Question

Make eqations

) A fraction becomes 4
5
if 1 is added to both numerator and denominator.
If, however, 5 is subtracted from both numerator and denominator, the fraction becomes 1by2
.

Answer 1

original fraction =
`(7)/(9)`

Explanation:

let x be the numerator and y be the denominator of the original fraction, then

`(x+1)/(y+1)` =
`(4)/(5)` ( cross- multiply )

5(x + 1) = 4(y + 1) ← distribute parenthesis on both sides

5x + 5 = 4y + 4 ( subtract 4y from both sides )

5x - 4y + 5 = 4 ( subtract 5 from both sides )

5x - 4y = - 1 → (1)

Also

`(x-5)/(y-5)` =
`(1)/(2)` ( cross- multiply )

2(x - 5) = y - 5

2x - 10 = y - 5 ( subtract y from both sides )

2x - y - 10 = - 5 ( add 10 to both sides )

2x - y = 5 → (2)

Solve the 2 equations simultaneously for x and y

Multiply (2) by - 4

- 8x + 4y = - 20 → (3)

Add (1) and (3) term by term to eliminate y , that is

- 3x = - 21 ( divide both sides by - 3 )

x = 7

Substitute x = 7 into either of the 2 equations and solve for y

Substituting into (1)

5(7) - 4y = - 1

35 - 4y = - 1 ( subtract 35 from both sides )

- 4y = - 36 ( divide both sides by - 4 )

y = 9

Thus the original fraction =
`(7)/(9)`

Answer 2

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