Ratio and Proportion — One Idea Behind Sharing, Partnership and Ages

Ratio — Parts, Not Amounts

A ratio a : b compares two quantities of the same kind by division: a : b means a/b. Multiplying or dividing both terms by the same non-zero number leaves it unchanged, so 6 : 9, 4 : 6 and 2 : 3 are the same ratio.

The single most useful habit: treat a ratio as a number of equal parts.

Worked example. Divide ₹1,200 between A and B in the ratio 3 : 5.

Total parts = 3 + 5 = 8. One part = 1200 ÷ 8 = ₹150.

A gets 3 × 150 = ₹450, B gets 5 × 150 = ₹750.

That 'find one part first' step removes almost all the algebra from the chapter.

A ratio is not a share. If A : B = 3 : 5, A does not get 3/5 of the money — A gets 3/8 of it. Confusing the second term with the total is the most common single error in this chapter, and examiners put the wrong value in the options on purpose.

1. If a : b = 2 : 3 and b : c = 4 : 5, then a : b : c is:

2. ₹5,200 is divided among A, B and C in the ratio 1/2 : 1/3 : 1/4. A's share is:

3. The mean proportional between 9 and 16 is:

4. The fourth proportional to 3, 5 and 12 is:

5. A starts a business with ₹9,000 and stays for 12 months. B joins with ₹12,000 and stays for 6 months. Out of a total profit of ₹15,000, B's share is:

6. The ratio of the present ages of A and B is 4 : 3. Six years from now the ratio will be 6 : 5. A's present age is:

7. If 40% of A is equal to 60% of B, then A : B is:

8. The third proportional to 4 and 8 is: