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 Page 1


 
  
Exercise 9.1 Page No: 9.6 
1. If x: y = 3: 5, find the ratio 3x + 4y: 8x + 5y
Solution: 
Given x: y = 3: 5 
We can write above equation as 
x/y = 3/5 
5x = 3y 
x = 3y/5 
By substituting the value of x in given equation 3x + 4y: 8x + 5y we get, 
3x + 4y: 8x + 5y = 3 (3y/5) + 4y: 8 (3y/5) + 5y 
= (9y + 20y)/5: (24y + 25y)/5 
= 29y/5: 49y/5 
= 29y: 49y  
= 29: 49 
2. If x: y = 8: 9, find the ratio (7x - 4y): 3x + 2y.
Solution: 
Given x: y = 8: 9 
We can write above equation as  
x/y = 8/9 
9x = 8y 
x = 8y/9 
By substituting the value of x in the given equation (7x - 4y): 3x + 2y we get, 
(7x - 4y): 3x + 2y = 7 (8y/9) – 4y: 3 (8y/9) + 2y 
= (56y – 36y)/9: (24y + 18y)/9 
= 20y/9: 42y/9 
= 20y: 42y 
= 20: 42 
= 10: 21 
3. If two numbers are in the ratio 6: 13 and their L.C.M is 312, find the numbers.
Solution: 
Given two numbers are in the ratio 6: 13 
Page 2


 
  
Exercise 9.1 Page No: 9.6 
1. If x: y = 3: 5, find the ratio 3x + 4y: 8x + 5y
Solution: 
Given x: y = 3: 5 
We can write above equation as 
x/y = 3/5 
5x = 3y 
x = 3y/5 
By substituting the value of x in given equation 3x + 4y: 8x + 5y we get, 
3x + 4y: 8x + 5y = 3 (3y/5) + 4y: 8 (3y/5) + 5y 
= (9y + 20y)/5: (24y + 25y)/5 
= 29y/5: 49y/5 
= 29y: 49y  
= 29: 49 
2. If x: y = 8: 9, find the ratio (7x - 4y): 3x + 2y.
Solution: 
Given x: y = 8: 9 
We can write above equation as  
x/y = 8/9 
9x = 8y 
x = 8y/9 
By substituting the value of x in the given equation (7x - 4y): 3x + 2y we get, 
(7x - 4y): 3x + 2y = 7 (8y/9) – 4y: 3 (8y/9) + 2y 
= (56y – 36y)/9: (24y + 18y)/9 
= 20y/9: 42y/9 
= 20y: 42y 
= 20: 42 
= 10: 21 
3. If two numbers are in the ratio 6: 13 and their L.C.M is 312, find the numbers.
Solution: 
Given two numbers are in the ratio 6: 13 
 
  
Let the required number be 6x and 13x 
The LCM of 6x and 13x is 78x 
= 78x = 312 
x = (312/78) 
x = 4 
Thus the numbers are 6x = 6 (4) = 24 
13x = 13 (4) = 52  
4. Two numbers are in the ratio 3: 5. If 8 is added to each number, the ratio becomes 
2:3. Find the numbers.
Solution: 
Let the required numbers be 3x and 5x 
Given that if 8 is added to each other then ratio becomes 2: 3 
That is 3x + 8: 5x + 8 = 2: 3 
(3x + 8)/ (5x + 8) = 2/3 
3 (3x + 8) = 2 (5x + 8)  
9x + 24 = 10x + 16 
By transposing 
24 – 16 = 10x – 9x 
x = 8 
Thus the numbers are 3x = 3 (8) = 24 
And 5x = 5 (8) = 40 
5. What should be added to each term of the ratio 7: 13 so that the ratio becomes 2: 3
Solution: 
Let the number to be added is x 
Then (7 + x) / (13 + x) = (2/3) 
(7 + x) 3 = 2 (13 + x) 
21 + 3x = 26 + 2x 
3x – 2x = 26 – 21 
x = 5 
Hence the required number is 5 
6. Three numbers are in the ratio 2: 3: 5 and the sum of these numbers is 800. Find the 
numbers
Page 3


 
  
Exercise 9.1 Page No: 9.6 
1. If x: y = 3: 5, find the ratio 3x + 4y: 8x + 5y
Solution: 
Given x: y = 3: 5 
We can write above equation as 
x/y = 3/5 
5x = 3y 
x = 3y/5 
By substituting the value of x in given equation 3x + 4y: 8x + 5y we get, 
3x + 4y: 8x + 5y = 3 (3y/5) + 4y: 8 (3y/5) + 5y 
= (9y + 20y)/5: (24y + 25y)/5 
= 29y/5: 49y/5 
= 29y: 49y  
= 29: 49 
2. If x: y = 8: 9, find the ratio (7x - 4y): 3x + 2y.
Solution: 
Given x: y = 8: 9 
We can write above equation as  
x/y = 8/9 
9x = 8y 
x = 8y/9 
By substituting the value of x in the given equation (7x - 4y): 3x + 2y we get, 
(7x - 4y): 3x + 2y = 7 (8y/9) – 4y: 3 (8y/9) + 2y 
= (56y – 36y)/9: (24y + 18y)/9 
= 20y/9: 42y/9 
= 20y: 42y 
= 20: 42 
= 10: 21 
3. If two numbers are in the ratio 6: 13 and their L.C.M is 312, find the numbers.
Solution: 
Given two numbers are in the ratio 6: 13 
 
  
Let the required number be 6x and 13x 
The LCM of 6x and 13x is 78x 
= 78x = 312 
x = (312/78) 
x = 4 
Thus the numbers are 6x = 6 (4) = 24 
13x = 13 (4) = 52  
4. Two numbers are in the ratio 3: 5. If 8 is added to each number, the ratio becomes 
2:3. Find the numbers.
Solution: 
Let the required numbers be 3x and 5x 
Given that if 8 is added to each other then ratio becomes 2: 3 
That is 3x + 8: 5x + 8 = 2: 3 
(3x + 8)/ (5x + 8) = 2/3 
3 (3x + 8) = 2 (5x + 8)  
9x + 24 = 10x + 16 
By transposing 
24 – 16 = 10x – 9x 
x = 8 
Thus the numbers are 3x = 3 (8) = 24 
And 5x = 5 (8) = 40 
5. What should be added to each term of the ratio 7: 13 so that the ratio becomes 2: 3
Solution: 
Let the number to be added is x 
Then (7 + x) / (13 + x) = (2/3) 
(7 + x) 3 = 2 (13 + x) 
21 + 3x = 26 + 2x 
3x – 2x = 26 – 21 
x = 5 
Hence the required number is 5 
6. Three numbers are in the ratio 2: 3: 5 and the sum of these numbers is 800. Find the 
numbers
 
  
Solution: 
Given that three numbers are in the ratio 2: 3: 5 and sum of them is 800 
Therefore sum of the terms of the ratio = 2 + 3 + 5 = 10 
First number = (2/10) × 800 
= 2 × 80 
= 160 
Second number = (3/10) × 800 
= 3 × 80 
= 240 
Third number = (5/10) × 800 
= 5 × 80 
= 400 
The three numbers are 160, 240 and 400 
7. The ages of two persons are in the ratio 5: 7. Eighteen years ago their ages were in
the ratio 8: 13. Find their present ages.
Solution: 
Let present ages of two persons be 5x and 7x 
Given ages of two persons are in the ratio 5: 7 
And also given that 18 years ago their ages were in the ratio 8: 13 
Therefore (5x – 18)/ (7x – 18) = (8/13) 
13 (5x – 18) = 8 (7x – 18) 
65x – 234 = 56x – 144 
65x – 56x = 234 – 144 
9x = 90 
x = 90/9 
x = 10 
Thus the ages are 5x = 5 (10) = 50 years 
And 7x = 7 (10) = 70 years 
8. Two numbers are in the ratio 7: 11. If 7 is added to each of the numbers, the ratio
becomes 2: 3. Find the numbers.
Solution: 
Let the required numbers be 7x and 11x 
If 7 is added to each of them then  
Page 4


 
  
Exercise 9.1 Page No: 9.6 
1. If x: y = 3: 5, find the ratio 3x + 4y: 8x + 5y
Solution: 
Given x: y = 3: 5 
We can write above equation as 
x/y = 3/5 
5x = 3y 
x = 3y/5 
By substituting the value of x in given equation 3x + 4y: 8x + 5y we get, 
3x + 4y: 8x + 5y = 3 (3y/5) + 4y: 8 (3y/5) + 5y 
= (9y + 20y)/5: (24y + 25y)/5 
= 29y/5: 49y/5 
= 29y: 49y  
= 29: 49 
2. If x: y = 8: 9, find the ratio (7x - 4y): 3x + 2y.
Solution: 
Given x: y = 8: 9 
We can write above equation as  
x/y = 8/9 
9x = 8y 
x = 8y/9 
By substituting the value of x in the given equation (7x - 4y): 3x + 2y we get, 
(7x - 4y): 3x + 2y = 7 (8y/9) – 4y: 3 (8y/9) + 2y 
= (56y – 36y)/9: (24y + 18y)/9 
= 20y/9: 42y/9 
= 20y: 42y 
= 20: 42 
= 10: 21 
3. If two numbers are in the ratio 6: 13 and their L.C.M is 312, find the numbers.
Solution: 
Given two numbers are in the ratio 6: 13 
 
  
Let the required number be 6x and 13x 
The LCM of 6x and 13x is 78x 
= 78x = 312 
x = (312/78) 
x = 4 
Thus the numbers are 6x = 6 (4) = 24 
13x = 13 (4) = 52  
4. Two numbers are in the ratio 3: 5. If 8 is added to each number, the ratio becomes 
2:3. Find the numbers.
Solution: 
Let the required numbers be 3x and 5x 
Given that if 8 is added to each other then ratio becomes 2: 3 
That is 3x + 8: 5x + 8 = 2: 3 
(3x + 8)/ (5x + 8) = 2/3 
3 (3x + 8) = 2 (5x + 8)  
9x + 24 = 10x + 16 
By transposing 
24 – 16 = 10x – 9x 
x = 8 
Thus the numbers are 3x = 3 (8) = 24 
And 5x = 5 (8) = 40 
5. What should be added to each term of the ratio 7: 13 so that the ratio becomes 2: 3
Solution: 
Let the number to be added is x 
Then (7 + x) / (13 + x) = (2/3) 
(7 + x) 3 = 2 (13 + x) 
21 + 3x = 26 + 2x 
3x – 2x = 26 – 21 
x = 5 
Hence the required number is 5 
6. Three numbers are in the ratio 2: 3: 5 and the sum of these numbers is 800. Find the 
numbers
 
  
Solution: 
Given that three numbers are in the ratio 2: 3: 5 and sum of them is 800 
Therefore sum of the terms of the ratio = 2 + 3 + 5 = 10 
First number = (2/10) × 800 
= 2 × 80 
= 160 
Second number = (3/10) × 800 
= 3 × 80 
= 240 
Third number = (5/10) × 800 
= 5 × 80 
= 400 
The three numbers are 160, 240 and 400 
7. The ages of two persons are in the ratio 5: 7. Eighteen years ago their ages were in
the ratio 8: 13. Find their present ages.
Solution: 
Let present ages of two persons be 5x and 7x 
Given ages of two persons are in the ratio 5: 7 
And also given that 18 years ago their ages were in the ratio 8: 13 
Therefore (5x – 18)/ (7x – 18) = (8/13) 
13 (5x – 18) = 8 (7x – 18) 
65x – 234 = 56x – 144 
65x – 56x = 234 – 144 
9x = 90 
x = 90/9 
x = 10 
Thus the ages are 5x = 5 (10) = 50 years 
And 7x = 7 (10) = 70 years 
8. Two numbers are in the ratio 7: 11. If 7 is added to each of the numbers, the ratio
becomes 2: 3. Find the numbers.
Solution: 
Let the required numbers be 7x and 11x 
If 7 is added to each of them then  
 
  
(7x + 7)/ (11x + 7) = (2/3) 
3 (7x + 7) = 2 (11x + 7) 
21x + 21 = 22x + 14 
22x – 21x = 21 – 14 
x = 21 – 14 = 7 
Thus the numbers are 7x = 7 (7) =49 
And 11x = 11 (7) = 77 
9. Two numbers are in the ratio 2: 7. 11 the sum of the numbers is 810. Find the
numbers.
Solution: 
Given two numbers are in the ratio 2: 7 
And their sum = 810 
Sum of terms in the ratio = 2 + 7 = 9 
First number = (2/9) × 810 
= 2 × 90 
= 180 
Second number = (7/9) × 810 
= 7 × 90 
= 630 
10. Divide Rs 1350 between Ravish and Shikha in the ratio 2: 3. 
Solution: 
Given total amount to be divided = 1350 
Sum of the terms of the ratio = 2 + 3 = 5 
Ravish share of money = (2/5) × 1350 
= 2 × 270 
= Rs. 540 
And Shikha’s share of money = (3/5) × 1350 
= 3 × 270 
= Rs. 810 
11. Divide Rs 2000 among P, Q, R in the ratio 2: 3: 5.
Solution: 
Page 5


 
  
Exercise 9.1 Page No: 9.6 
1. If x: y = 3: 5, find the ratio 3x + 4y: 8x + 5y
Solution: 
Given x: y = 3: 5 
We can write above equation as 
x/y = 3/5 
5x = 3y 
x = 3y/5 
By substituting the value of x in given equation 3x + 4y: 8x + 5y we get, 
3x + 4y: 8x + 5y = 3 (3y/5) + 4y: 8 (3y/5) + 5y 
= (9y + 20y)/5: (24y + 25y)/5 
= 29y/5: 49y/5 
= 29y: 49y  
= 29: 49 
2. If x: y = 8: 9, find the ratio (7x - 4y): 3x + 2y.
Solution: 
Given x: y = 8: 9 
We can write above equation as  
x/y = 8/9 
9x = 8y 
x = 8y/9 
By substituting the value of x in the given equation (7x - 4y): 3x + 2y we get, 
(7x - 4y): 3x + 2y = 7 (8y/9) – 4y: 3 (8y/9) + 2y 
= (56y – 36y)/9: (24y + 18y)/9 
= 20y/9: 42y/9 
= 20y: 42y 
= 20: 42 
= 10: 21 
3. If two numbers are in the ratio 6: 13 and their L.C.M is 312, find the numbers.
Solution: 
Given two numbers are in the ratio 6: 13 
 
  
Let the required number be 6x and 13x 
The LCM of 6x and 13x is 78x 
= 78x = 312 
x = (312/78) 
x = 4 
Thus the numbers are 6x = 6 (4) = 24 
13x = 13 (4) = 52  
4. Two numbers are in the ratio 3: 5. If 8 is added to each number, the ratio becomes 
2:3. Find the numbers.
Solution: 
Let the required numbers be 3x and 5x 
Given that if 8 is added to each other then ratio becomes 2: 3 
That is 3x + 8: 5x + 8 = 2: 3 
(3x + 8)/ (5x + 8) = 2/3 
3 (3x + 8) = 2 (5x + 8)  
9x + 24 = 10x + 16 
By transposing 
24 – 16 = 10x – 9x 
x = 8 
Thus the numbers are 3x = 3 (8) = 24 
And 5x = 5 (8) = 40 
5. What should be added to each term of the ratio 7: 13 so that the ratio becomes 2: 3
Solution: 
Let the number to be added is x 
Then (7 + x) / (13 + x) = (2/3) 
(7 + x) 3 = 2 (13 + x) 
21 + 3x = 26 + 2x 
3x – 2x = 26 – 21 
x = 5 
Hence the required number is 5 
6. Three numbers are in the ratio 2: 3: 5 and the sum of these numbers is 800. Find the 
numbers
 
  
Solution: 
Given that three numbers are in the ratio 2: 3: 5 and sum of them is 800 
Therefore sum of the terms of the ratio = 2 + 3 + 5 = 10 
First number = (2/10) × 800 
= 2 × 80 
= 160 
Second number = (3/10) × 800 
= 3 × 80 
= 240 
Third number = (5/10) × 800 
= 5 × 80 
= 400 
The three numbers are 160, 240 and 400 
7. The ages of two persons are in the ratio 5: 7. Eighteen years ago their ages were in
the ratio 8: 13. Find their present ages.
Solution: 
Let present ages of two persons be 5x and 7x 
Given ages of two persons are in the ratio 5: 7 
And also given that 18 years ago their ages were in the ratio 8: 13 
Therefore (5x – 18)/ (7x – 18) = (8/13) 
13 (5x – 18) = 8 (7x – 18) 
65x – 234 = 56x – 144 
65x – 56x = 234 – 144 
9x = 90 
x = 90/9 
x = 10 
Thus the ages are 5x = 5 (10) = 50 years 
And 7x = 7 (10) = 70 years 
8. Two numbers are in the ratio 7: 11. If 7 is added to each of the numbers, the ratio
becomes 2: 3. Find the numbers.
Solution: 
Let the required numbers be 7x and 11x 
If 7 is added to each of them then  
 
  
(7x + 7)/ (11x + 7) = (2/3) 
3 (7x + 7) = 2 (11x + 7) 
21x + 21 = 22x + 14 
22x – 21x = 21 – 14 
x = 21 – 14 = 7 
Thus the numbers are 7x = 7 (7) =49 
And 11x = 11 (7) = 77 
9. Two numbers are in the ratio 2: 7. 11 the sum of the numbers is 810. Find the
numbers.
Solution: 
Given two numbers are in the ratio 2: 7 
And their sum = 810 
Sum of terms in the ratio = 2 + 7 = 9 
First number = (2/9) × 810 
= 2 × 90 
= 180 
Second number = (7/9) × 810 
= 7 × 90 
= 630 
10. Divide Rs 1350 between Ravish and Shikha in the ratio 2: 3. 
Solution: 
Given total amount to be divided = 1350 
Sum of the terms of the ratio = 2 + 3 = 5 
Ravish share of money = (2/5) × 1350 
= 2 × 270 
= Rs. 540 
And Shikha’s share of money = (3/5) × 1350 
= 3 × 270 
= Rs. 810 
11. Divide Rs 2000 among P, Q, R in the ratio 2: 3: 5.
Solution: 
 
  
Given total amount to be divided = 2000 
Sum of the terms of the ratio = 2 + 3 + 5 = 10 
P’s share of money = (2/10) × 2000 
= 2 × 200 
= Rs. 400 
And Q’s share of money = (3/10) × 2000 
= 3 × 200 
= Rs. 600 
And R’s share of money = (5/10) × 2000 
= 5 × 200 
= Rs. 1000 
12. The boys and the girls in a school are in the ratio 7:4. If total strength of the school
be 550, find the number of boys and girls.
Solution: 
Given that boys and the girls in a school are in the ratio 7:4 
Sum of the terms of the ratio = 7 + 4 = 11 
Total strength = 550 
Boys strength = (7/11) × 550 
= 7 × 50 
= 350 
Girls strength = (4/11) × 550 
= 4 × 50 
= 200 
13. The ratio of monthly income to the savings of a family is 7: 2. If the savings be of
Rs. 500, find the income and expenditure.
Solution: 
Given that the ratio of income and savings is 7: 2 
Let the savings be 2x 
2x = 500 
So, x = 250 
Therefore, 
Income = 7x 
Income = 7 × 250 = 1750 
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FAQs on Ratio & Proportion (Exercise 9.1) RD Sharma Solutions - Mathematics (Maths) Class 7

1. What is the definition of ratio in mathematics?
Ans. In mathematics, a ratio is a comparison between two or more quantities. It represents the relationship between the sizes of different quantities and is expressed by dividing one quantity by another.
2. How do you simplify a ratio?
Ans. To simplify a ratio, you need to divide both the numerator and denominator of the ratio by their greatest common divisor (GCD). This will result in a simplified ratio with the smallest possible whole numbers.
3. What is the concept of proportion?
Ans. Proportion is a mathematical concept that relates two ratios or fractions to each other. It states that the two ratios are equal and can be represented by an equation. In a proportion, the product of the means is equal to the product of the extremes.
4. How do you solve problems involving ratio and proportion?
Ans. To solve problems involving ratio and proportion, you can use the cross-multiplication method. In this method, you set up a proportion equation with two ratios, cross-multiply the terms, and solve for the unknown variable.
5. Can ratios and proportions be used in real-life situations?
Ans. Yes, ratios and proportions are used in various real-life situations. They are commonly used in cooking recipes, financial calculations, construction projects, and even in understanding and analyzing data. Ratios and proportions help in comparing and scaling quantities accurately.
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