Ratio and Proportion is a very important topic with application in data interpretation, allegation and mixture. In this section you will learn about
- Ratio and its types
- Proportion and its relation with ratio
- Operations that will help in solving problems on Ratio and Proportion
- Solved question and answers to improve your skills
The first lesson in this topic is an introduction to ratio and proportion. We shared some formulae that will be required to tackle problems onratio and proportion. The exercises contain solved question and answers. When solving exercises, we recommend you to try solving the problems on your own, that way you will understand better and you can define your own approach to solving problems on ratio and proportion.
Introduction to Ratio and Proportion
Ratio :
Ratio is the comparison of two quantities of same kind. If A and B are two quantities then,
- The ratio of A to B is represented by A : B and its value is $\frac{A}{B}$
- The first term A is called the Antecedent and the second term B is called the consequent
- A : B has no unit
- If we multiply or divide the terms of the ratio by the same non-zero number, the resulting value will still be the same.
Proportion :
It is a statement that equates two ratios.
Example : $\frac12 =\frac24$
- A proportion is represented using the symbol “::”.
i.e A : B = C : D is represented as A : B :: C : D.
In the above statement,
- A and D are called Extremes, B and C are called Means.
- D is the fourth proportional to A, B and C
Note:
Product of the means = Product if the extremes.
=> A x D = B x C
- If $\frac AB = \frac BC$
Here, B is called the mean proportional and C is called the third proportional
- If $\frac AB =\frac CD$, then $\frac AB =\frac CD = \frac{A+C}{B+D}$
- Invertendo:
If $\frac AB = \frac CD$, then $\frac BA = D/C$
- Altrendo:
If $\frac AB = \frac CD$, then $\frac AC = \frac BD$
- Componendo:
If $\frac AB = \frac CD$, then $\frac{A+B}{B} = \frac{C+D}{D}$
- Dividendo:
If $\frac{A}{B} = \frac{C}{D}$, then $\frac{A-B}{B} = \frac{C-D}{D}$
- Componendo and Dividendo:
If $\frac AB = \frac CD$, then $\frac{A+B}{A-B} = \frac{C+D}{C-D}$
- Types of Ratio
- Duplicate Ratio:
Duplicate ratio of two numbers A and B is the ratio of their squares, i.e $\frac{A^2}{B^2}$
- Triplicate Ratio
Triplicate ratio of two numbers A and B is the ratio of their cubes, i.e $\frac{A^3}{B^3}$
- Sub-duplicate Ratio
Subduplicate ratio of two numbers A and B is the ratio of their square roots, i.e $\frac{\sqrt{a}}{\sqrt{b}}$
- Sub-triplicate Ratio
Sub-triplicate ratio of two numbers A and B is the ratio of their cube roots, i.e $\frac{\sqrt[3]{A}}{\sqrt[3]{B}}$
- Inverse or Reciprocal Ratio
Inverse ratio of A and B is $\frac{1}{A} : \frac{1}{B}$
Problems
Ratio and Proportion - Exercise - Blog
19 questions
- Income of A and B is in ratio 5:4 and their expenditure is in ratio 2:3. If the savings of A and B is Rs. 2000 & Rs. 900 respectively. Find the income of A.
- Rs. 3000
- Rs. 5000
- Rs. 3500
- Rs. 4000
Show the answer
Answer: Rs. 3000
Let income of A and B be 5$x$ and 4$x$ respectively and expenditure be 2$y$ and 3$y$ respectively.
Income - Expenditure = Savings
Given, savings of A is 2000 and savings of B is 900
5$x$ - 2$y$ = 2000
4$x$ - 3$y$ = 900
Solving for $x$ and $y$, we get $x$ = 600 and $y$ = 500
Income of A = 5$x$ = 5 x 600 = Rs. 3000
- Ratio of 2 numbers is 2:3. If the antecedent is increased by 6 and the consequent is decreased by 5 the ratio become 3:1. Find the antecedent.
- 15
- 12
- 20
- 18
Show the answer
Answer: 12
Let the 2 numbers be 2$x$ and 3$x$.
Given, the numerator is increased by 6 and denominator is decreased by 5
$\frac{2x + 6}{3x - 5} = \frac{3}{1}$
2$x$ + 6 = 9$x$ - 15
7$x$ = 21 => $x$ = 3
Antecedent = 2$x$ + 6 => 12
- If 3$x$+7$y$ : 5$x$+11$y$ = 8:13. What is $x$:$y$?
- 2 : 3
- 3 : 2
- 1 : 3
- 3 : 1
Show the answer
Answer: 3 : 1
$\frac{3x + 7y}{5x + 11y} = \frac{8}{13}$
39$x$ + 91$y$ = 40$x$ + 88$y$
$x$ = 3$y$
$x : y$ = 3 : 1
- If $x:y$ = 4:5. Find the value of ${(4x+5y)}/{(7x+3y)}$
- $\frac{16}{25}$
- $\frac{9}{16}$
- $\frac{21}{27}$
- $\frac{41}{43}$
Show the answer
Answer: $\frac{41}{43}$
Dividing the numerator and denominator of $\frac{4x + 5y}{7x + 3y}$ by $y$
=> $\frac{4(\frac xy) + 5}{7(\frac xy) + 3}$
=> $\frac{4(\frac45) + 5}{7(\frac45) + 3}$
=> $\frac{16 + 25}{28 + 15}$
=> $\frac{41}{43}$
- The price of Apple and Mango is in the ratio 5:4. If the price of apple decreases by 20% while that of Mango increases by 25%. What is the new ratio of the price of Apple and Mango.
- 3 : 4
- 4 : 5
- 4 : 3
- 5 : 4
Show the answer
Answer: 4 : 5
Let the price of Apple and Mango be 5$x$ and 4$x$ respectively.
Price of Apple after 20% decrease = 5$x$ (1-20/100) = 4$x$
Price of Mango after 25% increase = 4$x$ (1 + 25/100) = 5$x$
The new ratio of the price of Apple and Mango = 4$x$ : 5$x$ = 4:5
- An amount of Rs. 1000 is distributed between A, B and C such that A gets Rs. 100 more than B and B gets 4 times the amount C gets. Find the amount received by A.
- Rs. 700
- Rs. 400
- Rs. 900
- Rs. 500
Show the answer
Answer: Rs. 500
Let amount received by C be $x$. Then,
B gets 4$x$ and A gets 4$x$ + 100
Now,
4$x$ + 100 + 4$x$ + $x$ = 1000
9$x$ = 900
$x$ = 100
Therefore A receives 4x100 + 100 = Rs. 500
- If the marks obtained by A, B and C are in the ratio 3 : 5 : 2. If B's score is 120. Find the total score by A, B and C.
- 300
- 240
- 320
- 250
Show the answer
Answer: 240
Let the total score be $x$.
Given,
B's score = 120. Then,
$\frac{5}{3+5+2}$ x $x$ = 120
$x$ = 240.
- If one fourth of $x$ is equal to three fifths of $y$. Find ${(2x + 5y)}$/${(3x + 4y)}$
- $\frac{9}{13}$
- $\frac{9}{11}$
- $\frac{7}{13}$
- $\frac{7}{9}$
Show the answer
Answer: $\frac{9}{13}$
1/4 x $x$ = 3/5 x $y$
$\frac{x}{y} = \frac{12}{4}$
$\frac{x}{y} = \frac{3}{1}$
$\frac{2x + 3y}{3x + 4y} =\frac{2(\frac{x}{y}) + 3}{3(\frac xy) + 4}$
= $\frac{2x3 + 3}{3x3 + 4}$
= $\frac{9}{13}$
Therefore, $\frac{2x + 5y}{3x + 4y} =\frac{9}{13}$
- In 30 lts of mixture of milk and water, the ratio of milk and water is 5:1. If 20 lts of water is added to this mixture. What will be the new ratio of milk and water.
- 2 : 3
- 4 : 5
- 3 : 5
- 1 : 1
Show the answer
Answer: 1 : 1
Amount of milk in 30 lts of mixture = $\frac{5}{6}$ x 30 = 25 lts.
Hence remaining amount in the mixture is water = 5 lts.
If 20 lts is added to the mixture, the amount of water will be = 5 + 20 = 25 lts.
The ratio of milk and water after adding 20lts of water = 25 : (20+5) = 1 : 1
- An amount of Rs. 3250 is distributed between A, B and C in the ratio 1/2 : 1/3 : 1/4. What is B's share?
- Rs. 1750
- Rs. 1000
- Rs. 1500
- Rs. 1250
Show the answer
Answer: Rs. 1000
$\frac12 : \frac13 : \frac14$ = 6 : 4 : 3 (By taking the lcm of denominators)
B's share = $\frac{4}{6 + 4 + 3}$ x 3250 = 4 x 250 = Rs. 1000.
- What is the third proportional to 1 and 3?
- 60%
- 9
- 12
- 15
Show the answer
Answer: 9
Let the third proportional be $x$.
Then, 1:3 = 3:$x$
1/3 = 3/$x$ => $x$ = 9
- If a:b :: c:d, given that a=4 and d=6 what is the product of b and c?
- 10% profit
- 14
- 16
- 24
Show the answer
Answer: 24
In a proportional, product of means = product of extremes.
Therefor b x c = 4 x 6
b x c = 24
- Find the fourth proportional to 2, 3, 4.
- 6
- 8
- 10
- 12
Show the answer
Answer: 6
Let d be the 4th proportional. Then
$\frac{2}{3} = \frac{4}{d}$
d = $\frac{4 x 3}{2}$ = 6
- If A:B = 2:3, B:C = 6:7 and C:D = 9:11. Find A:B:C:D
- 36 : 55 : 63 : 77
- 36 : 54 : 63 : 77
- 33 : 56 : 63 : 77
- 32 : 54 : 66 : 77
Show the answer
Answer: 36 : 54 : 63 : 77
$\frac AB = \frac23, \frac BC = \frac67$ and $\frac CD = \frac911$
First, lets make the consequent in $\frac BC$ same as the antecedent in $\frac CD$
$\frac CD = \frac{\text{9x7}}{\text{11x7}} =\frac{63}{77}$
$\frac BC = \frac{\text{6x9}}{\text{7x9}} = \frac{54}{63}$
B : C : D = 54 : 63 : 77
Next let the consequent in A/B be same as the value of B in the above ratio
$\frac AB = \frac{\text{2 x 18}}{\text{3 x 18}} = \frac{36}{54}$
Now,
A : B : C : D = 36 : 54 : 63 : 77
- If A/2 = B/3 = C/4. Find A:B:C.
- 1 : 2 : 3
- 2 : 3 : 4
- 3 : 4 : 5
- 4 : 5 : 6
Show the answer
Answer: 2 : 3 : 4
$\frac A2 = \frac B3$ => A : B = 2 : 3
$\frac B3 = \frac C4$ => B : C = 3 : 4
The consequent in A : B is same as the antecedent in B : C
A : B : C = 2 : 3 : 4
- A bag contains Rs. 10, Rs. 5 and Rs. 1 coins in the ratio 1:2:5. All the coins together amount to Rs. 350. Find the number of Rs. 1 coins in the bag.
- 70
- 90
- 140
- 180
Show the answer
Answer: 70
Let the number of Rs 10, Rs 5 and Rs 1 coins be 1$x$, 2$x$ and 5$x$ respectively.
Given
10 x 1$x$ + 5 x 2$x$ + 1 x 5$x$ = 350
$25x$ = 350
$x$ = 14
Number of Rs. 1 coins = 5$x$ = 5 x 14 = 70
- If AxD=BxC. Find (A+B)/(C+D) Given B:D=2:3
- $\frac35$
- $\frac25$
- $\frac23$
- $\frac13$
Show the answer
Answer: $\frac23$
A X D = B x C => $\frac AB = \frac CD$
Adding 1 on both sides.
$\frac AB + 1 = \frac CD + 1$
$\frac{A+B}{B} = \frac{C+D}{D}$
$\frac{A+B}{C+D} = \frac{B}{D}$
Given $\frac{B}{D} = \frac23$
$\frac{A+B}{C+D} = \frac23$
- What is the sub-duplicate ratio of 16/25
- 2 : 3
- 4 : 7
- 1 : 3
- 4 : 5
Show the answer
Answer: 4 : 5
Sub duplicate ratio of two numbers A and B is $\sqrt{A}$ and $\sqrt{B}$ => $\sqrt{16}:\sqrt{25}$
= 4 : 5
- Two terms are in ratio 5:4. If the difference in terms is 9. Find the sum of the terms.
- 70
- 81
- 90
- 65
Show the answer
Answer: 81
Let the two terms be A and B. Then
A : B = 5 : 4
Using Componendo and Dividendo
$\frac{A+B}{A-B} = \frac{5+4}{5-4}$
$\frac{A+B}{A-B} = 9$
Given A-B = 9
A+B = 9x9 = 81
Hence the sum of the terms = 81
