The numerator of a fraction is 1 less than the denominator.?

Question

If the numerator and denominator are both increased by 4, the new fraction will be 1/8 more than the original fraction. find the original fraction

Answer 1

Let x be the denominator. It follows that x - 1 is the numerator.

Therefore, the fraction will be

(x - 1) / x

We're given that if we increase the numerator and denominator by 4, i.e.

[(x - 1) + 4] / [x + 4]

This new fraction will be 1/8 more than the original fraction.

[(x - 1) + 4] / [x + 4] = (x - 1)/x + 1/8

Now, we just solve this equation. First, let's simplify.

[x + 3] / [x + 4] = (x - 1)/x + 1/8

Multiply both sides by 8x(x + 4) to get rid of all fractions.

8x(x + 3) = 8(x - 1)(x + 4) + x(x + 4)

Expand everything.

8x^2 + 24x = 8(x^2 + 3x - 4) + x^2 + 4x

8x^2 + 24x = 8x^2 + 24x - 32 + x^2 + 4x

Move everything to the left hand side,

8x^2 + 24x - 8x^2 - 24x + 32 - x^2 - 4x = 0

Now, group like terms.

32 - x^2 - 4x = 0

Rewrite in descending x power.

-x^2 - 4x + 32 = 0

Multiply both sides by -1,

x^2 + 4x - 32 = 0

Now, factor

(x + 8)(x - 4) = 0

Therefore, x = -8 or x = 4

Remember that our original fraction is

(x - 1)/x

That means, if x = -8, our original fraction is

(-8 - 1)/(-8) = -9/(-8) = 9/8

If x = 4, our original fraction is

(4 - 1)/4 = 3/4

So we have *two* original fractions: 9/8 and 3/4

Due to a technicality involving the reduction of the negative sign of fractions, we reject the solution 9/8, and our only fraction is
3/4.

Answer 2

Original fraction:
(x-1)/x

New fraction;
(x-1+4)/(x+4)
Simplifying:
(x+3)/(x+4)

(old fraction) + (1/8) = (new fraction)
(x-1)/x + 1/8 = (x+3)/(x+4)
8(x-1)/8x + x/8x = (x+3)/(x+4)
(8x - 8)/8x + x/8x = (x+3)/(x+4)
(9x - 8)/8x = (x+3)/(x+4)

cross multiply:
(9x - 8)(x+4) = 8x(x+3)
9x^2 - 8x + 36x - 32 = 8x^2 + 24x
9x^2 + 28x - 32 = 8x^2 + 24x
x^2 + 4x - 32 = 0
(x - 4)(x + 8) = 0
x = 4, -8


So... either it's 3/4 or -9/-8. We can eliminate -9/-8 because that simplifies to 9/8 and no longer meets the requirements for our old fraction.

So, the answer is 3/4

Answer 3

(a-1+4)/(a+4) = 1/8 + (a-1)/a
8a(a + 3) = (a+4)a + 8(a-1)(a+4)
a^2 + 4a - 32 = 0
a1= -8
a2 = 4

Answer : 3/4 or (-9)/(-8)