What Are Ratios and How Are They Written?
A ratio compares two (or more) quantities of the same kind. If a bag contains 3 red balls and 5 blue balls, the ratio of red to blue is 3:5. You read this as 'three to five'.
Ratios can be written in three ways:
- Using a colon: 3:5
- As a fraction: ⅗
- In words: 3 to 5
The order matters. The ratio 3:5 is not the same as 5:3.
Simplifying ratios
Ratios can be simplified by dividing both numbers by their highest common factor. The ratio 12:8 simplifies to 3:2 (dividing both by 4). Simplified ratios are easier to work with and compare. A ratio in its simplest form has no common factors other than 1.
Part-to-part and part-to-whole ratios
A part-to-part ratio compares one part of a group to another part — like 3 red balls to 5 blue balls. A part-to-whole ratio compares one part to the total — like 3 red balls out of 8 total. Part-to-whole ratios can be expressed as fractions, decimals, or percentages.
Equivalent Ratios and Proportion
Two ratios are equivalent if they simplify to the same form. The ratios 2:4, 3:6, and 10:20 are all equivalent to 1:2. This is the same concept as equivalent fractions.
Scaling with ratios
Ratios make scaling straightforward. Suppose a recipe for 4 people uses 200g of rice. To scale it for 10 people, multiply both quantities by the same factor. The ratio of people to rice stays 4:200 = 1:50, so 10 people need 500g.
Direct proportion
Two quantities are in direct proportion if their ratio stays constant as both values change. If you earn £12 per hour, doubling your hours doubles your pay. The ratio of hours to pay (1:12) stays fixed. Direct proportion means more of one always means proportionally more of the other.
Inverse proportion
Two quantities are in inverse proportion if one increases as the other decreases. More workers completing a job means fewer hours needed. If 3 workers take 12 hours, 6 workers take 6 hours — the product stays constant at 36. Ratios and proportion appear throughout algebra and science.
Ratios in Everyday Life
Ratios are everywhere once you start looking.
Maps and scale models
A map scale of 1:50,000 means every 1 cm on the map represents 50,000 cm (500 m) in reality. Architects use scale drawings at ratios like 1:100. Model makers replicate objects at ratios like 1:72 for aircraft models.
Finance and mixtures
Interest rates, tax rates, and profit margins all involve ratios. A company might allocate costs in a ratio of 3:2:1 across departments. Chemists mix solutions using ratios — diluting a concentrated acid to a precise ratio ensures safe and effective use.
The golden ratio
The golden ratio — approximately 1:1.618, often written as φ (phi) — appears in nature, art, and architecture. The spiral of a nautilus shell, the proportions of the Parthenon, and the arrangement of sunflower seeds all approximate this ratio. Percentages are another way to express ratios relative to 100.
Frequently asked questions
- What is the difference between a ratio and a fraction?
- Both compare quantities, but differently. A fraction always shows a part of a whole — ¾ means 3 parts out of 4. A ratio compares any two quantities — 3:4 could mean 3 red and 4 blue (total 7). Part-to-whole ratios can be written as fractions, but part-to-part ratios cannot. The context determines which form is appropriate.
- How do you divide a quantity in a given ratio?
- Add the parts of the ratio to find the total number of shares. Divide the quantity by that total to find one share. Then multiply each part by one share. Example: divide £40 in the ratio 3:5. Total shares = 8. One share = £5. The split is £15 and £25.
- What does 1:1 ratio mean?
- A 1:1 ratio means equal quantities. One part A for every one part B. Mixing equal volumes of two solutions gives a 1:1 ratio. A square has a 1:1 ratio of width to height. Many symmetrical designs and balanced formulas use this proportion.
- Why do ratios not have units?
- Ratios compare quantities of the same kind, so the units cancel out. The ratio 4 metres : 8 metres simplifies to 4:8, then 1:2 — no metres needed. This is why they can describe any relationship regardless of what is being measured: length, mass, time, or money.