Question

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. The third proportional of 5, 10 is 40.
2. If three quantities are in continued proportion, then first is to the third is the sub-duplicate ratio of first to the second.

Options

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Neither (i) nor (ii)

Explanation:

i.

Let x be the third proportional to 5, 10.

⇒ Third proportional to 5, 10 is the same thing as fourth proportional to 5, 10, 10, then

5 : 10 :: 10 : x

⇒ `5/10 = 10/x`

⇒ 5x = 10 × 10

⇒ 5x = 100

⇒ x = `100/5`

⇒ x = 20

Therefore, the (i) statement is invalid.

ii.

If a, b, c are in continued proportion,

a : b = b : c

`a/b = b/c`

ac = b²

c = `b^2/a`

a : c = a : `b^2/a`

Simplify the ratio

a : c = a² : b²

Duplicate ratio of a : b is:

a² : b²

Sub-duplicate ratio of a : b is:

`sqrta : sqrtb`

a : c = a² : b²

So, it is the duplicate ratio,

Therefore, the (ii) statement is invalid.