How to calculate ratio part II: some worked examples and problems to solve
My first hub on ratio (entitled How To Work Out Ratios - Calculating Ratios Made Simple) has generated lots of response from people who have been given ratio problems to solve as part of their coursework or whatever. Rather than continue to give answers in the comments section at the bottom of the first hub, I thought I'd compile a few typical questions here, together with step-by-step solutions. At the end, there will be a section of new questions for you to try to solve…. followed by a third hub with the answers!
When it comes to solving problems involving ratio and proportion, half the battle is in making sense of the question. Sometimes it can help to write down - without any maths at all - what you will actually need to do, before attempting to solve the question itself. That's the approach I'll adopt here - hope it helps.
Worked example 1
In an office, the ratio of staff who like sport to those who don't is 3:2. If 50 staff gave their opinions, how many of them like sport?
Ask yourself this question: do the 50 staff members correspond to one number of the ratio (either the "3" or the "2"), or both? If you read the question carefully, you'll realise it's both, i.e. the 50 covers EVERYONE in the office, both sport lovers and sport haters.
So what you then need to do is add the 3 and the 2 together, giving 5 "parts".
50 people = 5 parts.
Now you know this, you need to work out how many people correspond to one "part". To do this, divide 5 into 50, which gives you 10.
If 1 part = 10 people, then
3 parts = 3 x 10 = 30 people and
2 parts = 2 x 10 = 20 people.
So there are 30 people in the office who like sport, and 20 people who don't.
This is a classic example of a "splitting ratio" type of question.
Worked example 2
A vet has advised his client that the ideal ratio of rams to ewes is 1:45. If the client has 1,800 ewes, how many rams does he need?
Ask the question posed in worked example 1, i.e. "does the number of ewes (1,800) correspond to the "1", the "45" or both added together? In this case, it corresponds to the 45.
In other words:
Rams : ewes = 1 : 45 = ? : 1800
To solve this question, you need to ask yourself "What number do I have to multiply the 45 by in order to get 1800 (the actual number of ewes)? Because that's what I'll have to multiply the 1 by to get the "?", which is the actual number of rams."
To get this mystery number, you just divide the 45 into 1800, which gives you 40.
Rams : ewes = 1 : 45 = 40 : 1800
Sometimes it helps to think of this as two equivalent fractions:
1 = 40
You have to multiply the 45 by 40 to get 1800, so you have to multiply the 1 by 40 as well.
This is an example of an "equivalent ratio" question, in other words.
Worked example 3
Cathy works in a hairdressers and needs to make some Rich Auburn dye. This is done by mixing Chestnut Brown, Flaming Red and Blue Lagoon in a 9:5:2 ratio. She has a 500 ml container and wants to make 400 ml of dye so that she has enough room in the container to mix the ingredients together without spilling the dye everywhere! How much of each colour does Cathy need?
Sometimes when answering questions of this kind, you'll find that some of the information they give you is completely irrelevant. In this case, it's the fact that the container is 500 ml - so just ignore that bit! The crucial number you need to work with is the 400 ml of actual dye that you're making. Like worked example 1, this is the sort of question where the final amount of dye (400 ml) doesn't correspond to any single part of the ratio, but to all the parts added together.
Thus 400 ml = 9 + 5 + 2 = 16 parts
1 part = 400 divided by 16 = 25 ml.
You therefore need:
9 x 25 = 225 ml of Chestnut Brown
5 x 25 = 125 ml of Flaming Red, and
2 x 25 = 50 ml of Blue Lagoon.
You should find that if you add the amounts together, it comes to 400 ml. Lo and behold:
225 + 125 + 50 = 400. This is a useful check to do at the end of a question like this one, to see whether you've done the calculations correctly.
Some problems for you to try… answers to follow in another hub
Stuart and Ali run a printing business together. When they started the business, Stuart put in £1,500 and Ali put in £2,500. What is the ratio of Ali's to Stuart's contribution, in its simplest form?
Five years later Stuart and Ali decide to sell the business to a Mrs Khan. They split the proceeds (£24,000) in the same ratio as the ratio of their original contributions. How much does Ali get?
To make Green Shimmer paint, you need to mix Yellow Sun, Blue Topaz and Silver Moon paints in a ratio of 7:7:4. How much of each colour do you need in ml if you're making 36 litres of Green Shimmer? Hint: 1 litre = 1,000 ml.
Broadview Animal Rescue Centre has 270 dogs and 195 cats. What is the ratio of cats to dogs, expressed in its simplest form?
When making chocolate cake, Andy needs a 5:3:2 weight ratio of flour, chocolate powder and caster sugar. If he uses 28 grams of sugar, how much of the other ingredients does he need?
Kylie is having a party and wants to make a punch bowl of Sea Breeze cocktail, which consists mainly of 2 parts vodka, 3 parts cranberry juice and 3 parts grapefruit juice. How much of each ingredient (in ml) would she need to make:
a) 4 litres of cocktail?
b) 1 litre of cocktail?
Click here for the answers to these questions... no cheating though!
© Empress Felicity August 2010