Fractions in Ratio Problems

Q: Why does multiplying the fractions work to combine ratios?

When you multiply $\frac{g}{r}$ by $\frac{r}{y}$, the $r$ in the numerator of the first fraction and the $r$ in the denominator of the second fraction cancel out algebraically, leaving you with $\frac{g}{y}$.

Q: Can I use this method for more than three variables?

Yes. If you have a chain of ratios such as $a:b$, $b:c$, and $c:d$, you can find $a:d$ by multiplying the three fractions: $\frac{a}{b} \times \frac{b}{c} \times \frac{c}{d}$.

Updated July 2026

This section teaches how to convert ratios into fractions to solve complex proportion problems. By understanding the algebraic relationship between $x : y$ and $x / y$ , students can link multiple ratios together. This technique is vital for solving multi part problems involving shared quantities in the ESAT.

Core concept

A ratio $x : y$ in the proportion $p : q$ can be expressed as the equation $\frac{x}{y} = \frac{p}{q}$ . This equivalence allows students to manipulate ratios using the rules of algebraic fractions.

Ratios and Fractions

The relationship between two quantities in a ratio can always be expressed as a fraction. If the ratio of $x : y$ is given as $p : q$ , it follows mathematically that:

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

This principle is the foundation for solving complex ratio problems, especially those where multiple different ratios must be combined to find the relationship between variables that are not directly compared.

Worked Example: Combining Multiple Ratios

Consider a bag containing $y$ yellow counters, $g$ green counters, and $r$ red counters. We are given the following information:

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

What is the ratio of green to yellow counters ( $g : y$ )?

Method 1: The Multiplication Method

One efficient way to solve this is to express each ratio as a fraction and use multiplication to cancel out the common variable. First, we write the known ratios as fractional equations:

$\frac{r}{y} = \frac{2}{3}$

$\frac{g}{r} = \frac{4}{5}$

We want to find the ratio $g : y$ , which is equivalent to the fraction $\frac{g}{y}$ . We can derive this by multiplying the two fractions we already have:

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

Substituting the numerical values into the equation:

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

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

Method 2: The Common Link Method

Another way to combine these ratios is to find a common link between them. To do this, one ratio must be expressed in a way that aligns its common variable with the value in the other ratio. In our example, the common variable is $r$ .

In the first ratio, $r$ corresponds to 2. In the second ratio, $r$ corresponds to 5. To link them, we scale the ratios so that $r$ has the same value in both. One way to do this is to scale the first ratio so that $r$ matches the value of 5, or to find a common multiple like 10.

If we scale both ratios to make $r = 10$ :

Multiply $r : y = 2 : 3$ by 5 to get $10 : 15$ .
Multiply $g : r = 4 : 5$ by 2 to get $8 : 10$ .

Now that $r$ is 10 in both expressions, we can write a single combined ratio for green, red, and yellow:

$g : r : y = 8 : 10 : 15$

From this unified ratio, we can identify that the specific relationship between green and yellow counters is $g : y = 8 : 15$ .

Key takeaways

A ratio $x : y = p : q$ is equivalent to the fraction $\frac{x}{y} = \frac{p}{q}$ .
To find the relationship between two variables linked by a third common variable, you can multiply their respective fractions together.
Combining multiple ratios requires scaling the shared terms until they have a common value.
The order of the variables in a ratio is crucial: $g : y = 8 : 15$ is different from $y : g = 8 : 15$ .

Tips

In the ESAT, Method 1 is typically faster for multiple choice questions where you only need to compare two specific variables. Method 2 is better if the question asks for the total number of items or a ratio involving all parts of the system.

Cautions

Be careful not to confuse a ratio with the fraction of a total. While $r : y = 2 : 3$ means $\frac{r}{y} = \frac{2}{3}$ , it means that red counters make up $\frac{2}{2 + 3} = \frac{2}{5}$ of the total bag.

Insight

This topic bridges the gap between basic arithmetic and algebraic manipulation. Understanding that ratios are multiplicative comparisons allows you to treat them as linear transformations, which is useful when solving related problems in Physics or Chemistry.

Frequently asked questions

Why does multiplying the fractions work to combine ratios?

When you multiply $\frac{g}{r}$ by $\frac{r}{y}$ , the $r$ in the numerator of the first fraction and the $r$ in the denominator of the second fraction cancel out algebraically, leaving you with $\frac{g}{y}$ .

How do I know which variable should be the common link?

The common link is always the variable that appears in both of the given ratios. In the example provided, $r$ (red counters) appeared in both the comparison with yellow and the comparison with green.

Can I use this method for more than three variables?

Yes. If you have a chain of ratios such as $a:b$ , $b:c$ , and $c:d$ , you can find $a:d$ by multiplying the three fractions: $\frac{a}{b} \times \frac{b}{c} \times \frac{c}{d}$ .

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