Definition, Formulas, and Questions and answers of Ratio and Proportion

Ratio and Proportion are explained majorly based on fractions.A ratio is when a fraction is written in the form a:b, whereas a proportion says that two ratios are equal. In this case, a and b can be any two numbers. The two key notions of ratio and proportion provide the foundation for understanding numerous concepts in mathematics and science.

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The Definition of a Ratio

In some cases, comparing two amounts using the division approach is quite efficient. A ratio can be defined as the comparison or simplified form of two quantities of the same sort. This relationship tells us how many times one quantity equals another. To put it another way, a ratio is a number that may be used to indicate one quantity as a percentage of another.

Important Points to Keep in Mind:

Between amounts of the same sort, there should be a ratio

.

When comparing two objects, the units must be comparable.

There should be a logical sequence to the terms.

If the ratios are equal, such as fractions, a comparison of two ratios can be made.

Proportion Definition

Proportion is a mathematical expression that states that two ratios are comparable. In other terms, the proportion declares that the two fractions or ratios are equal. In proportion, two sets of provided numbers are said to be directly proportional to one other if they increase or decrease in the same ratio.

For example, the time it takes a train to go 100 kilometres per hour is the same as the time it takes to travel 500 kilometres in 5 hours. For example, 100 km/hr = 500 km/5 hours.

 

The proportion can be classified into the following categories, such as:

• Direct Proportion

• Inverse Proportion

• Continued Proportion

Difference Between Ratio and Proportion

1.The ratio is used to compare the sizes of two objects that have the same unit of measurement.

The percentage is a mathematical expression that expresses the relationship between two           ratios.

2. Ratio is written with a colon (:) and a slash (/).

   Proportion is written with a double colon (::) or the equal sign (=).

3. Ratio is a phrase.

    Proportion is a formula.

4. "To every" is a keyword for identifying ratio in an issue.

     "Out of" is a keyword for identifying proportion in an issue.

Ratio and Proportion Tricks

Let's look at some guidelines and strategies for solving problems involving ratios and proportions.

If uy = vx and u/v = x/y, then uy = vx.

If u/v equals x/y, then u/x equals v/y.

If u/v = x/y, v/u must be y/x.

If u/v equals x/y, (u+v)/v equals (x+y)/y.

(u-v)/v = (x-y)/y if u/v = x/y.

The componendo -Dividendo Rule states that if u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y).

If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c 0 are true, then a =b = c is true.

Questions and answers:

 

Question 1:

Find the number which, when subtracted from the terms of the ratio 19:23 makes it equal to the ratio of 3:4 ?

 

Answer:

3: 7

Explanation: 

The two numbers are in the ratio 19:23

the new ratio = 3:4

let the number required to be added be x

then according to the question,

19-x/23-x= 3/4

by cross multiplying

4(19-x)= 3(23-x)

76-4x= 69-3x

x= 7

Question 2: 

Which least number must be subtracted from each of the term 14, 17, 34, 42 to make them proportional? 3

Answer: 

2 :2

Explanation:

Let the number which should be subtracted from 14, 17, 34, 42 such that the remainders are proportional is x

Now as given

(14-x): (17-x):: (34-x): (42-x) are proportional

Therefore

product of means= product of extremes

Hence, 2 should be subtracted from each number

Question 3:

Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Sumit's salary? 

Answer: 

4: 38000

Explanation: 

Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively

then (2x+4000/3x+4000)= 40/57

⇒ 57(2x + 4000) = 40(3x + 4000)

⇒ 6x = 68,000

⇒ 3x = 34,000

Sumit's present salary

= (3x + 4000)

= Rs. (34000 + 4000)

= Rs. 38,000

Question 4:

Seats for Mathematics, Physcis and Biology in a school are in the ratio 5:7:8. There is a proposal to increase these seats by 40%,50% and 75% respectively. What will be the ratio of increased seats?

Answer:

1: 2 : 3 : 4 

 

Explanation:

  
Questions 5:
 

A and B have money in the ratio 2:1. If A gives Rs. 2 to B, the money will be in the ratio of 1:1. What were the initial money they had?

Option 1:  ₹ 12 and ₹ 6
Option 2:  ₹ 16 and ₹ 8
Option 3:  ₹ 8 and ₹ 4 
Option 4:  ₹ 6 and ₹ 3

Answer: 

3: ₹ 8 and ₹ 4

Explanation:

Let A and B have ₹ 2x and ₹ x initially.

∴ 2x–2 = x + 2

⇒ x = 4 

∴ Initial amount with A = ₹ 8 ∴ Initial amount with B = ₹ 4.

Question 6:

The ratio of income of A, B and C is 3 : 7 : 4 and the ratio of their expenditure is 4 : 3 : 5.If A saves Rs. 300 out of Rs. 2400 then find savings of C.

Option 1: 675 

Option 2: 775 

Option 3: 575 

Option 4: 875

Answer:
3: 575
 

Explanation:

The ratio of income of a,b, c is 3:7:4 respectively. Let the income be 3x, 7x, 4x respectively.

The ratio of expenses of a,b, c is 4:3:5 respectively. Let the expenditure be 4y, 3y, 5y respectively. It is given that the income of a is 2400.

 

Question 7:

The incomes of a and b are in the ratio 3:2 and their expenditures are in the ratio 5:3. If each save Rs. 1000 then b's income is :

Option 1: 4000 

Option 2: 6000

 Option 3: 8000 

Option 4: 12000

Answer: 

1: 4000

Explanation:

Let income of a be 3x and Be 2x

and according to the question their investment be 5y and 3y respectively.

ATQ........

3x - 5y = 1000 ......... (1) 2x - 3y =1000.......... (2)

x = [1000+3y]/ 2 (from eq 2) putting the value of x in eq1 wage:-

3* [1000+3y]/2 - 5y = 1000 =>y=1000

x=(1000+3000)/2 = 2000

incomeofb=2x=4000

 

 

Question 8:

The income of A and B are in the ratio of 9 : 4 and their expenditure in the ratio 7 : 3. If each saves Rs. 2000, find income of A.

Option 1: 36000 

Option 2: 32000

Option 3: 72000

Option 4: 76000

Answer: 

3: 72000

 

Explanation:

Let,

Income of A=9x

Income of B=4x

Expenses of A = 7x

Expenses of B = 3x

A & B both saves ₹2000

then we find the value of x through this equation. (9x-2000)/(4x-2000) = 7/3

27x-6000 = 28x-14000

x=8000

Now put the values and find INCOME Income of A = 9x = 9*8000 = ₹72000 Income of B = 4x = 4*8000 = ₹32000

 

Question 9:

A cat leaps 5 leaps for every 4 leaps of a dog, but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speed of the cat to that of the dog?

Option 1: 11:15 

Option 2: 15:16

Option 3: 11:16

 Option 4: 15:11

 

Answer: 

3: 11:16

 

Explanation:

 

Question 10:

Ratio of water and milk in a container is 2 : 3. If 40 liter mixture removed from the container and same quantity of milk is added to it then the ratio of water to milk becomes 1 : 4. Find the initial quantity of mixture ?

Option 1: 75 

Option 2: 80 

Option 3: 85 

Option 4: 90

Answer: 

2: 80

 

Explanation:

Let the initial quantity of the mixture = 5x

Then, =>2x−16/3x−24+40=1/4 =>8x−64=3x+16x =>5x=80

=> 16 lit

Then the initial quantity of the mixture = 5x = 5 x 16 = 80 lit.

 

Question 11:

The ratio of successful and unsuccessful examinees in an examination in a school is 6 : 1. The ratio would have been 9 : 1 if 6 more examinees had been successful. The total number of examinees is

Option 1: 110

Option 2: 120 

Option 3: 140

Option 4: 160

Answer: 

3: 140

 

Explanation:

Total students = 6x+x=7x

6x + x=9 x−6 1

⇒ 6x+6=9x–54

⇒ 9x–6x=54+6=60

⇒ 3x=60⇒x=20

∴ Total number of students =7×20=140

 

Question 12:

The ratio of number of balls in bags x,y is 2 : 3. Five balls are taken from bag y and are dropped in bag x. Number of balls are equal in each bag now. Number of balls in each bag now is

Option 1: 45 

Option 2: 20 

Option 3: 30

Option 4: 25

Answer: 

2 : 20 

Explanation:

Number of balls in bag x and y respectively = 2a and 3a 

∴ 3a–5=2a+3

⇒ a=5+3=8

∴ Total number of balls

=5a=40

∴ Balls in each bag = 20

 

Question 13:

The sum of two numbers is 28 and their quotient is 3. What are the two numbers?

Option 1: 6,22 

Option 2: 19,9 

Option 3: 7,21 

Option 4: 20,8

Answer: 

3: 7,21

 

Explanation:

x+y=28 

x/y=3,

x=3y 

Substitute: 3y+y=28 4y=28

y=7

 x=21

 

Question 14

The same type of work is assigned to three groups of men. The ratio of person in the groups is 3:4:5.Find Ratio Days Which they will complete the work.

Option 1: 20 : 15 : 12 

Option 2: 15 : 12 : 18

Option 3: 20 : 12 : 15

Option 4: 15 : 25 : 18

1: 20:15:12

Explanation:

The same type of work is assigned to three groups of men the ratio person in the groups is 3 ratio 4 ratio 5. As we know the number of days is inversely proportional to the number of person.

Thus, the number of days is:

1/3: 1/4: 1/5

find the LCM of 3,4 and 5. 

The LCM of 3,4and5 is60. 

60/3: 60/4: 60/5

20:15:12
 

Question 15:

What must be added to two numbers that are in the ratio of 21: 17 so that they become in the ratio of 6 : 5 ?

Option 1: 8 

Option 2: 6 

Option 3: 4

Option 4: 16

Answer: 

1: 8

Explanation:

The two numbers are in the ratio 21:17 the new ratio = 6:5

let the number required to be added be x then according to the question, 21+x/17+x =6/5

by cross multiplying 5(21+x)= 6(17+x) 

110+5x= 102+6x

 x=8

 

Question 16:

A mixture contains milk and water in a ratio of 6:5. When 33 litres of the mixture is replaced by the same quantity of water, the ratio becomes 3:4. What is the initial quantity of the water?

Option 1: 40

Option 2: 50

Option 3: 60

Option 4: 70

Answer: 

4: 70

 

Explanation:

let the quantity of mixture at the start = 11X+33 litre Quantity of mixture 33 litre are removed = 11 X Quantity of milk = 6X

Quantity of water = 5X

So when 33 litre of water is added, the equation will be 6X:5X+33::3:4

X= 11

Quantity of mixture at the start

=11×11+33

= 154 litre

Initial quantity of water = 154 × 5/11 = 70 litre

 

Question 17:

A vessel contains liquid A and B in the ratio 5:3. If 16 litres of the mixture are removed and the same quantity of liquid B is added, the ratio becomes 3:5. What quantity does the vessel hold?

Option 1: 20 

Option 2: 40 

Option 3: 50 

Option 4: 60

Answer: 

2: 40

 

Explanation:

Let A is 5x,and B is 3x.

In removal of 16l of mixture,

A removed=(5/8)×16=10l,

B removed=(3/8)×16=6l.

Now,as per question,16l of B is added.

 Hence,new content of B=3x-6+16=3x+10.

newA:new B=3:5 (5x-10):(3x+10)=3:5

Solving,we get X=5

Hence, volume of vessel=8x=40l

 

Question 18:

A container contains a mixture of two liquids A and in the ratio of 5:3,if16 of mixture is replaced by liquid B then the ratio of two liquids becomes 3 : 5. How much of the liquid B was there in the bucket?

Option 1: 4 

Option 2: 6

Option 3: 8

Option 4: 10

Answer: 

2: 6

Explanation:

Let bucket contains 5x and 3x liquids A and B respectively.

When 16 litres of mixture are drawn off, quantity of A in mixture left:

 

 

Question 19:

A mixture contains milk and water in the ratio 4 : 3. On adding 6 liters of water the ratio becomes 8 : 7. Find the total quantity of the final mixture.

Option 1: 80 

Option 2: 90

Option 3: 70

Option 4: 65

Answer: 

2: 90

 

Explanation:

A mixture contains milk and water in the ratio 4 : 3 Amount of milk = 4x

Amount of water = 3x

On adding 6 litres of water the ratio becomes 8 : 7 Aer adding water:-

Amount of milk, m = 4x

Amount of water, w = 3x+6

The ratio becomes 8 : 7

Amount Of Water=3x+6=3*12+6=42 Amount Of Milk=4x=4*12=48

Total quantity of the final mixture is 90 litres

 

Question 20:

₹ 10,000/- has to be distributed among 3 craftsmen, 5 helpers and 6 labourers such that each helper receives the amount twice as much as a labourer receives and each craftsman receives the amount thrice as much as a labourer receives. What is the amount received by the three craftsmen?

Option 1: ₹4000 

Option 2: ₹2400 

Option 3: ₹2700

Option 4: ₹3600

Answer: 

4: ₹3600

 

Explanation:

Let the labour receives = Rs x

So, helper receives = Rs 2x

And craftsman receives = Rs 3x Ratio of Labour:Helper:3Crasmen=6:10:9 Amount received by 3 craftsman = 10000×9/25=Rs 3600

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ratios and proportions are foundational to student understanding across multiple topics in mathematics and science. In mathematics, they are central to developing concepts and skills related to slope, constant rate of change, and similar figures, which are all fundamental to algebraic concepts and skills.