Ratios

A ratio is a mathematical comparison of two or more quantities that expresses how many times one value contains or is contained within another. Written as a:b or as the fraction a/b, ratios appear everywhere in daily life — from mixing paint colours and scaling recipes to calculating speeds and comparing prices. Understanding ratios is the gateway to proportional reasoning, one of the most powerful tools in mathematics and science. Ratios can be part-to-part or part-to-whole. A part-to-part ratio compares one portion of a group to another portion (for example, 3 red marbles to 5 blue marbles gives a ratio of 3:5). A part-to-whole ratio compares one portion to the entire group (3 red marbles out of 8 total gives 3:8). Both types can be simplified by dividing both terms by their greatest common factor (GCF), just like reducing a fraction. A proportion states that two ratios are equal: a:b = c:d. Solving proportions — finding an unknown quantity that makes two ratios equal — underpins commercial mathematics, including currency exchange, unit pricing, scale drawings, and percentage calculations. Mastering ratios and proportions gives students a flexible, transferable problem-solving strategy applicable across every quantitative discipline.

Key takeaways
  • Writing a Ratio
  • Simplifying Ratios
  • Equivalent Ratios
  • Setting Up a Proportion
  • Dividing a Quantity in a Given Ratio

Writing a Ratio

A ratio compares quantity A to quantity B in three equivalent notations: the colon form a:b, the fraction form a/b, and the word form 'a to b'. All three mean the same thing. Example: if a class has 12 girls and 18 boys, the ratio of girls to boys is 12:18, which simplifies to 2:3 by dividing both numbers by the GCF of 6.

Simplifying Ratios

Always reduce a ratio to its lowest terms by finding the GCF of both quantities and dividing. For 36:48, the GCF is 12, so the simplified ratio is 3:4. This makes comparison easier and is considered the standard form. Ratios with units (e.g. 150 km : 3 h) simplify to a rate: 50 km/h. Ensure both quantities use the same unit before simplifying — convert 1.5 m : 60 cm to 150 cm : 60 cm = 5:2.

Equivalent Ratios

Two ratios are equivalent when they simplify to the same value, just as equivalent fractions represent the same portion. You can generate equivalent ratios by multiplying or dividing both terms by the same non-zero number. For example, 2:3, 4:6, 10:15, and 100:150 are all equivalent. This idea is fundamental to proportions: a/b = c/d means the ratios a:b and c:d are equivalent.

Setting Up a Proportion

A proportion is an equation of two equal ratios: a/b = c/d. If one of the four values is unknown, use cross-multiplication to solve. Cross-multiplying gives a × d = b × c, then isolate the unknown. For example, if 3/4 = x/20, then 3 × 20 = 4 × x, so 60 = 4x, giving x = 15. Always label quantities carefully so you compare like units in the numerator and denominator.

Dividing a Quantity in a Given Ratio

To divide an amount in a ratio like 3:5, first find the total number of parts: 3 + 5 = 8. Each part equals the total amount divided by 8. Multiply each ratio term by the value of one part to get each share. For example, to divide R240 in the ratio 3:5: one part = R240 ÷ 8 = R30; first share = 3 × R30 = R90; second share = 5 × R30 = R150. Check: R90 + R150 = R240. ✓

The Key Equation

ab=cd  ⟹  a×d=b×c\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c
  • a — First term (first extreme) (any unit): The numerator of the first ratio in the proportion.
  • b — Second term (first mean) (any unit): The denominator of the first ratio; must be the same type of quantity as d.
  • c — Third term (second mean) (any unit): The numerator of the second ratio; same type of quantity as a.
  • d — Fourth term (second extreme) (any unit): The denominator of the second ratio. Often the unknown to be solved.

Worked Example

Problem. A car travels 270 km using 18 litres of fuel. How many litres of fuel are needed to travel 450 km at the same rate?

x=18×450270=8100270=30 Lx = \frac{18 \times 450}{270} = \frac{8100}{270} = 30 \text{ L}

Answer. 30 litres of fuel are needed to travel 450 km.

Figure
Diagram
Dividing R360 in the Ratio 2 : 3 : 4Total parts = 2 + 3 + 4 = 9  |  Value of 1 part = R360 ÷ 9 = R40Share A2 partsShare B3 partsShare C4 partsR80R120R1602 × R403 × R404 × R40Verification: R80 + R120 + R160 = R360 ✓Ratio Check80 : 120 : 160 → divide by 40 → 2 : 3 : 4 ✓Simplified ratio matches the original ratio given in the problem.
Origins · 1 minEuclid of Alexandria and how this was discovered
EO
Euclid of Alexandria
-300–-260 · Ancient Greek

Euclid systematically defined ratio and proportion in Book V of his monumental work 'Elements' (c. 300 BCE). His definition of equal ratios — later called the Eudoxan theory of proportion — was so rigorous that it remained the standard treatment for over 2,000 years. He defined the ratio of two magnitudes as a relationship with respect to size, and proved properties of proportions that underpin all of modern ratio arithmetic. Euclid's 'Elements' is the second most printed book in history after the Bible, with editions published continuously from 1482 to the present day.

  • Proportion and Direct Variation — Direct variation (y = kx) is the algebraic form of a proportion, extending ratio reasoning to linear functions and graphs.
  • Percentages — A percentage is a ratio expressed out of 100, so all percentage calculations rely directly on proportional reasoning.
  • Rates and Unit Rates — A rate is a special ratio comparing quantities in different units (e.g. km/h or price per item), a core concept in commercial mathematics.
  • Scale Drawings and Maps — Scale drawings use ratios to represent real-world distances on paper, requiring the same proportion-solving skills developed here.