Introduction to Ratio

1Summary 重點摘要

Learn what ratio is, know how to simplify and relationship between ratios.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What a ratio isA ratio compares the relative sizes of two quantities of the same kind, written a:b, where a is the antecedent and b the consequent. For example, with 20 boys and 30 girls, the ratio of boys to girls is 20:30. Because a ratio expresses relative size, it carries no unit — the word 'people' cancels away when 20 is compared with 30. The order of a ratio is meaningful: 20:30 and 30:20 mean opposite things, so always follow the order the question asks for.
2. Step 2: How to simplify a ratioSimplifying a ratio works like reducing a fraction: divide both terms by their highest common factor until they are coprime. For example, 20:30 divided by 10 gives 2:3, and the worked example 15:25 divided by 5 gives 3:5. The simplified ratio expresses the same relation and is called an equivalent ratio — 2:3, 4:6 and 6:9 are all equivalent. Simplify fully whenever the question asks for the simplest form, and multiply through to clear decimals, such as turning 1.5:2 into 3:4 by multiplying both terms by 2.
3. Step 3: Units must agreeBefore comparing two quantities with units, convert them into the same unit, otherwise the whole ratio is wrong. For example, 1 m : 50 cm should first become 100 cm : 50 cm, giving 100:50 = 2:1; writing 1:50 directly is a serious error. Know the common conversions well: 1 m = 100 cm, 1 kg = 1000 g, 1 minute = 60 seconds. Once simplified, the ratio itself carries no unit, which beginners often confuse with quantities such as speed that do require units.
4. Step 4: Continued ratios with three or more termsA ratio can compare more than two quantities, written a:b:c. For example, three strings of lengths 2 m, 3 m and 5 m give the ratio 2:3:5. Simplify a continued ratio by dividing every term by a common factor, such as turning 4:6:8 into 2:3:4. Another useful skill is merging two two-term ratios: if A:B = 2:3 and B:C = 3:4, then B matches directly and A:B:C = 2:3:4; if the B values differ, scale one ratio first so they agree. Continued-ratio questions of this kind are very common in examinations.
5. Step 5: Sharing in a given ratioTo share a total in a given ratio, find the total number of parts first, then the value of one part. Sharing $100 between A and B in 2:3: total parts = 2 + 3 = 5, each part = 100 ÷ 5 = $20, so A receives 2 × 20 = $40 and B receives 3 × 20 = $60. The check is that the shares add back to the total: 40 + 60 = 100. The classic mistakes are forgetting to add every term of the ratio, or treating parts as people; remember 'total ÷ total parts = one part' and the method stays orderly.
6. Step 6: Common applicationsRatios appear everywhere in daily life. Sugar water mixed in the ratio sugar:water = 1:9, using 5 g of sugar, means one part = 5 g, so water = 9 parts = 9 × 5 = 45 g. A scale of 1:100 on a plan means 1 cm on paper equals 100 cm in reality. In such questions, first work out what 'one part' is worth, multiply out the shares, then confirm the parts add up to the total given by the question — this habit prevents most arithmetic slips.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Ratio compares two quantities
• Symbol :
• Simplify: divide by common factor
• Units must be same
• Share in ratio: first calculate total parts