## RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7

**Other Exercises**

- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.1
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.2
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.3
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.4
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.5
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.6
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.8
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.9
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.10
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.11
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables VSAQS
- RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables MCQS

**Question 1.**

The sum of two numbers is 8. If their sum is four times their difference, find the numbers.

**Solution:**

Let first number = x

and second number = y

x + y = 8 ….(i)

and x + y = 4 (x – y)

=> 4 (x – y) = 8

=> x – y = 2 ….(ii)

Adding (i) and (ii),

2x = 10 => x = 5

Subtracting (ii) from (i),

2y = 6 => y = 3

Numbers are 5, 3

**Question 2.**

The sum of digits of a two digit number is 13. If the number is subtracted from the one obtained by interchanging the digits, the result is 45. What is the number ?

**Solution:**

Let unit’s digit = x

and ten’s digit = y

Number = x + 10y

Now according to the condition

x + y = 13 ….(i)

Number after interchanging their digits,

y + 10x

Now y + 10x – x – 10y = 45

9x – 9y = 45

=> x – y = 5

x – y = 5 ….(ii)

Adding (i) and (ii),

2x = 18 => x = 9

subtracting 8

2y = 8 => y = 4

Number = x + 10y = 9 + 4 x 10 = 9 + 40 = 49

**Question 3.**

A number consists of two digits whose sum is five. When the digits are reversed, the number becomes greater by nine. Find the number.

**Solution:**

Let units digit = x

and ten’s digit = y

Number = x + 10y

and number by reversing their digits = y+ 10x

Now according to the conditions,

x + y = 5 ….(i)

and y + 10x = x + 10y + 9

=> y + 10x – x – 10y = 9

=> 9x – 9y = 9x – y = 1 ….(ii)

(Dividing by 9)

Adding we get:

2x = 6 => x = 3

and subtracting,

2y = 4 => y = 2

Number = x + 10y = 3 + 10 x 2 = 3 + 20 = 23

**Question 4.**

The sum of digits of a two digit number is 15. The number obtained by reversing the order of digits of the given number exceeds the given number by 9. Find the given number. **(C.B.S.E. 2004)**

**Solution:**

Let the ones digit = x

and tens digit = y

Number = x + 10y

and number by reversing the order of digits = y +10x

According to the conditions,

x + y = 15 ….(i)

y + 10x = x + 10y + 9

=> y + 10x – x – 10y = 9

=> 9x – 9y = 9

=>x – y = 1 ……..(ii)

(Dividing by 9)

Adding (i) and (ii)

2x = 16

x = 8

and subtracting, 2y = 14 => y = 7

Number = x + 10y = 8 + 10 x 7 = 8 + 70 = 78

**Question 5.**

The sum of a two-digit number and the number formed by reversing the order of digits is 66. If the two digits differ by 2, find the number. How many such numbers are there ?** [NCERT]**

**Solution:**

Sum of two-digit number and number formed by reversing its digits = 66

Let units digit = x

Then tens digit = x + 2

Number = x + 10 (x + 2) = x + 10x + 20 = 11x + 20

and by reversing its digits

Unit digit = x + 2

and tens digit = x

Number = x + 2 + 10x = 11x + 2

11x + 20 + 11x + 2 = 66

=> 22x + 22 = 66

=> 22x = 66 – 22 = 44

=> x = 2

Number = 11x + 20 = 11 x 2 + 20 = 22 + 20 = 42

and number by reversing its digits will be 11x + 2 = 11 x 2 + 2 = 22 + 2 = 24

Hence numbers are 42 and 24

**Question 6.**

The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers.

**Solution:**

Let first number = x

and second number = y

x + y = 1000 ……..(i)

**Question 7.**

The sum of a two digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number. **(C.B.S.E. 2002)**

**Solution:**

Let the unit’s digit of the number = x

and ten’s digit = y

Number = x + 10y

By reversing the digits, the new number will be = y +10x

According to the condition,

x + 10y + y + 10x = 99

=> 11x + 11y = 99

=> x + y = 9 ….(i)

and x – y = 3 ….(ii)

Adding we get,

2x = 12

x = 6

and subtracting, 2y = 6

y= 3

Number = x + 10y = 6 + 10 x 3 = 6 + 30 = 36

**Question 8.**

A two-digit number is 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number. **(C.B.S.E. 2001C)**

**Solution:**

Let the unit digit of the number = x

and tens digit = y

Number = x + 10y

and number after reversing the order of digits = y + 10x

According to the conditions,

x + 10y = 4 (x + y)

=> x + 10y = 4x + 4y

=> 4x + 4y – x – 10y = 0

=> 3x – 6y = 0

=> x – 2y = 0

=> x = 2y ….(ii)

and x + 10y + 18 = y + 10x

=> x + 10y – y – 10x = -18

=> – 9x + 9y = -18

=> x – y = 2 ….(ii)

(Dividing by – 9)

=> 2y – y = 2 {From (i}

=> y = 2

x = 2y = 2 x 2 = 4

Number = x + 10y = 4 + 10 x 2 = 4 + 20 = 24

**Question 9.**

A two-digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number. **(C.B.S.E. 2001C)**

**Solution:**

Let unit digit of the number = x

and ten’s digit = y

Number = x + 10y

and number after reversing the digits = y + 10x

According to the conditions,

x + 10y = 4 (x + y) + 3

=> x + 10y = 4x + 4y + 3

=> x + 10y – 4x – 4y = 3

=> -3x + 6y = 3

=> x – 2y = -1 ….(i)

(Dividing by -3)

and x + 10y + 18 = y + 10x

=> x + 10y – y – 10x = -18

=> -9x + 9y = -18

=>x – y = 2 ….(ii)

(Dividing by 9)

Subtracting (i) from (ii)

y = 3

x – 3 = 2

=>x = 2 + 3 = 5 {From (ii)}

Number = x + 10y = 5 + 10 x 3 = 5 + 30 = 35

**Question 10.**

A two digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed. Find the number. **(C.B.S.E. 2001C)**

**Solution:**

Let units digit of the number = x

and ten’s digit = y

then number = x + 10y

The number by reversing the digits = y+ 10x

According to the condition given,

x + 10y = 6 (x + y) + 4

=> x + 10y = 6x + 6y + 4

=> x + 10y – 6x – 6y = 4

=> -5x + 4y = 4 ….(i)

and x + 10y – 18 = y + 10x

=> x + 10y – y – 10x = 18

=> -9x + 9y = 18

=> x – y = -2 ….(ii)

(Dividing by 9)

=> x = y – 2

Substituting in (i),

-5 (y – 2) + 4y = 4

-5y + 10 + 4y = 4

-y = 4 – 10 = – 6

y = 6

Number = x + 10y = 4 + 10 x 6 = 4 + 60 = 64

**Question 11.**

A two-digit number is 4 times the sum of its digits and twice the product of the digits. Find-the number. **(C.B.S.E. 2005)**

**Solution:**

Let the units digit of the number = x

and tens digit = y

Number = x + 10y

According to the conditions given,

**Question 12.**

A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number. **(C.B.S.E. 2005)**

**Solution:**

Let the units digit of the number = x

and tens digit = y

Number = x + 10y

and number after interchanging its digits = y + 10x

According to the conditions,

xy = 20

**Question 13.**

The difference between two pumbers is 26 and one number is three times the other. Find them.

**Solution:**

Let first number = x

and second number = y

x – y = 26 ……….(i)

x = 3y ….(ii)

Substituting the value of x in (i)

3y – y = 26

=> 2y = 26

=>y = 13

x = 3y = 3 x 13 = 39

Numbers are 39, 13

**Question 14.**

The sum of the digits o,f a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

**Solution:**

Let the units digit of the number = x

and tens digit number = y

Number = x + 10y

and the number by reversing the order of the digits = y + 10x

According to the condition;

x + y = 9 …..(i)

9 (x + 10y) = 2 (y + 10x)

=> 9x + 90y = 2y + 20x

=> 9x + 90y – 2y – 20x = 0

=> -11x + 88y = 0

=> x – 8y = 0 (Dividing by -11)

=> x = 8y

Substituting the value of x in (i)

8y + y = 9

=> 9y = 9

=> y= 1

x = 8y = 1 x 8 = 8

Number = x + 10y = 8 + 10 x 1 = 8 + 10 = 18

**Question 15.**

Seven times a two-digit number is equal to four times the number obtained by reversing the digits. If the difference between the digits is 3. Find the number.

**Solution:**

Let the units digit of the number = x

and tens digit = y

Number = x + 10y

and number after reversing the digits = y + 10x

According to the conditions,

x – y = 3 ….(i)

and 7 (x + 10y) = 4 (y + 10x)

=> 7x + 70y = 4y + 40x

=> 7x + 70y – 4y – 40x = 0

=> -33x + 66y = 0

=> x – 2y = 0 (Dividing by -33)

=> x = 2y

Substituting the value of x in (i),

2y – y = 3 => y = 3

x = 2y = 2 x 3 = 6

and number = x + 10y = 6 + 10 x 3 = 6 + 30 = 36

**Question 16.**

Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers. **[NCERT Exemplar]**

**Solution:**

Let the two numbers be x and y.

Then, by the first condition, ratio of these two numbers = 5 : 6

x : y = 5 : 6

**Question 17.**

A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.** [NCERT Exemplar]**

**Solution:**

Let the two-digit number = 10x + y

Case I : Multiplying the sum of the digits by 8 and then subtracting 5 = two-digit number

=> 8 x (x + y) – 5 = 10x + y

Hope given RD Sharma Class 10 Solutions Chapter 3 Pair of Linear Equations in Two Variables Ex 3.7 are helpful to complete your math homework.

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