Whether you are using units from the Metric system (as we do in this post) or US measurement system (the GMAT being an American test), the concepts don’t change.

## Introduction to Ratios

#### Definition

Ratio is the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.

(Reference : Oxford dictionary)

Notation: Ratio of two values a and b is written as a:b or a/b or a to b.

For instance, the ratio of number of boys in a class to the number of girls is 2:3. Here, 2 and 3 are not taken as the exact count of the students but a multiple of them, which means the number of boys can be 2 or 4 or 6…etc and the number of girls is 3 or 6 or 9… etc. It also means that in every five students, there are two boys and three girls.

Question: In a certain room, there are 28 women and 21 men. What is the ratio of men to women? What is the ratio of women to the total number of people?

Solution:

Men : women = 21 : 28 = 3:4

Women : total number of people = 28 : 49 = 4 : 7

Question: In a group, the ratio of doctors to lawyers is 5:4. If the total number of people in the group is 72, what is the number of lawyers in the group?

Solution:

Let the number of doctors be 5x and the number of lawyers be 4x.

Then 5x+4x = 72 → x=8.

So the number of lawyers in the group is 4*8 = 32.

Question: In a bag, there are a certain number of toy-blocks with alphabets A, B, C and D written on them. The ratio of blocks A:B:C:D is in the ratio 4:7:3:1. If the number of ‘A’ blocks is 50 more than the number of ‘C’ blocks, what is the number of ‘B’ blocks?

Solution:

Let the number of the blocks A,B,C,D be 4x, 7x, 3x and 1x respectively

4x = 3x + 50 → x = 50.

So the number of ‘B’ blocks is 7*50 = 350.

Question:

If the ratio of chocolates to ice-cream cones in a box is 5:8 and the number of chocolates is 30, find the number of ice-cream cones.

Solution:

Let the number of chocolates be 5x and the number of ice-cream cones be 8x.

5x = 30 → x = 6.

Therefore, number of ice-cream cones in the box = 8*6 = 48.

## Introduction to Proportion

A lot of questions on ratio are solved by using proportion.

#### Definition & Notation

A proportion is a comparison of two ratios. If a : b = c : d, then a, b, c, d are said to be in proportion and written as a:b :: c:d or a/b = c/d.

a, d are called the extremes and b, c are called the means.

For a proportion a:b = c:d, product of means = product of extremes → b*c = a*d.

Let us take a look at some examples:

Question:

In a mixture of 45 litres, the ratio of sugar solution to salt solution is 1:2. What is the amount of sugar solution to be added if the ratio has to be 2:1?

Number of litres of sugar solution in the mixture = (1/(1+2)) *45 = 15 litres.

So, 45-15 = 30 litres of salt solution is present in it.

Let the quantity of sugar solution to be added be x litres.

Setting up the proportion,

sugar solution / salt solution = (15+x)/30 = 2/1 → x = 45.

Therefore, 45 litres of sugar solution has to be added to bring it to the ratio 2:1.

Question:

A certain recipe calls for 3kgs of sugar for every 6 kgs of flour. If 60kgs of this sweet has to be prepared, how much sugar is required?

Solution:

Let the quantity of sugar required be x kgs.

3 kgs of sugar added to 6 kgs of flour constitutes a total of 9 kgs of sweet.

3 kgs of sugar is present in 9 kgs of sweet. We need to find the quantity of sugar required for 60 kgs of sweet. So the proportion looks like this.

3/9 = x/60 → x=20.

Therefore, 20 kgs of sugar is required for 60 kgs of sweet.

## Problems on Mixtures / Blends

Question:

If a 60 ml of water contains 12% of chlorine, how much water must be added in order to create a 8% chlorine solution?

Solution:

Let x ml of chlorine be present in water.

Then, 12/100 = x/60 → x = 7.2 ml

Therefore, 7.2 ml is present in 60 ml of water.

In order for this 7.2 ml to constitute 8% of the solution, we need to add extra water. Let this be y ml.

Then, 8/100 = 7.2/y → y = 90 ml.

So in order to get a 8% chlorine solution, we need to add 90-60 = 30 ml of water.

Question:

There is a 20 litres of a solution which has 20% of bleach. Extra bleach is added to it to make it to 50% bleach solution. How much water has to be added further to bring it back to 20% bleach solution?

This question has 3 parts.

In the first part, there is 20% of bleach in 20 L of solution → 4 L of bleach in 16 L of water = 20 L of solution. Let’s note the details in a table for better clarity and understanding.

 Bleach Water Total % Quantity in L % Quantity in L 20% 4 L 80% 16 L 20 L

In the second part, Extra bleach is added to bring it to 50% of total solution. Let the amount of bleach added be x litres.

 Bleach Water Total % Quantity in L % Quantity in L 20% 4 L 80% 16 L 20 L 50% 4+x 50% 16 L 20+x

Then, (4+x)/(20+x) = 50/100 → x = 12 L of bleach is added.

Now, there is 4+12 = 16 L of bleach in 16 L of water in a total of 32 L of solution.

Now, to bring the bleach percentage back to 20%, extra water is added and the amount of bleach remains the same. Let this extra amount of water be y litres.

 Bleach Water Total % Quantity in L % Quantity in L 20% 4 L 80% 16 L 20 L 50% 16 L 50% 16 L 32 L 20% 16 L 80% 16+y 32+y

16 L of bleach constitutes 20% of the solution →

16/(32+y) = 20/100 → y = 48.

Therefore, 48 litres of water has to be added to the solution if bleach has to be 20% of the whole solution.

Question:

1 kg of cashews costs Rs. 100 and 1 kg of walnuts costs Rs. 120. If a mixture of cashews and walnuts is sold at Rs. 105 per kg,then what fraction of the total mixture are walnuts?

Solution:

For this type of problems, first step is to determine how much each of the items is above or below the target.

Our target price is Rs. 105. Cashews price is Rs. 5 below the target price and walnuts price is Rs. 15 above the target price.

So, for each kg of cashews added, let’s consider it as ‘-5’ and for each kg of walnuts added, let’s consider it as ‘+15’. These two have to be added in such a way that they cancel out each other. Adding ‘-5’ thrice gives a ‘15’ and adding ‘+15’ once results in cancellation of the terms.

This means that adding 3 kgs of cashews and 1 kg of walnuts gives a mixture that can be sold at Rs. 105 per kg.

So, 3 kgs of cashews present for every 1 kg of walnuts. The ratio of cashews to walnuts is 3:1. Fraction of walnuts in the mixture = 1/(3+1) = 1/4 of the total mixture

## Practice Questions in Ratio and Proportion

Problem 1

On a certain map, 1 cm = 12 km actual distance. If two places are 96 km apart, what is their distance on map?

A. 10 cm
B. 12 cm
C. 96 cm
D. 8 cm

D.

Explanation:

1cm/12 km = x cm/100 km → x = 8 cm

Problem 2

A person types 360 words in 4 minutes. How much time does he take to type 900 words?
A. 15
B. 90
C. 10
D. 9

C.

Explanation

4/360 = x/900 → x=10

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1. Raisa says:

Hey can you please emphasize on the working of the cashews and walnuts question?

I’m stuck on how we’re assuming this:
So, for each kg of cashews added, let’s consider it as ‘-5’ and for each kg of walnuts added, let’s consider it as ‘+15’. These two have to be added in such a way that they cancel out each other. Adding ‘-5’ thrice gives a ‘15’ and adding ‘+15’ once results in cancellation of the terms.

Why do they have to be added to cancel out eaach other?

Would appreciate the help.

Thanks!

• Prodigy says:

I think if they equal zero, then is the only time a ratio can be found. If it isn’t equal to zero you won’t get a ratio.

• vikas says:

100x +120y=105(x+y).
15y=5x
x/y=3/1
we have to find. y/(x=y)=1/4

• geethanjalikotaru says:

given 1 KG of cashews cost 100 and walnuts was 120 given their mixture costs 105 rupees
this 105 rupees of mixture cost is for just 1 kg includes X grams of cashews and Y grams of walnuts
so, X+Y=1 KG or X+Y=1000 grams——1
1 KG of cashews cost is 100 so ,1000 grams=100 rupees. Hence 1 gram=1/10 rupees same way 1 gram of walnuts costs 12/100 rupees
so, X/10 +Y*(12/100)=105——-2
solve equation 1 and 2 . Quantity of X and Y will be 750 grams and 250 grams so their ratio is 3:1
walnuts to the total ratio will be 1/4

Simplest form of 7.5 :11/2.

3. caterina damico says:

Hello I have this peoblem
The ratio of the area of the dining room to the family room is 2 to 3. After remodeling the family room is now 1/2 as large as it used to be and has 60 Sq less than the dining room. How many Sq feet is the isning room?

• Erika says:

Hi
Let the area of the dining room be d and the family room be f. d/2 = b
f/3 = k
d= 2k
f=3k
Total area = 5k
5k=3k +60(1/2*3k +1/2*3k + 60)
2k= 60
k = 30
Area of family = 3*30 /2 = 45
Area of dining = 105
45+105 = 150 and total area remains the same

• Theo Jackson says:

Hi Caterina,

I had a look at Erica’s answer, and unfortunately it doesn’t add up. However, I will stick to the same format she used:

dining room = d
family room = f

d : f = 2 : 3

Lets make that ratio easier to handle: d : f = 4 : 6

Now what does 4 : 6 mean? It means that the dining room represents 4 parts whilst the family room represents 6 parts. A part can be any size in a ratio, what matters is the proportion that the ratio describes.

Lets say 1 part = k, therefore d : f = 4k : 6k

Now if the family room halves in size, the ratio becomes 4k : 3k
because the family room used to be 6 parts but is now 3 parts.

It is now also 60sq ft less than the dining room, so ‘size of f’ – ‘size of d’ = 60 sq ft
therefore 4k – 3k = 60 sq ft
i.e 1k = 60sq ft

Remember that the dining room is 4k so its size is:

4k = 4 x 60 sq ft = 240 sq ft

A Telecom Service operator announced that between 8 PM and 8 AM all outgoing calls and text
messages shall be charged @ 50 % of the charge rate as applicable from 8 AM to 8 PM. After availing
of this benefit, my bill dues reduced by 20 %. What is the ratio of my calls and SMS during day and
night time?

5. Forget it says:

3:4;
Actual bill b4 availing the discount 100
60:40 =100
50 % of 40 = 20
60 + 20 = 80
Reduction of 20%

6. Ramanujan says:

let the total area be X, so Dining room (D) = 2/5 X and Family room (F) = 3/5 X
so 2/5 X + 3/5 X = X

Now, After remodeling Family room (F) = 3/10 X, since it is half of original
Dining room (D) = 3/10 X + 60
So, 3/10 X + 3/10 X + 60 = X
therefore, X = 150 and Dining room (D) = 105

7. Ramanujan says:

let the Calls and SMS in the morning time be X and the same in night time be Y and assuming the original rates to be 1, the total bill is X+Y

now after the announcement by the telecom operator the total bill is down by 20% i.e. X+ 0.5 Y = 0.8*(X+Y)

therefore X / Y = 3 / 2

• Nischal says:

For every 2 boy students there are 3 girl students and for every one teacher there are 10 students whereas for every 4 male teachers there are 5 female teachers. Which of the following is the ratio of number of boy students to the number of male teachers?

• Sandosh Chemban says:

For 1 teacher, there are 10 students which contains 4 boys, and 6 girls(because ratio of boys to girls is 2:3)

Therefore ratio of boys to teacher is 4:1

Now for 9 (4 male+5 female) teachers, there are total 90 students which has 36(90*0.4) boys

This means for a group of 36 boys, there are 4 male teachers.

Hence the ratio of boys to male teacher is 36:4 or 9: 1

Ratio is 1:6:8; amount to be shared is 52,500.00

9. emman ces says:

Three similar lamps use 4 liters of oil in 80 hours. How much oil will 6 lamps of the same kind use in 40 hours?

• John says:

Worker*Rate*Time=Object

Worker(lamp)=3
Time=80 hours
Object = 4 liters

3R80=4. R=4/240 or R=1/60

Worker(lamp)=6
Time=40 hours
Object= x

6R40=X (Rate is 1/60 from previous). 6(1/60)40=x (240/60)=x and x=4

Object or Liters is 4 liters.

10. Bariisoo Abraahim says:

THE RATIO OF MALE STUDENTS TO FEMALE STUDENTS IS 5:3. IF THE NUMBER OF MALE STUDENTS TWENTY FOUR MORE THAN TWO-THIRD OF THE FEMALE, HOW MANY FEMALE AND MALE STUDENTS ARE THEIR IN A CERTAIN CLASS?

11. Lekha says:

What is the ratio of 99,729

12. abhishek says:

if total population is 40,000 and 40% of adult are illiterate and 85% children are literate . if ratio of adult and childre is 2:3 then how people are literate and how many number of people are illitate in that area?

13. Paul Lee says:

I want to create a concrete that weighs 20kg. using sand, cement & water. So, the ratio of these substances is 1: 3: 3. Therefore, how many kg of the sand is required to carry out the process?

• Sandosh Chemban says:

The given ratio is 1:3:3.

Now let us multiply the ratio by x and add together, which gives 1x+3x+3x=7x

or 7x = 20

or x=2.857

The required qty of sand is 1x=2.857 kg

14. Steven says:

Hello i have this problem …….2.1Determine which is better buy-cheddar cheese that costs R19,95 for 350g, or Gouda cheese that costs R34.32 for a block that weighs 528g
2.2Determine the price per Kg of the Gouda cheese in question 2.1 above
2.3A sports shop is selling Nike running shoes at R2100.At the end of the season ,the manager of the shop decreases the price in the ratio of 9:8 what will the decreased price of these shoes Nike shoes be?
2.4 Anne, A factory worker earns R58/hour.Anne works 8hours a day, 5 days a week , and there are 22 working days a month .Determine the number of hours worked and Annes total salary at the end of the month.
2.5 Anne works 12 hours overtime in a month. According to Annes contract,the pay for overtime is 20% higher. Determine Annes hourly rate for overtime .
2.6 What will Anne earn for just her over time work hours
2.7 A new school is bieng built.it will require a team of 12 builders to work for 62 days to complete the project .If two of the wokers cannot work on this particular project how long will it take for the school to be built with the remaining builders

15. Jakes says:

A certain island contains 550 cats and 675 dogs. What is the ratio of of dogs in this island

16. Uma Katie R. Macapagal says:

Thank you for that resources I’ve learned many things.

17. Synt Nynyu says:

The mathematical problems and the solutions are very helpful.