WaywardHouse

9  Fractions and decimals

Arithmetic with parts of wholes

A recipe needs \(\frac{3}{4}\) of a cup of flour and you want to make half the recipe. How much flour do you use?

Four friends split a restaurant bill of £37.60 equally. What does each person pay?

You run 2.5 km on Monday and 1.75 km on Wednesday. How far have you run in total?

These are all fraction and decimal problems. The numbers sit between the integers — and the arithmetic for handling them is a small set of rules you can learn once and use everywhere.

9.1 What the notation is saying

The fraction \(\frac{a}{b}\) means “\(a\) divided by \(b\).” The top number is the numerator (how many parts you have); the bottom number is the denominator (how many equal parts make the whole).

Decimals are fractions whose denominators are powers of ten: \[0.3 = \frac{3}{10}, \quad 0.47 = \frac{47}{100}, \quad 0.006 = \frac{6}{1000}\]

They are interchangeable. \(\frac{3}{4} = 0.75\). \(\frac{1}{3} = 0.\overline{3}\) (the bar means the digit repeats forever).

Equivalent fractions: \(\frac{2}{4} = \frac{1}{2}\) because you can divide numerator and denominator by 2. A fraction is in lowest terms when numerator and denominator share no common factor other than 1.

Use the controls below to explore what a fraction looks like — as a divided rectangle and as a position on the number line.

If the denominator input goes below the numerator, the interactive clamps it to keep the fraction valid (≤ 1). Try setting numerator and denominator to the same value to see what “1 whole” looks like in both views.

9.2 The method

Multiplying fractions

Multiply numerators together; multiply denominators together — that’s it: \[\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\]

Simplify before or after. Before is usually easier.

Dividing fractions

Flip the second fraction and multiply — this is called multiplying by the reciprocal: \[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\]

“Flip and multiply” works because \(\frac{c}{d} \times \frac{d}{c} = 1\): the two fractions cancel each other out, leaving you with a clean multiplication.

Adding and subtracting fractions

Mixed numbers: convert first. Multiply the whole number by the denominator and add the numerator: \(1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}\).

You need a common denominator before you can add. The simplest route: multiply both denominators to get one that works for both. \[\frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{bc}{bd} = \frac{ad + bc}{bd}\]

Simplify the result.

Converting between fractions and decimals

Fraction → decimal: divide numerator by denominator. Decimal → fraction: write the decimal over its place-value denominator, then simplify.

Why this works

A fraction \(\frac{a}{b}\) is a number. The rules for arithmetic with fractions follow from the same rules that govern whole numbers — distribution, commutativity, and the fact that multiplying by 1 changes nothing. To add \(\frac{a}{b} + \frac{c}{d}\), rewrite the first fraction as \(\frac{a}{b} \times \frac{d}{d} = \frac{ad}{bd}\) and the second as \(\frac{c}{d} \times \frac{b}{b} = \frac{bc}{bd}\). Both multiplications are by 1, so neither fraction’s value changes — only its representation. The result is \(\frac{ad + bc}{bd}\). Division by a fraction is multiplication by its reciprocal because \(\frac{a}{b} \times \frac{b}{a} = 1\) — they are inverses of each other.

Use the calculator below to check any fraction arithmetic. Choose two fractions and an operation; the working is shown step by step.

9.3 Worked examples

Example 1 — Halving a recipe. A recipe needs \(\frac{3}{4}\) cup of flour, \(1\frac{1}{2}\) cups of oats, and \(\frac{5}{8}\) cup of sugar. You want to make half the recipe. What quantities do you need?

Multiply each ingredient by \(\frac{1}{2}\).

Convert the mixed number first: \(1\frac{1}{2} = \frac{3}{2}\).

\[\frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \text{ cup flour}\]

\[\frac{3}{2} \times \frac{1}{2} = \frac{3}{4} \text{ cup oats}\]

\[\frac{5}{8} \times \frac{1}{2} = \frac{5}{16} \text{ cup sugar}\]

Example 2 — Splitting costs. Three friends split the cost of a camping trip. They agree to share costs in proportion to how many nights each stays: one friend stays \(\frac{1}{3}\) of the nights, another \(\frac{5}{12}\), and the third \(\frac{1}{4}\). The total cost is £120. What does each pay?

Check the fractions sum to 1 first — to make sure everyone’s share is accounted for: \[\frac{4}{12} + \frac{5}{12} + \frac{3}{12} = \frac{12}{12} = 1 \checkmark\]

Friend A: \(\frac{1}{3} \times 120 = £40\)

Friend B: \(\frac{5}{12} \times 120 = £50\)

Friend C: \(\frac{1}{4} \times 120 = £30\)

Total: £120. Check.

Example 3 — Running distances. You run 2.5 km on Monday, 1.75 km on Wednesday, and \(\frac{7}{4}\) km on Friday. What is your total distance for the week?

Convert \(\frac{7}{4}\) to a decimal so all three are in the same form: \(\frac{7}{4} = 1.75\) km.

\[2.5 + 1.75 + 1.75 = 6.0 \text{ km}\]

Example 4 — What fraction is left? A bag of rice is 0.38 used up. What fraction remains? Express it as a decimal and as a fraction in lowest terms.

Remaining: \(1 - 0.38 = 0.62\).

As a fraction: \(0.62 = \frac{62}{100} = \frac{31}{50}\).

\(\frac{31}{50}\) of the bag is left.

9.4 Where this goes

Fraction arithmetic is the foundation of algebra. When you solve \(\frac{2x}{3} = 8\), you multiply both sides by \(\frac{3}{2}\) — a fraction operation. Every algebraic fraction, rational expression, and limit in calculus is built on the rules in this chapter.

In calculus, the derivative is defined as a limit of a fraction: \(\frac{\Delta y}{\Delta x}\) as \(\Delta x \to 0\). That fraction needs to be understood, not just computed. The conceptual step from “fraction” to “rate of change” is not large; it requires only that you see the notation clearly.

Where this shows up

  • Any time you split something equally — a bill, a recipe, a distance — you’re doing fraction arithmetic.
  • A structural engineer applies a safety factor of \(\frac{3}{2}\) (or 1.5) to every load — multiplying by a fraction.
  • A financial model apportions costs, equity, and interest using decimal fractions at every step.
  • A GIS analyst computing land-cover proportions is adding fractions that must sum to 1.

The arithmetic is identical. The precision matters.

9.5 Exercises

  1. A metal bar is \(4\frac{3}{8}\) m long. A \(1\frac{5}{8}\) m section is cut off. How much remains?

  2. A recipe calls for \(\frac{2}{3}\) cup of milk. You want to make \(2\frac{1}{2}\) times the recipe. How much milk do you need?

  3. A sports pitch is 0.75 km long and 0.48 km wide. What is its area in km²? Express the answer as a decimal and as a fraction in lowest terms.

  4. Three taps fill a tank. Each supplies \(\frac{3}{8}\) of the tank’s capacity per hour. A drain removes \(\frac{5}{8}\) of the tank’s capacity per hour. Is the tank filling or emptying? At what net fraction per hour?

  5. A bag of trail mix is analysed: 0.44 nuts, 0.29 dried fruit, 0.18 chocolate chips, and the rest is seeds. What fraction is seeds? Express as a decimal and a simplified fraction.

  6. You have a 500 mL bottle of juice concentrate. The instructions say to mix it at \(\frac{1}{4}\) strength. What fraction of the final drink will be concentrate? How many mL of concentrate goes into a 250 mL glass?

  7. Convert each to a decimal, then order from least to greatest: \(\frac{7}{12}\), \(\frac{3}{5}\), \(0.58\), \(\frac{11}{20}\), \(0.\overline{6}\).