The numerator of a fraction is 3 less than the denominator. If the numerator and the denominator are each increased by 1, the value of the fraction becomes ¾. What is the original fraction?  

Let the original fraction be \( \frac{x - 3}{x} \), where \(x\) represents the denominator. According to the problem, if we increase both the numerator and denominator by 1, the fraction becomes \( \frac{3}{4} \). Therefore, the equation is:

\[
\frac{(x - 3) + 1}{x + 1} = \frac{3}{4}
\]

Simplify the numerator on the left-hand side:

\[
\frac{x - 2}{x + 1} = \frac{3}{4}
\]

Now, we can cross-multiply to solve for \(x\):

\[
4(x - 2) = 3(x + 1)
\]

Expanding both sides:

\[
4x - 8 = 3x + 3
\]

Now, subtract \(3x\) from both sides:

\[
x - 8 = 3
\]

Add 8 to both sides:

\[
x = 11
\]

Now that we know the denominator \(x = 11\), the numerator is:

\[
x - 3 = 11 - 3 = 8
\]

Thus, the original fraction is:

\[
\frac{8}{11}
\]

Would you like more details on this solution or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would the solution change if the fraction became \( \frac{2}{3} \) instead of \( \frac{3}{

The numerator of a fraction is 3 less than the denominator. If the numerator and the denominator are each increased by 1, the value of the fraction becomes ¾. What is the original fraction?

Discover how to find the original fraction where the numerator is 3 less than the denominator and increasing both by 1 results in the fraction ¾. This algebraic problem covers key concepts such as equation solving and fraction manipulation, ideal for Grades 6-8.

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