Working with Fractions in Ratio Problems

Converting Ratios to Fractions

In mathematics, a ratio is a way of comparing two or more quantities. However, for the purpose of algebraic manipulation, it is often more useful to express these relationships as fractions. If the ratio of two quantities $x : y$ is given as $p : q$, this can be interpreted as a multiplicative relationship where the value of $x$ divided by $y$ is equal to $p$ divided by $q$.

Formally, we write:

$\frac{x}{y} = \frac{p}{q}$

This conversion is the foundation for solving complex ratio problems. Once expressed as a fraction, you can rearrange the equation to find one variable in terms of another, such as $x = \frac{p}{q}y$ or $y = \frac{q}{p}x$.

Combining Multiple Ratios

Many university admissions problems involve three or more variables linked by separate ratios. For example, you might be given the ratio of $A$ to $B$ and the ratio of $B$ to $C$, and then be asked to find the ratio of $A$ to $C$. There are two primary methods for solving these problems: the fractional method and the scaling method.

Worked Example: The Counter Problem

A bag contains $y$ yellow counters, $g$ green counters, and $r$ red counters. We are given the following information:

1. The ratio of red counters to yellow counters ($r : y$) is $2 : 3$.
2. The ratio of green counters to red counters ($g : r$) is $4 : 5$.

Find the ratio of green counters to yellow counters ($g : y$).

Method 1: Using Fractional Equations

First, convert the given ratios into fractional form:

$\frac{r}{y} = \frac{2}{3}$ (Equation i)

$\frac{g}{r} = \frac{4}{5}$ (Equation ii)

We want to find the ratio $g : y$, which is equivalent to finding the fraction $\frac{g}{y}$. We can achieve this by multiplying the two fractions together, as the variable $r$ will cancel out:

$\frac{g}{y} = \frac{g}{r} \times \frac{r}{y}$

Substituting our values into this equation:

$\frac{g}{y} = \frac{4}{5} \times \frac{2}{3} = \frac{8}{15}$

Therefore, the ratio $g : y$ is $8 : 15$.

Method 2: Scaling for a Common Variable

To combine ratios directly, we must ensure the common variable ($r$) is represented by the same number of parts in both ratios.

In the first ratio, $r : y = 2 : 3$. In the second ratio, $g : r = 4 : 5$.

The common variable is $r$. In the first ratio, it has 2 parts, and in the second, it has 5 parts. To align them, we find a common multiple of 2 and 5, which is 10.

Scale the first ratio by multiplying both parts by 5: $r : y = 10 : 15$

Scale the second ratio by multiplying both parts by 2: $g : r = 8 : 10$

Now that $r$ is 10 in both, we can combine them into a single three-part ratio for $g : r : y$: $8 : 10 : 15$

By looking at the first and last parts of this sequence, we find the ratio $g : y$ is $8 : 15$. This confirms our result from Method 1.