost per litre (or millilitre) for each size so that you can make a vaild comparison.
8) This question introduces the idea of an inverse proportion, the more men that are employed the less hours are required. It is easy to get confused about this sort of problem. To avoid this just break it down into logical steps:-
- Calculate the total man hours to paint the fence with the original number of men (number of men x hours taken)
- Take the total man hours and divide by the new number of men.
9) Remember that ∝ means “is propotional to”.
10) There are a number of ways that two variables can be proportional to each other. Remember that ∝ means “is proportional to” and “k” is the constant or propotionality
y is directly proportional to x y ∝ kx
y is directly proportional to x² y ∝ kx²
y is directly proportional to x³ y ∝ kx³
Note that all of these involve a variation k/x
y is inversely proportional to x y ∝ k/x
y is inversely proportional to x² y ∝ k/x²
y is inversely proportional to x³ y ∝ k/x³
In this question you are told 3 possible types of proportionality and asked to match to them to some data sets. For each set of data work out 1/x, x² and 1/x² and see how the answers are related to y.
11) This is testing your knowledge of k (the constant of proportionality) and how to write an equation.
Ratio and Proportion Answers
1) Ratio of boys to girls is 30:20, which simplifies to 3:2.
2) No Mary is incorrect. To convert a ratio into a fraction you need to add the parts to calculate the denominator (the bottom half of the equation) and then both sides of the ratio become numerators to give fractions that add up to 1. In this case the ratio of girls to boys is 3:1. So the sum of parts is 3 + 1 = 4. So 3/4 of the class are girls and 1/4 (not 1/3) of the class are boys.
3) Divide both sides of 5:4 by 4 gives 1.25:1
4) Arsenals squad has a total of 24 players, of which there are 5 British players. So Arsenal’s squad has 5/24 British players. Chelsea squad has a total of 30 players of which there are 6 British players. So Chelsea’s squad has 6/30 British players
Arsenal = 5/24 & Chelsea = 6/30 = 1/5 = 5/25
Arsenal = 5/24 and Chelsea = 5/25 and Arsenal has greater proportion of British players.
Or using common denominator method
Simplify– Arsenal = 5/24 and Chelsea = 1/5
Common denominator = 5 x 24 = 120
Arsenal = 25/120 & Chelsea = 24/120
25/120 > 24/120 so Arsenal has greater proportion of British players.
5) Van Persie = 29÷(29+8=3+6+15) = 29/61 x £1m = 475.409.84
Walcott = 8/61 x £1m = £131,147.54
Chamakh = 3/61 x £1m = £49,180.33
Oxlade Chamberlain = 6/61 x £1m = £98,360.66
Gervinho = 15/61 x £1m = £245,901.64
Total = £1,000,000.01 (rounding error)
6) The ratio to convert Euros to Pounds sterling is €1.20:£1. Therefore to convert £13,500 to Euros = £13,500 x 1.20 = €16,200. Therefore Mr. Wenger should reject Gervinho’s complaint as when you convert his salary to Euros you see that he is actually paid €200 per week more than his brother.
7) The smaller size costs £1.50/o.75 = £2 per litre, the larger size costs £1.95/1 = £1.95 per litre. Therefore Gervinho should buy the larger size.
8) Man hours to creosote the fence = 3 men x 6 hours = 18 hours. Therefore it would take 10 men 18 hours/10 men = 1.8 hours to creosote the fence.
9) Goal scored (g) = 32/56 games played ℗
Therefore g ∝ .5714p
Based on last season if he plays 65 games he might expect to score 65 x .5714 = 37 goals (round down 37.141)
10) First set of data:
Applying 1/x gives 1/2, 1/3 & 1/4 = 0.5, 0.33 & 0.25. Checking the movement of 1/x against the values of y (2.0, 1.33 & 1.0)
Movement in 1/x = 0.33/0.5= 0.66 and 0.25÷0.33 = 0.75
Movement in y = 1.33÷2 = 0.66 and 1/1.33 = 0.75
Therefore for the first set of data y is inversely proportional to x.
Second set of data:
Applying 1/x² gives 0.25, 0.111 and 0.0625. Checking the movement of 1/x² against the movement in y.
Movement in 1/x² = 0.111÷0.25 = 0.44, .0625÷0.111 = 0.56
Movement in y = .44÷1 = 0.44, 0.25÷0.44 = 0.56
Therefore for the second set of data y is inversely proportional to x².
For the third set of data
Applying x² gives 4,9 & 16. Checking against y gives:
Movement in x² = 9/4 = 2.25, 16/9 = 1.77
Movement in y = 36/16 = 2.25, 64/36 = 1.77
Therefore for the third set of data y is directly proportional to x²
11) First set of data y = 4/x
Second set of data y= 4/x²
Third set of data y = 4x²