Select the correct alternative from the given choices. If...

Select the correct alternative from the given choices. If three is subtracted from the numerator and five is added to the denominator of a fraction, the new fraction formed is $1 / 2$. If two is added to the numerator of the initial fraction, the ratio of the new numerator to the denominator becomes $1: 1$. Find the original fraction. (A) $11 / 13$ (B) $18 / 23$ (C) $13 / 15$ (D) $13 / 11$

Select the correct alternative from the given choices.
If three is subtracted from the numerator and five is added to the denominator of a fraction, the new fraction formed is $1 / 2$. If two is added to the numerator of the initial fraction, the ratio of the new numerator to the denominator becomes $1: 1$. Find the original fraction.
(A) $11 / 13$
(B) $18 / 23$
(C) $13 / 15$
(D) $13 / 11$

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Key Concepts

Algebraic Equations

Algebraic equations involve using variables to represent unknown quantities and creating equations based on conditions provided in a problem. By setting up equations that relate the modified fraction's numerator and denominator to desired values, the solver can determine the values of the unknowns.

System of Equations

A system of equations consists of multiple equations that share common variables. In this problem, the different conditions lead to two separate equations that must be solved together to find the original fraction. This concept is central in problems where several constraints must hold simultaneously.

Fraction Manipulation

Fraction manipulation is the process of performing operations such as addition or subtraction on the numerator or denominator of a fraction to create a new fraction. Understanding how modifying the fraction's parts affects its overall value is key to forming correct relationships and equations based on given transformations.

Ratios and Proportions

Ratios express a relationship between two quantities, and when such relationships are set equal, we have a proportion. In this problem, one condition presents the resulting fraction in terms of a specific ratio, which is then used to form one of the algebraic equations necessary for solving the original fraction.

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