Question

when 3 is added to the denominators and 2 is subtractfrom numerator a fraction becomes 1/4 and when 6 added to numerator and denominators multiplied by 3 & become 2/ 3​

Answer 1

Question :-

when 3 is added to the denominator and 2 is subtract from numerator a fraction becomes 1/4 and when 6 added to numerator and denominators multiplied by 3 & become 2/ 3.

Find the fraction.

Solution :-

• Let the numerator be x.
• Let the denominator be y.

Now applying first condition i.e. When 3 is added to the denominator and 2 is subtracted from the numerator a fraction becomes 1/4.

We get,

$\sf:\longmapsto{ \frac{x - 2}{y + 3} = \frac{1}{4} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{4(x - 2) = 1(y + 3)} \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{4x - 8 = y + 3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{4x - 8 - 3 = y} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{4x - 11 = y \: \: \: \: \: \: \: \: - - (1)}$

And applying second condition i.e. when 6 is added to numerator and the denominator is multiplied by 3, it becomes 2/3.

We get,

$\sf:\longmapsto{ \frac{3(x + 6)}{3y } = \frac{2}{3} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{x + 6 = 2y \: \: \: \: - - - (2)}$

Now put value of equation 1 in equation 3

We get,

$\sf: \longmapsto{x + 6 = 2(4x - 11)} \\ \\ \sf: \longmapsto{x + 6 = 8x - 22} \: \: \: \: \\ \\ \sf: \longmapsto{8x - x = 6 + 22} \: \: \: \: \\ \\ \sf: \longmapsto{7x = 28} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf: \longmapsto{x = \frac{28}{7} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf: \longmapsto \red{x = 4} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Now substituting the value of x in equation 1

We get-

$\sf:\longmapsto{y = 4(4) - 11} \\ \\ \sf:\longmapsto{}y = 16 - 11 \: \: \: \: \\ \\ \bf:\longmapsto \red{y = 5} \: \: \: \: \: \: \: \: \: \: \: \: \:$

This implies

Required fraction : $\rm{ \frac{x}{y} = \frac{4}{5} }$

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Answer 2

Step-by-step explanation:

let denominator be b

numerator be a

3 is added to denominator = b + 3

2 is subtracted from numerator = a - 2

fraction becomes 1/4

(a - 2) / ( b + 3 ) = 1/4

4 (a - 2) = (b + 3)

4a - 8 = b + 3

4a - b = 11==============eq1

6 added to numerator = 6 + a

denominator is multiplied by 3 = 3b

fraction becomes 2/3

(6 + a)/3b = 2/3

3 (6 + a) = 2 × 3b

18 + 3a = 6b

3a - 6b = - 18

3 (a - 2b) = - 3×6

a - 2b = - 6==============eq 2

eq2 can be multiplied with 4

4 (a - 2b = - 6) ======4a - 8b= - 24 =======eq3

eq3 - eq1 ====4a - 8b - 4a + b = - 24 - 11 = 13

- 7b = -35

b = 5

from eq1 4a - (5) = 11

4a = 11 + 5

4a = 16

a = 16/4

a = 4

the fraction is 4/5