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Third Proportion

If a, b, and c are in continued proportion, then c is called the third proportion of a and b.
Example 23: Find the value of x if 15, 30, and x are in continued proportion.
Solution: Since 15, 30 and x are in continued proportion, then, x will be the third proportion.

So, 15 : 30 :: 30 : x
Therefore, 15 × x = 30 × 30 (Product of extremes = Product of means)
×
30 30
x = = 60
15
Thus, the value of x is 60.

Practice Time 9B

1. Verify whether the following ratios are in proportion.

( a) 15 : 45 and 40 : 120 (b) 33 : 121 and 9 : 96 (c) 24 : 28 and 36 : 48 (d) 32: 48 and 70 : 210
( e) 4 : 6 and 8 : 12 (f) 33 : 44 and 75 : 100
2. Which of the following numbers are in proportion?

( a) 35, 7, 10 and 2 (b) 20, 15, 40 and 35 (c) 16, 18, 22 and 27 (d) 9, 7, 63 and 49
3. Find the value of x:
( a) 15 : 90 : : x : 78 (b) 45 : 27 : : 15 : x ( c) x : 44 : : 94 : 22 (d) 28 : x : : 16 : 12
4. If 9, 18, x are in continued proportion, find the value of x.

5. Find the mean proportion between 27 and 3?
6. Find the value of x if 25, 35, and x are in continued proportion.
7. The incomes of Subhash and Shekhar are in the ratio 5 : 9. If Subhash earns `38,450 per month. How
much does Shekhar earn per month?
8. Iqra covers 14 km in 25 minutes at a constant speed. How much time will it take her to cover 56 km?

9. The ratio of the length and breadth of a rectangular ground is 12 : 7. If the length is 108 m, find its
breadth.

A Pinch of History

The concept of ratio and proportion was developed and studied by ancient Greek mathematicians,
including Thales of Miletus, Pythagoras, and Euclid. Thales of Miletus, was being one of the
earliest mathematicians to recognize the importance of ratios and proportions. He used the
concept of similar triangles to measure the height of the Great Pyramid of Giza by comparing
the lengths of its shadow to a stick's shadow.

Pythagoras and the Pythagorean school further advanced the study of ratios. They discovered
the concept of proportionality, which states that the ratio of two quantities is equal to the ratio
of the corresponding quantities.

Euclid, systematically presented the principles of ratio and proportion in his seminal work
"Elements." He defined proportions and established fundamental theorems, laying the
foundation for the formal study of this mathematical concept.

211 Ratio and Proportion