## Overview

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

 Name Goals Van Persie 29 Walcott 8 Chamakh 3 Oxlade Chamberlain 6 Gervinh 15

### 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

Directly Pro­por­tional

y is directly pro­por­tional to x            y ∝ kx

y is directly pro­por­tional to x²          y ∝ kx²

y is directly pro­por­tional to x³           y ∝ kx³

Inversely Pro­por­tional

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.

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