Question

The numerator of a fraction is 1 less than the denominator.
When both numerator and denominator are increased by 2,
the fraction is increased by . Find the original fraction.

Answer 1

There is missing data in the problem, I have added it to solve the problem. You can use your real data once you understand the explanation.

Answer:

Original fraction:

$\displaystyle \frac{5}{6}, \ \frac{-9}{-8}$

Step-by-step explanation:

The numerator of a fraction is 1 less than the denominator and when both parts are increased by 2, the value of the fraction increases by 1/24. Find the original fraction.

Let's call:

x=original numerator

x+1=original denominator

x+2=increased numerator

x+3=increased denominator

Original fraction:

$\displaystyle \frac{x}{x+1}$

Increased fraction:

$\displaystyle \frac{x+2}{x+3}$

The difference between both is 1/24:

$\displaystyle \frac{x+2}{x+3}-\frac{x}{x+1}=\frac{1}{24}$

Multiply by (x+3)(x+1):

$\displaystyle (x+3)(x+1)\frac{x+2}{x+3}-(x+3)(x+1)\frac{x}{x+1}=\frac{(x+3)(x+1)}{24}$

Simplifying each fraction when possible:

$\displaystyle (x+1)(x+2)-(x+3)(x)=\frac{(x+3)(x+1)}{24}$

Operating:

$\displaystyle x^2+3x+2-x^2-3x=\frac{x^2+4x+3}{24}$

Simplifying:

$\displaystyle 2=\frac{x^2+4x+3}{24}$

Multiplying by 24:

$48=x^2+4x+3$

Rearranging:

$x^2+4x-45=0$

Factoring:

$(x-5)(x+9)=0$

We have two solutions:

$x=5, x=-9$

Selecting the first solution:

Original fraction:

$\displaystyle \frac{5}{6}$

Increased fraction:

$\displaystyle \frac{7}{8}$

The difference between both is:

$\displaystyle \frac{7}{8}-\frac{5}{6}=\frac{21-20}{24}=\frac{1}{24}$

This is a valid solution

Selecting the second solution:

Original fraction:

$\displaystyle \frac{-9}{-8}=\frac{9}{8}$

Increased fraction:

$\displaystyle \frac{-7}{-6}=\frac{7}{6}$

The difference between both is:

$\displaystyle \frac{7}{6}-\frac{9}{8}=\frac{28-27}{24}=\frac{1}{24}$

This solution is only valid if we express the fractions with their negative values. When we simplify them, the first condition is not met. Thus, the solutions are:

Original fraction:

$\displaystyle \frac{5}{6}, \ \frac{-9}{-8}$

Note: The last solution can be arguable because the real simplified fraction is not a solution. My opinion is that it's valid as long as it's expressed with their original signs.