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.

Ignore this picture, it's just an illustration and has nothing to do with the text next to it!
Ignore this picture, it's just an illustration and has nothing to do with the text next to it!

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.

Therefore:

Rams : ewes = 1 : 45 = 40 : 1800

Sometimes it helps to think of this as two equivalent fractions:

1 = 40
45 1800


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

Problem 1

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?


Problem 2

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?


Problem 3

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.


Problem 4

Broadview Animal Rescue Centre has 270 dogs and 195 cats. What is the ratio of cats to dogs, expressed in its simplest form?


Problem 5

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?


Problem 6

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

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Comments 10 comments

Varsha 5 years ago

Hi

Can you please help me with this problem..

Two ingots are melted together in a furnace - the first one weighing 12 kg and containing 64% iron while the second weighs 9 kg and contains 99% iron. What percentage of iron is present in the resultant large ingot?


EmpressFelicity profile image

EmpressFelicity 5 years ago from Kent, England, UK Author

@Varsha "Two ingots are melted together in a furnace - the first one weighing 12 kg and containing 64% iron while the second weighs 9 kg and contains 99% iron. What percentage of iron is present in the resultant large ingot?"

Whew! That's a bit different from your normal ratio question. I make the answer 79%. There may be an easier way of doing it than the method I used, but here goes:

Call the weight of iron in the large ingot "y". The percentage of iron in the large ingot can therefore be written as:

(y x 100)/21

...since 21 kg is the total weight of material in the large ingot (12 + 9 = 21)

y = [(64 x 12)/100] + [(99 x 9)/100]

= (768 + 891)/100

= 1659/100

Therefore the percentage of iron in the large ingot is:

[(1659/100) x 100]/21

which cancels down to

1659/21

= 79%.


Anne mary Joseph 5 years ago

can you please assist me with this poblem?The parent group of a local head start promoted recycling in their neighborhood one saturday in march by distributing sets of blue and green recycling bins to grocery shoppers at Central Marketplace.Out of the 248 shoppers who stopped by head start's recycling information table,168 took home a set of bins:What is the ratio of the shoppers who took bins to the total number of shoppers who stopped at the information table?It is appoximately?


Nicola Rowley 5 years ago

HI

I,ve read your site, it's very good. but i can't get the answer to a question that my 10 yr olds have been given for homework. can you help?

"Work out the masses A, B, C and D in kilograms. Each mass is a whole number and is less than 10kg. There are 2 possible solutions:

One of A is as heavy as 3 of c

Two of B are as heavy as 1 of D

Two of C are as heavy as one of B

One of D is as heavy as 4 of C."

I'm completely stuck, as the comparisons above are all different?


EmpressFelicity profile image

EmpressFelicity 5 years ago from Kent, England, UK Author

@Nicola:

My first thought was "simultaneous equations" when I looked at your ten year olds' homework question, but then I realised I'd just end up going round in circles (like you did!). In the end, I just worked it out by saying that as D = 4C and D is both a whole number and less than 10kg, D's got to be either 4 kg or 8 kg. Sure enough, if D = 8 kg, then C = 2 kg and B = 4 kg (because 2C = B). Which means that A is 6 kg.

If D = 4 kg, then C = 1 kg, B = 2 kg and A = 3 kg. No algebra or ratios needed.


EmpressFelicity profile image

EmpressFelicity 5 years ago from Kent, England, UK Author

@Anne Mary:

Ratio of people who took bins home to people who stopped at the table is

168:248

You can then cancel it down by dividing by 4, to give

42:62

and again by dividing by 2, to give

21:30

Dividing by 3 gives

7:10

Alternatively, you could round the 21:30 off to 20:30, which cancels down to give 2:3.


cee 4 years ago

pls the answer is wrong, how do u get 30 from dividing 62 by 2, if that is so, what's the answer of 60 divided by 2?


David 4 years ago

If you look at worked example 3 there is or seems to be a simple typo, where you carry the ml measurements down from you therefore need section you have 25 + 125 + 50 = 400. The initial 25 should be 225 shouldn't it?


EmpressFelicity profile image

EmpressFelicity 4 years ago from Kent, England, UK Author

@David: thanks for pointing it out, you're quite right! I have corrected the typo.


EmpressFelicity profile image

EmpressFelicity 4 years ago from Kent, England, UK Author

@cee, you're right also (sorry it's taken so long to reply).

My comment to Anne Marie should read:

"@Anne Mary:

Ratio of people who took bins home to people who stopped at the table is

168:248

You can then cancel it down by dividing by 4, to give

42:62

and again by dividing by 2, to give

21:31

Since 31 is a prime number, you can't cancel any further than this but you can round the 21:31 off to 20:30, which cancels down to give 2:3."

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