# GMAT / GRE Mathematics Tutorial Series: Ratio And Proportion

Contents

Ratio and Proportion Questions & Word Problems

The concepts always remain the same and do not differ based on the metric system used.

## Introduction to Ratios

According to the Oxford Dictionary, a ratio is defined as “Ratio is the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.”

Notation: Ratio of two values m and n is written as:

m: n or m/n or m to n

For instance, in a classroom, the ratio of the number of boys to the number of girls is 2:5. Here, 2 and 5 do not represent the exact count of the students but is multiple of them, which means the number of boys can be 2 or 4 or 6…etc. and the number of girls is 5 or 10 or 15… etc. It also means that for every seven students, there are two boys and five girls.

Question: In a pet shop, there are 28 cats and 21 dogs. What is the ratio of cats to dogs? What is the ratio of cats to the total number of pets in the pet shop?

Solution:

Cats: Dogs = 28: 21 = 4:3

Cats: total number of pets = 28: 49 = 4: 7

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

Solution:

Let the number of men be 5x and the number of women be 4x.

Then 5x+4x = 72 → x=8.

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

Question: In a playschool, there are a certain number of socks with alphabets L, M, N and O written on them. The ratio of socks, L:M:N:O is in the ratio 4:7:3:1. If the number of ‘L socks is 50 more than the number of ‘N socks, what is the number of ‘M socks?

Solution:

Let the number of the socks L,M,N,O be 4x, 7x, 3x and 1x respectively

4x = 3x + 50 → x = 50.

So the number of ‘M socks is 7*50 = 350.

Question:

If the ratio of pens to pencils in a box is 5:8 and the number of pens is 30, find the number of pencils.

Solution:

Let the number of pens be 5x and the number of pencils be 8x.

5x = 30 → x = 6.

Therefore, number of pencils in the box = 8*6 = 48.

## Introduction to Proportion

A lot of questions on ratios can be solved by using proportions.

A proportion is a relative comparison between two equivalent ratios. If q : w = t : y,

then q, w, t, y are said to be in proportion

and this is written as

q:w :: t:y or q/w = t/y

q, y are called the extremes and w, t are called the means.

For a proportion q:w = t:y, product of means = product of extremes → w*t = q*y.

Here are a few examples:

Question:

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

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

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

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

Setting up the proportion,

petrol solution / oil solution = (15+x)/30 = 2/1 → x = 45.

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

Question:

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

Solution:

Let the quantity of cocoa powder required be x kgs.

3 kgs of cocoa powder added to 6 kgs of flour constitute a total of 9 kgs of sweet.

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

3/9 = x/60 → x=20.

Therefore, 20 kgs of cocoa powder is required for 60 kgs of sweet.

Also Check  GMAT / GRE Tutorial Series Set Theory

## Problems on Mixtures / Blends

Question:

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

Solution:

Let x ml of potassium be present in water.

Then, 12/100 = x/60 → x = 7.2 ml of potassium.

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

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

Which means, 8/100 = 7.2/y → y = 90 ml of water.

So in order to get 8% potassium solution, we need to add 90-60ml, which is 30 ml of water.

Question:

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

This question has 3 parts.

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

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

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

 Phenyl 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 phenyl is added.

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

Now, to bring the phenyl percentage back to 20%, extra water is added and the amount of phenyl remains the same. Let’s consider the extra amount of water be y litres.

 Phenyl 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 phenyl 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 phenyl has to be 20% of the whole solution.

Question:

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

Solution:

For these types of problems, the first step is to find out by how much each item is above or below the target price/value.

Our target price is Rs. 105. Rice price is Rs. 5 below the target price and pulses price is Rs. 15 above the target price.

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

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

So, 3 kgs of rice present for every 1 kg of pulses. The GMAT exam ratio and proportion questions of rice to pulses is 3:1. Fraction of pulses in the mixture = 1/(3+1) = 1/4 of the total mixture

## Practice Questions in Ratio and Proportion

Problem 1

In a certain graph, 1 cm represents 12 km of the actual distance. If the distance between point A and point B is 96 km, what is their distance on map?

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

Explanation:

From the given statement, 1cm/12 km = x cm/100 km → x = 8 cm

Problem 2

A person types 360 words in 4 minutes. What will be the time taken for typing 900 words?
A. 15
B. 90
C. 10
D. 9