Let’s look at two skills: writing a ratio as a fraction and using a ratio to form a linear function.
Writing a ratio as a fraction
A ratio compares two or more quantities.
For example, the ratio of boys to girls in a class is $2:3$ This means there are 2 parts boys and 3 parts girls.
Example 1: Write the ratio of boys to girls as a fraction.
Step 1: The ratio is $2:3$
Step 2: To write boys as a fraction of girls, put boys on top and girls on the bottom.
\[ \frac{\text{Boys}}{\text{Girls}}=\frac{2}{3} \]
Answer: $\boxed{\frac{2}{3}}$
Be careful: the total is different!
If the question asks for the fraction of the class who are boys, you must add the ratio parts together.
\[ 2+3=5\text{ parts in total} \]
Therefore,
\[ \frac{\text{Boys}}{\text{Total}}=\boxed{\frac25} \]
And the fraction who are girls is $\frac35$.
Remember:
- Boys as a fraction of girls: $\frac23$
- Boys as a fraction of the total:$\frac25$
Writing a ratio as a linear function
A linear function is an equation such as
\[ y=3x \]
It shows how one quantity changes in relation to another.
Example
The ratio of $y$ to $x$ is $3:1$
Write $y$ as a function of $x$.
Step 1: Write the ratio.
\[ y:x=3:1 \]
Step 2: Turn it into a fraction.
\[ \frac{y}{x}=\frac31=3 \]
Step 3: Multiply both sides by $x$.
\[ \boxed{y=3x} \]
This is the linear function.
If $x=4$, then $y=3\times4=12$.
Example: A harder Grade 5 question
The ratio of $y$ to $x$ is $5:2$. Write $y$ as a function of $x$.
Step 1: Write the ratio as a fraction.
\[ \frac{y}{x}=\frac52 \]
Step 2: Multiply both sides by $x$.
\[ \boxed{y=\frac52x} \]
You can also write this as $y=2.5x$.
What if the ratio is given the other way around?
Suppose the ratio of $x$ to $y$ is $2:5$.
\[ x:y=2:5 \]
This means
\[ \frac{x}{y}=\frac25 \]
Rearranging to make $y$ the subject:
\[ 2y=5x \]
\[ \boxed{y=\frac52x} \]
Notice that the fraction is reversed when making $y$ the subject.
The key rules to remember
- Fraction of the total: add all the ratio parts first.
- Ratio of $y$ to $x$: divide $y$ by $x$ to find the multiplier.
- Linear function: write the equation in the form $y=kx$, where $k$ is the ratio multiplier.
- Making $y$ the subject: if the ratio is $x:y=a:b$, then $y=\frac ba x$.