5.2 Ratio, fractions and linear functions — Learn

Prerequisites

You should be able to multiply and divide whole numbers and fractions. Check units before comparing quantities.

Key terms

A linear relationship has a constant rate of change. A constant of proportionality is the multiplier in a relationship $y=kx$.

Objectives

• Express ratio parts as fractions of a total and identify proportional linear relationships.
• Check that the result matches the units and direction of change.

Learn the method

A ratio can describe a fraction of a total. For $a:b$, the first amount is $a/(a+b)$ of the total. Proportion means equality of ratios. A linear relationship has form $y=mx+c$. Direct proportion is $y=kx$, whose graph passes through $(0,0)$.

Worked example. The ratio of red to blue counters is $2:3$. If the ratio stays the same, how many blue counters go with 8 red counters?

• Set the ratios equal: $\frac{2}{3}=\frac{8}{b}$.
• The first ratio has been multiplied by 4, so multiply the second part by 4: $b=12$.
• The equivalent ratio is $8:12$.

Worked example. 4 cakes use 300 g of flour. Find the flour for 10 cakes.

• Find flour for one cake: $300\div4=75$ g.
• Multiply by 10 cakes: $75\times10=750$ g.
• Check the rate: $750\div10=75$ g per cake.

The amounts are in direct proportion because their ratio stays constant: $300:4=750:10$. If $n$ is the number of cakes and $f$ is flour in grams, the relationship is $f=75n$.

Watch out. A straight line is not always direct proportion. Check its $y$-intercept: direct proportion passes through $(0,0)$.

You should now be able to

• I can write each ratio part as a fraction of the total.
• I can use a constant multiplier to complete a proportional relationship.
• I can explain why a non-zero intercept is not direct proportion.