Question

If two times the larger of the two numbers is divided
by the smaller one, we get 3 as quotient and 5 as the
remainder. Also, if ten times the smaller number is
divided by the larger one, we get 4 as quotient and
2 as remainder. Find the numbers.

Answer 1

Let x and y be the two numbers. such that x > y

Now,

According to the Question

• Case 1

Two times the larger number divided by the smaller one we get 3 as Quotient and 5 as Remainder.

This can be represented in the form of linear equation as:

$</p><p>\rightarrow \quad 3y + 5 = 2x \\ \rightarrow \quad 2x - 3y = 5 \quad \mathtt{(1)} </p><p>$

• Case 2

Ten times the smaller number divided by the larger one, we get 4 as Quotient and 2 as Remainder.

This can be represented in the form of linear equation as:

$</p><p>\rightarrow \quad 10y = 4x + 2 \\ \rightarrow \quad 4x - 10y = -2 \\ \rightarrow \quad 2x - 5y = -1 \quad \mathtt{(2)} </p><p>$

Now,

Subtract (2) from (1)

$</p><p>\rightarrow \quad \cancel{2x} - 3y \cancel{- 2x} + 5y = 5 + 1 \\ \rightarrow \quad 2y = 6 \\ \rightarrow \quad y = 3 </p><p>$

We get y = 3, Put value of y in (1)

-- 2x - 3(3) = 5

-- 2x - 9 = 5

-- 2x = 14

-- x = 7

We get x = 7

Therefore, Numbers are 7 and 3.

Answer 2

AnswEr :

let first no.(larger) be n and second no.(smaller) be m.

Dividend = Divisor × Quotient + Remainder

⋆ If two times the larger of the two numbers is divided by the smaller one, we get 3 as quotient and 5 as the remainder.

⇒ 2 times Larger No. = Smaller No. × 3 + 5

⇒ 2 × n = m × 3 + 5

⇒ 2n = 3m + 5

⇒ 2n - 3m = 5⠀ ⠀⠀⠀⠀— eq.( I )

⋆ If ten times the smaller number is divided by the larger one, we get 4 as quotient and 2 as remainder.

⇒ 10 times Smaller No. = Larger No. × 4 + 2

⇒ 10 × m = n × 4 + 2

⇒ 10m = 4n + 2

⇒ 10m = 2(2n + 1)

• Dividing Both term by 2

⇒ 5m = 2n + 1

⇒ 2n - 5m = - 1⠀ ⠀⠀⠀— eq.( II )

_________________________________

• Subtracting eq.( I ) from eq.( II ) :

⇝ 2n - 5m = - 1

⇝ 2n - 3m = 5⠀⠀

⠀⠀-⠀⠀+⠀⠀⠀-

_________________

⇝ - 5m + 3m = - 1 - 5

⇝ - 2m = - 6

• Dividing Both term by - 2

⇝ m = 3

• Putting the value of m in ( I ) :

⇝ 2n - 3m = 5

⇝ 2n - 3 × 3 = 5

⇝ 2n - 9 = 5

⇝ 2n = 5 + 9

⇝ 2n = 14

• Dividing Both term by 2

⇝ n = 7

∴ So, Larger No. is 7 and Smaller No. is 3.