Some problems involving ratio and proportion


Example (1)
The prize money of £180 from a competition is to be divided between three children.
Adam won the first prize, Beth won the second prize and Chris won the third prize.
It is decided that the division will be in the ratio of 1st : 2nd : 3rd = 5 : 3 :1.
How much does each child receive?
The £180 is being divided into   5 + 3 + 1 = 9   equal parts.
Each part is   £180/9 = £20.
It is being divided so 1st : 2nd : 3rd = 5 : 3 : 1.
Therefore Adam gets 5 x £20 = £100 and Beth gets 3 x £20 = £60 and Chris gets 1 x £20 = £20.

( Useful check   The total of the prizes is £180 as it should be.)

Example (2)
A glass contains 25ml of whisky which is 40% alcohol v/v. (v/v means the % of alcohol to total volume.)
If I add 60 ml of water and stir the mixture, what is the alcohol content to the nearest 0.1%?
The 25 ml of whisky is 40% alcohol so it contains   (40/100) x 25 = 10 ml of alcohol.
After I've added the 60 ml of water, I've got 85 ml of fluid of which 10 ml is alcohol.
Therefore the fraction of alcohol is 10/85 and the % of alcohol is   (10/85) x 100 = 11.8% v/v to the nearest 0.1%.
Example (3)
32 tonnes of a mixture of sand and gravel is 25% sand. How many tonnes of sand must be added to produce a mixture which is 40% gravel?
25% of the 32 tonnes is sand so the quantity of sand is   (25/100) x 32 = 8   tonnes.
Therefore the remaining 24 tonnes of the present mixture is gravel.
We want this 24 tonnes to be 40% of the new mixture since we are only adding sand.
Now 40% = 40/100 = 2/5 so 24 tonnes is 2/5 of the mixture.
Therefore 12 tonnes is 1/5 of the mixture, and the whole mixture is 5 x 12 = 60 tonnes.
Or you could work out this answer by using percentages and saying
Now 24 tonnes of this mixture is gravel so 36 tonnes needs to be sand.
We already have 8 tonnes of sand so we must add another 28 tonnes of sand to get the required mixture.
Example (4)
An eccentric person leaves instructions in his will that a certain sum of money is to be divided between his relatives so that Patrick will receive half and Quentin will receive one fifth. The remainder is to be divided so that Rufus receives three quarters and the rest goes to Simon.
If Simon eventually received £324 how much did each of the rest receive? How much money had the eccentric person earmarked to be divided between his relatives?
The money for Rufus and Simon was divided in the ratio 3:1 so Simon's £324 was one part of it.
Therefore Rufus received 3 x £324 = £972 and between them they received £1296.
Now we need to find what fraction this was of the total amount of money left to the relatives.
We know that half went to Patrick and one fifth to Quentin. For the next step, we need to write these fractions as tenths. I've shown how this works in the drawing below.
The £1296 to be divided between Rufus and Simon was 3/10 of the total.
Therefore £1296/3 = £432 is 1/10 of the total and the eccentric left 10 x £432 = £4320 to be divided between his relatives.
Of this, Patrick received £4320/2 = £2160 and Quentin received £4320/5 = £864.
Example (5)
How many litres of each of a 90% v/v solution of alcohol and a 75% v/v solution of alcohol will we need to mix together in order to make 30 litres of an 80% v/v solution of alcohol?
Much the easiest way to deal with this problem is to use letters for the quantities we want to find out.
So suppose that we need p litres of the 90% solution and q litres of the 75% solution.
We must have   p + q = 30   since we want 30 litres altogether.
Also, this 30 litres is 80% alcohol so it contains   (80/100) x 30 = 24 litres of alcohol.
Now the p litres of 90% v/v alcohol solution contains   (90/100)p litres of alcohol.
The q litres of 75% v/v alcohol solution contains   (75/100)q litres of alcohol.
So   (90/100)p + (75/100)q = 24   or   90p + 75q = 2400   multiplying the equation all through by 100.
But q = 30 - p so we can say
90p + 75(30 - p) = 2400   so   15p = 2400 - 2250 = 150.
Therefore p = 10 and q = 20. We need 10 litres of the 90% v/v solution and 20 litres of the 75% v/v solution to make 30 litres of an 80% v/v solution of alcohol.
Back to the main percentage page