Ratio and Proportion (Including Sharing in a Ratio)
ICSE Class 7 · Mathematics · Chapter 6
What this chapter covers
- Ratio: comparing two quantities
- Equivalent ratios and comparing ratios
- Proportion
- Sharing a quantity in a given ratio
- Continued ratio
- Continued proportion
- Word problems on ratio and proportion
Key definitions
Continued proportion is the relation among three quantities a, b and c of the same kind in which a : b = b : c.
Key points
- A ratio compares two quantities of the same kind by division; it is written a : b and has no unit.
- Both quantities must be in the same unit before a ratio is formed.
- A ratio is simplified by dividing both terms by their HCF; multiplying or dividing both terms by the same number gives an equivalent ratio.
- Adding the same number to both terms changes a ratio.
- Two ratios are compared by cross-multiplying or by writing them as fractions.
- A proportion states that two ratios are equal: a : b :: c : d.
- In a proportion, the product of the extremes equals the product of the means: a × d = b × c.
- To divide a quantity in the ratio a : b, divide it by a + b to get one part, then multiply by each term.
- Shares found by dividing in a ratio must add up to the whole and simplify back to the given ratio.
- A continued ratio a : b : c compares three quantities; two ratios are combined by making the common term equal using the LCM.
- In a continued proportion a : b :: b : c, the middle term b is the mean proportional and b × b = a × c.
Questions answered
Express 45 minutes to 2 hours as a ratio in its simplest form.
2 hours = 120 minutes, so the ratio is 45 : 120 = 3 : 8.
Which is greater, 4 : 9 or 3 : 7?
Cross-multiply: 4 × 7 = 28 and 9 × 3 = 27. Since 28 > 27, 4 : 9 is greater.
Explain why the ratio 3 kg : 500 g cannot be simplified to 3 : 500.
The two quantities are in different units. Changing 3 kg to 3000 g gives 3000 : 500 = 6 : 1.
Are 8, 12, 18 and 27 in proportion?
Yes. Product of extremes = 8 × 27 = 216 and product of means = 12 × 18 = 216.
Find x if x : 15 :: 6 : 9.
9x = 15 × 6 = 90, so x = 10.
Divide 96 marbles between Asha and Gopal in the ratio 5 : 3.
Total parts = 8, one part = 12. Asha gets 60 marbles and Gopal gets 36.
Divide ₹1,200 among three children in the ratio 1 : 2 : 3.
Total parts = 6, one part = ₹200. The shares are ₹200, ₹400 and ₹600.
If a : b = 3 : 5 and b : c = 2 : 3, find a : b : c.
Make b equal to 10 (the LCM of 5 and 2): a : b = 6 : 10 and b : c = 10 : 15. So a : b : c = 6 : 10 : 15.