A field is 80 metres in length and 60 metres in width. The ratio between its…

Concept — A ratio a : b compares two quantities in a fixed order — the quantity named first supplies the first term. Because the order is part of the meaning, a : b and b : a describe two different comparisons. A ratio is in its lowest terms once both terms have been divided by their highest common factor (HCF).

Applying it here

1. The comparison asked for is width : length, so the width supplies the first term and the length the second.

2. Width = 60 m and length = 80 m, so the comparison is 60 : 80.

3. HCF(60, 80) = 20, the largest number that divides both terms.

4. Divide each term by the HCF: 60 ÷ 20 = 3 and 80 ÷ 20 = 4.

5. Therefore width : length = 3 : 4.

Cross-check and contrast

• Scaling 3 : 4 back up by 20 returns 60 : 80, the pair we started from, so the simplification is sound.

• Reading the same two quantities in the opposite order gives length : width = 80 : 60 = 4 : 3. That is a perfectly valid ratio, but it answers a different question: here the width is named first, so the width must appear first.

• As a size check, the width of this field is smaller than its length, so the first term of the required ratio must be the smaller of the two terms.

Answer — the ratio of the width to the length is 3 : 4.