Quantities as Fractions

Expressing One Quantity as a Fraction of Another

To express quantity A as a fraction of quantity B, write \( \dfrac{A}{B} \) and simplify. This fraction can be less than 1 (if A is smaller than B) or greater than 1 (if A is larger than B). The result describes how A relates proportionally to B — it is the foundation of percentage calculations, ratio comparisons, and rate problems.

Requirements

  • Same units: both quantities must be expressed in the same unit before forming the fraction. Do not mix grams with kilograms, or minutes with hours.
  • Simplify: cancel the fraction to its lowest terms using the HCF of numerator and denominator.
  • Fractions greater than 1: these are valid and common — for example, expressing a doubled quantity as a fraction of the original gives \( \frac{2}{1} = 2 \) or \( \frac{5}{3} \) where the result exceeds 1.

Connection to Percentage

Once a fraction is formed, multiplying by 100 converts it to a percentage. This is the most reliable method for any percentage calculation:

\[ \text{Percentage} = \frac{A}{B} \times 100 \]

Connection to Ratio

If \( A : B = 3 : 7 \), then A as a fraction of B is \( \frac{3}{7} \), and A as a fraction of the total \( (A+B) \) is \( \frac{3}{10} \). Distinguishing "as a fraction of B" from "as a fraction of the total" is important.

Worked Examples

Express 45 minutes as a fraction of 3 hours. Simplify fully.

Convert to the same unit: 3 hours = 180 minutes.

\[ \frac{45}{180} = \frac{1}{4} \]
In a class of 32 students, 20 are girls. Express the number of boys as a fraction of the total.
\[ \text{Boys} = 12 \qquad \frac{12}{32} = \frac{3}{8} \]
Express 350 g as a fraction of 2 kg. Express 80 cm as a fraction of 3 m.
\[ \frac{350}{2000} = \frac{7}{40} \qquad \frac{80}{300} = \frac{4}{15} \]
A recipe originally needs 200 g of flour. A baker uses 350 g. Express the amount used as a fraction of the original. Is this greater than 1?
\[ \frac{350}{200} = \frac{7}{4} = 1\frac{3}{4} \]

Yes — greater than 1, meaning the baker used more than the original amount.

A share price rises from 80p to 96p. Express the new price as a fraction of the old. Hence find the percentage increase.
\[ \frac{96}{80} = \frac{6}{5} \implies 120\% \text{ of original} \implies 20\% \text{ increase} \]

 Key Takeaways

  • To express A as a fraction of B: write \( \frac{A}{B} \) and simplify. Both must be in the same unit first.
  • The result can exceed 1 — this is valid and means A is larger than B.
  • Percentage = fraction × 100.
  • "As a fraction of B" and "as a fraction of the total \((A+B)\)" are different calculations — be precise about the denominator.
  • Always cancel to lowest terms using the HCF.