Ratios, Fractions and Functions

Ratios, Fractions and Linear Functions

A ratio \( a : b \) is directly related to fractions and to linear functions of the form \( y = kx \). Understanding how these three representations connect allows the same proportional relationship to be described algebraically, graphically and numerically — and makes it possible to move between them fluently on any problem.

Ratio to Fraction

  • A as a fraction of B: \( \frac{a}{b} \)
  • A as a fraction of the total \( (A+B) \): \( \frac{a}{a+b} \)
  • B as a fraction of the total: \( \frac{b}{a+b} \)

Ratio to Linear Function

If \( y : x = b : a \), then \( y = \frac{b}{a} x \). The coefficient \( \frac{b}{a} \) is the gradient of the line through the origin, and it equals the ratio \( b : a \) written as a fraction. Any proportional relationship can therefore be plotted as a straight line through the origin with gradient \( = \frac{b}{a} \).

\[ y : x = 3 : 4 \implies y = \frac{3}{4}x \implies \text{gradient} = 0.75 \]

Reading Ratio from a Graph

If a straight-line graph through the origin has gradient \( k \), then \( y : x = k : 1 \). Read any point \( (x, y) \) and write the ratio \( y : x \), then simplify.

Worked Examples

In a bag, red : blue marbles = 5 : 3. What fraction of the marbles are red? What fraction are blue?
\[ \text{Red fraction} = \frac{5}{5+3} = \frac{5}{8} \qquad \text{Blue fraction} = \frac{3}{8} \]
The ratio of boys to girls in a class is 4 : 5. Express girls as a fraction of boys.
\[ \frac{\text{girls}}{\text{boys}} = \frac{5}{4} \]
The ratio of distance to time for a cyclist is 5 : 1 (km : h). Write the equation of the linear function and find the gradient.
\[ d = 5t \qquad \text{gradient} = 5 \text{ km/h} \]
Orange juice is diluted with water in ratio 1 : 4. Write an equation for the volume of juice \( j \) in terms of total volume \( T \).
\[ j = \frac{1}{5}T \qquad \text{(since juice is } \frac{1}{1+4} = \frac{1}{5} \text{ of the total)} \]
A straight-line graph through the origin passes through \( (4, 10) \). Write the ratio \( y : x \) in simplest form and state the gradient.
\[ y : x = 10 : 4 = 5 : 2 \qquad \text{gradient} = \frac{10}{4} = 2.5 \]

 Key Takeaways

  • Ratio \( a : b \) → fraction of total: \( \frac{a}{a+b} \) and \( \frac{b}{a+b} \). Fraction of each other: \( \frac{a}{b} \) or \( \frac{b}{a} \).
  • Ratio \( y : x = b : a \) → linear function \( y = \frac{b}{a}x \) → gradient \( = \frac{b}{a} \).
  • Reading ratio from a graph: any point \( (x, y) \) gives ratio \( x : y \); the gradient equals \( \frac{y}{x} \).
  • All proportional relationships graph as straight lines through the origin.