Problems on Simultaneous Linear Equations — Multiple Choice Questions

Sum of digits of a two digit number is 8. If the number obtained by reversing the digits is 18 more than the original number, then the original number is

1. 35

2. 53

3. 26

4. 62

Answer

Let the digit at ten's place be x and digit at one's place be y.

Given, sum of digits = 8.

x + y = 8 .......(i)

Number = 10 × x + y = 10x + y

On reversing digits number = 10 × y + x = 10y + x.

Given, number obtained by reversing the digits is 18 more than the original number.

⇒ 10y + x = 10x + y + 18

⇒ 10y - y + x - 10x = 18

⇒ 9y - 9x = 18

⇒ y - x = 2 .........(ii)

Adding (i) and (ii) we get,

⇒ x + y + (y - x) = 8 + 2

⇒ 2y = 10

⇒ y = 5.

Substituting value of y in (i) we get,

⇒ x + 5 = 8

⇒ x = 3.

Number = 10x + y = 10(3) + 5 = 35.

Hence, Option 1 is the correct option.

The sum of two natural numbers is 25 and their difference is 7. The numbers are

1. 17 and 8

2. 16 and 9

3. 18 and 7

4. 15 and 10

Answer

Let the two numbers be x and y.

Given, sum = 25.

x + y = 25 ........(i)

Given, difference = 7.

x - y = 7 .......(ii)

Adding (i) and (ii) we get,

⇒ (x + y) + (x - y) = 25 + 7

⇒ 2x = 32

⇒ x = 16.

Substituting value of x in (i) we get,

⇒ 16 + y = 25

⇒ y = 9.

Hence, Option 2 is the correct option.

The sum of two natural numbers is 240 and their ratio is 3 : 5. Then the greater number is

1. 180

2. 160

3. 150

4. 90

Answer

Let the two numbers be x and y.

Given, sum = 240.

∴ x + y = 240

⇒ x = 240 - y ........(i)

$\therefore \dfrac{x}{y} = \dfrac{3}{5}$ ......(ii)

Substituting value of x from (i) in (ii) we get,

$\Rightarrow \dfrac{240 - y}{y} = \dfrac{3}{5} \\[1em] \Rightarrow 5(240 - y) = 3y \\[1em] \Rightarrow 1200 - 5y = 3y \\[1em] \Rightarrow 8y = 1200 \\[1em] \Rightarrow y = 150.$

Substituting value of y in (i) we get,

⇒ x = 240 - 150 = 90.

Hence, Option 3 is the correct option.

The sum of the digits of a two digit number is 9. If 27 is added to it, the digits of the number get reversed. The number is

1. 27

2. 72

3. 63

4. 36

Answer

Let the digit at ten's place be x and digit at one's place be y.

Given, sum of digits = 9.

⇒ x + y = 9 .......(i)

Number = 10 × x + y = 10x + y

On reversing digits, number = 10 × y + x = 10y + x.

⇒ 10x + y + 27 = 10y + x

⇒ 10x - x + y - 10y + 27 = 0

⇒ 9x - 9y + 27 = 0

⇒ 9(x - y + 3) = 0

⇒ x - y + 3 = 0

⇒ y - x = 3 .......(ii)

Adding (i) and (ii) we get,

⇒ x + y + (y - x) = 9 + 3

⇒ 2y = 12

⇒ y = 6.

Substituting value of y in (i) we get,

⇒ x + 6 = 9

⇒ x = 3.

Number = 10 × x + y = 10 × 3 + 6 = 36.

Hence, Option 4 is the correct option.

The sum of the digits of a two digit number is 12. If the number is decreased by 18, its digits get reversed. The number is

1. 48

2. 84

3. 57

4. 75

Answer

Let the digit at ten's place be x and digit at one's place be y.

Given, sum of digits = 12.

x + y = 12 .......(i)

Number = 10 × x + y = 10x + y

On reversing digits, number = 10 × y + x = 10y + x.

⇒ 10x + y - 18 = 10y + x

⇒ 10x - x + y - 10y - 18 = 0

⇒ 9x - 9y = 18

⇒ 9(x - y) = 18

⇒ x - y = 2 .......(ii)

Adding (i) and (ii) we get,

⇒ x + y + (x - y) = 12 + 2

⇒ 2x = 14

⇒ x = 7.

Substituting value of x in (i) we get,

⇒ 7 + y = 12

⇒ y = 5.

Number = 10 × x + y = 10 × 7 + 5 = 75.

Hence, Option 4 is the correct option.

Aruna has only ₹1 and ₹2 coins with her. If the total number of coins that she has is 50 and the amount of the money with her is ₹75, then the number of ₹1 and ₹2 coins are respectively

1. 35 and 15

2. 35 and 20

3. 15 and 75

4. 25 and 25

Answer

Let ₹1 coins be x and ₹2 coins be y.

According to first condition,

x + y = 50 .......(i)

According to second condition,

1.x + 2.y = 75

x + 2y = 75 ......(ii)

Subtracting (i) from (ii) we get,

⇒ x + 2y - (x + y) = 75 - 50

⇒ 2y - y = 25

⇒ y = 25.

Substituting value of y in (i) we get,

⇒ x + 25 = 50

⇒ x = 25.

Hence, Option 4 is the correct option.

The age of a woman is four times the age of her daughter. Five years hence, the age of the woman will be three times the age of her daughter. The present age of the daughter is

1. 40 years

2. 20 years

3. 15 years

4. 10 years

Answer

Let age of daughter be x years so, the age of woman is 4x years.

After 5 years,

Age of daughter = (x + 5) years

Age of woman = (4x + 5) years.

According to question,

⇒ 4x + 5 = 3(x + 5)

⇒ 4x + 5 = 3x + 15

⇒ 4x - 3x = 15 - 5

⇒ x = 10.

Hence, Option 4 is the correct option.

Father's age is six times his son's age. Four years hence, the age of the father will be four times his son's age. The present age in years of the son and the father are, respectively,

1. 4 and 24

2. 5 and 30

3. 6 and 36

4. 3 and 24

Answer

Let age of son be x years so, the age of father is 6x years.

After 4 years,

Age of son = (x + 4) years

Age of father = (6x + 4) years.

According to question,

⇒ 6x + 4 = 4(x + 4)

⇒ 6x + 4 = 4x + 16

⇒ 2x = 16 - 4

⇒ 2x = 12

⇒ x = 6.

Age of father = 6x = 36.

Hence, Option 3 is the correct option.

Consider the following two statements:

Statement 1: A husband is 2 years older than his wife, and sum of their ages is 52 years. Then the wife is 25 years old.

Statement 2: A father is twice as old as his daughter, and difference of their ages is 26 years. Then the father is 50 years old.

Which of the following is valid?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and Statement 2 is false.

4. Statement 1 is false, and Statement 2 is true.

Answer

Let x years be the husband's age and y years be the wife's age.

Given,

Husband is 2 years older than his wife.

⇒ x = y + 2 ..................(1)

Sum of husband's age and wife's age = 52 years

⇒ x + y = 52 ..................(2)

Substituting the value of x from equation (1) in equation (2), we get

⇒ (y + 2) + y = 52

⇒ 2y + 2 = 52

⇒ 2y = 52 - 2

⇒ 2y = 50

⇒ y = $\dfrac{50}{2}$

⇒ y = 25 years.

∴ Statement 1 is true.

Let the age of daughter be a years.

Given,

Father's age is twice that of daughter's age.

Father's age = 2a years

Difference of father's age and daughter's age = 26 years

⇒ 2a - a = 26

⇒ a = 26

Father's age = 2a = 2 x 26 = 52 years.

∴ Statement 2 is true.

∴ Both the statements are true.

Hence, option 1 is the correct option.