Question

The numerator of a fraction in 7 less than the denominator. If the denominato is increased by 9 and the numerator by 2, the fraction is not changed. Find the fraction.​

Answer 1

Answer:-

Let the fraction be x/y.

Given:-

Numerator is 7 less than the denominator.

⟹ Numerator = Denominator - 7

⟹ x = y - 7 -- equation (1)

And,

If denominator is increased by 9 and numerator by 2 ; there will be no change in the fraction.

i.e.,

$\sf \: \dfrac{x + 2}{y + 9} = \dfrac{x}{y}$

Substitute x = y - 7 from equation (1).

$\implies \sf \: \dfrac{y - 7 + 2}{y + 9} = \dfrac{y - 7}{y} \\ \\ \\ \implies \sf \: \dfrac{y - 5}{y + 9} = \dfrac{y - 7}{y} \\ \\ \\ \implies \sf \:y(y - 5) = (y + 7)(y - 9) \\ \\ \\ \implies \sf \: {y}^{2} - 5y = y(y - 9) + 7(y - 9) \\ \\ \\ \implies \sf \: {y}^{2} - 5y = {y}^{2} - 9y + 7y - 63 \\ \\ \\ \implies \sf \: - 5y + 9y - 7y = - 63 \\ \\ \\ \implies \sf \: - 3y = - 63 \\ \\ \\ \implies \sf \:y = \frac{ - 63}{ - 3} \\ \\ \\ \implies \boxed{ \sf y = 21}$

Substitute y = 21 in equation (1).

⟹ x = 21 - 7

⟹ x = 14

∴ Required fraction x/y is 14/21.

Answer 2

Information provided with us:

• The numerator of a fraction is 7 less than the denominator.
• Denominator is increased by 9
• Numerator is also increased but by2
• The fraction was not changed

What we have to calculate:

• Find the fraction?

Required Calculations:

• Let us assume that the fraction is a / b. So in the question it is given that the numerator of that fraction is 7 less than the denominator we would subtract 7 from b

$:\longmapsto$ a = b - 7

• Now it is given that numerator is increased by 2 and denominator is increased by 9 so we would add 2 in the numerator and 9 in the denominator.

$:\longmapsto \sf{\dfrac{a + 2}{ b + 9} = \dfrac{a}{b}}$

• Here we would be substituting the value of a which we got in the above step in this equation.

$:\longmapsto$ (b - 7 ) + 2 / b + 9 = (b - 7) / b

$:\longmapsto$ b - 7 + 2 / b + 9 = b - 7 / b

$:\longmapsto$ b - 5 / b + 9 = b - 7 / b

Cross multiplying,

$:\longmapsto$ b (b - 5) = (b + 9) (b - 5)

$:\longmapsto$ b × (b - 5) = b² - 9b + 7b - 63

$:\longmapsto$ b² - 5b = b² - 9b + 7b - 63

$:\longmapsto$ b² - 5b = b² - 2b - 63

$:\longmapsto$ b² - b² = 5b - 2b - 63

Now,

$:\longmapsto$ 5b - 2b - 63 = 0

On subtracting 5b and 2b we would get,

• 3b - 63 = 0

Changing the sides,

$:\longmapsto$ 3b = 63

$:\longmapsto$ b = 63/3

$:\longmapsto$ b = 21

• Now, we got the value of b which is 21. So we would evaluate this value in (a = b - 7) inorder to get the value of a.

$:\longmapsto$ (21) - 7

$:\longmapsto$ a = 21 - 7

$:\longmapsto$ a = 14

• Henceforth, fraction is 14/21!