Question

a fraction becomes 8/11 if three is both numerator and denominator if 3 is subtracted from the numerator and denominator it becomes 2/
5 find the fraction​

Answer 1

$\bf{\huge{\underline{\boxed{\rm{\red{Answer:}}}}}}$

Given:-

• a fraction becomes 8/11 if three is both numerator and denominator
• when 3 is subtracted from the numerator and denominator it becomes 2/5

To find :-

• The fraction.

Solution :-

Let the numerator of the fraction be x.

Let the denominator of the fraction be y.

Hence, the fraction = $\bf\large\frac{x}{y}$

As per the first condition,

• a fraction becomes 8/11 if three is both numerator and denominator

$\bf\large\frac{x + 3}{y + 3}$ = $\bf\large\frac{8}{11}$

Cross multiplying,

11 (x + 3) = 8 (y + 3)

11x + 33 = 8y + 24

Transporting the terms,

11x - 8y = 24 - 33

11x - 8y = - 9 ------> 1

As per the second condition,

• if 3 is subtracted from the numerator and denominator it becomes 2/5

Numerator = x

Denominator = y

If 3 is subtracted from both,then,

Numerator = x - 3

Denominator = y - 3

Fraction = $\bf\large\frac{2}{5}$

$\bf\large\frac{x - 3}{y - 3}$ = $\bf\large\frac{2}{5}$

Cross multiplying,

5(x - 3) = 2 (y - 3)

5x - 15 = 2y - 6

Transporting the terms,

5x - 2y = -6 + 15

5x - 2y = 9 ---->2

Multiply equation 2 by 4

20x - 8y = 36 -----> 3

Solve equations 1 and 3 simultaneously by elimination method,

.......20x - 8y = 36 ----> 3

- .....11x - 8y = -9 ----->1

........9x = 45

x = $\bf\large\frac{45}{9}$

x = 5

Substitute the value of x in equation 2

5x - 2y = 9 ---->2

5 × 5 - 2y = 9

25 - 2y = 9

- 2y = 9 - 25

- 2y = - 16

y = $\bf\large\frac{-16}{-2}$

y = 8

$\bf{\large{\underline{\boxed{\rm{\blue{Solution :\: (x, y) = (5,8)}}}}}}$

Fraction formed = $\bf\large\frac{5}{8}$

$\bf{\huge{\underline{\boxed{\rm{\pink{Verification:}}}}}}$

For first case :-

• a fraction becomes 8/11 if three is added to both numerator and denominator

Numerator = 5

Denominator = 8

If three is added,

Numerator = 5 + 3 = 8

Denominator = 8 + 3 = 11

As per the question, the fraction becomes 8/11, hereby, by adding 3 to both the numerator and denominator we satisfy the first condition.

$\bf\large\frac{5+3}{8 + 3}$ = $\bf\large\frac{8}{11}$

$\bf\large\frac{8}{11}$ = $\bf\large\frac{8}{11}$

LHS = RHS.

For second case :-

• if 3 is subtracted from the numerator and denominator it becomes 2/5

Numerator = 5 - 3 = 2

Denominator = 8 - 3 = 5

Fraction = $\bf\large\frac{2}{5}$

We subtracted 3 from both numerator and denominator and hence satisfied the condition of fraction becoming 2/5.

$\bf\large\frac{5-3}{8-3}$ = $\bf\large\frac{2}{5}$

$\bf\large\frac{2}{5}$ = $\bf\large\frac{2}{5}$

LHS = RHS

Hence verified!

Note :- In the first part of the question, a fraction becomes 8/11 if three is both numerator and denominator, the word added is missing. I carried on with the question just with an assumption of this word. So the first condition is a fraction becomes 8/11 if three is " added" both numerator and denominator and we were even able to verify the answer, hence our assumption was right.

Answer 2

Step-by-step explanation:

let the fraction be x/y

As per condition 1

x+3/y+3=8/11

11(x+3)=8(y+3)

11x+33=8y+24

11x-8y+33-24=0

11x-8y+8=0

11x-8y=-9....... eq(1)

As per condition 2

x-3/y-3=2/5

5(x-3)=2(y-3)

5x-15=2y-6

5x-2y-15+6=0

5x-2y-9=0

5x-2y=9........eq(2)

solving eq (1) and (2)by elimination method

11x-8y=-9....... eq(1)×5

5x-2y=9....... eq(2)×11

55x-40y=-45

55x-22y=99

(-)____(+)_(-)__

-18y=-144

y=144/18

y=8

put y=8 in eq (1)

11x-8y=-9

11x-8(8)=-9

11x-64=-9

11x=-9+64

11x=55

x=55/11

x=5

The fraction is 5/8

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