Question

From the following expressions 3ab, ac, bc, a, 5ab, -2ab, 2ad the three like terms are:

3ab, 5ab, -2ab

2a, a, 2ac

3ab, a, b

None of these
​

Answer 1

Explanation:

We will learn how to find the mean and third proportional of the set of three numbers.

If x, y and z are in continued proportion then y is called the mean proportional (or geometric mean) of x and z.

If y is the mean proportional of x and z, y^2 = xz, i.e., y = +\(\sqrt{xz}\).

For example, the mean proportion of 4 and 16 = +\(\sqrt{4 × 16}\) = +\(\sqrt{64}\) = 8

If x, y and z are in continued proportion then z is called the third proportional.

For example, the third proportional of 4, 8 is 16.

Solved examples on understanding mean and third proportional

1. Find the third proportional to 2.5 g and 3.5 g.

Solution:

Therefore, 2.5, 3.5 and x are in continuous proportion.

\(\frac{2.5}{3.5}\) = \(\frac{3.5}{x}\)

⟹ 2.5x = 3.5 × 3.5

⟹ x = \(\frac{3.5 × 3.5}{2.5}\)

⟹ x = 4.9 g

2. Find the mean proportional of 3 and 27.

Solution:

The mean proportional of 3 and 27 = +\(\sqrt{3 × 27}\) = +\(\sqrt{81}\) = 9.

3. Find the mean between 6 and 0.54.

Solution:

The mean proportional of 6 and 0.54 = +\(\sqrt{6 × 0.54}\) = +\(\sqrt{3.24}\) = 1.8

4. If two extreme terms of three continued proportional numbers be pqr, \(\frac{pr}{q}\); what is the mean proportional?

Solution:

Let the middle term be x

Therefore, \(\frac{pqr}{x}\) = \(\frac{x}{\frac{pr}{q}}\)

⟹ x\(^{2}\) = pqr × \(\frac{pr}{q}\) = p\(^{2}\)r\(^{2}\)

⟹ x = \(\sqrt{p^{2}r^{2}}\) = pr

Therefore, the mean proportional is pr.

5. Find the third proportional of 36 and 12.

Solution:

If x is the third proportional then 36, 12 and x are continued proportion.

Therefore, \(\frac{36}{12}\) = \(\frac{12}{x}\)

⟹ 36x = 12 × 12

⟹ 36x = 144

⟹ x = \(\frac{144}{36}\)

⟹ x = 4.

6. Find the mean between 7\(\frac{1}{5}\)and 125.

Solution:

The mean proportional of 7\(\frac{1}{5}\)and 125 = +\(\sqrt{\frac{36}{5}\times 125} = +\sqrt{36\times 25}\) = 30

7. If a ≠ b and the duplicate proportion of a + c and b + c is a : b then prove that the mean proportional of a and b is c.

Solution:

The duplicate proportional of (a + c) and (b + c) is (a + c)^2 : (b + c)^2.

Therefore, \(\frac{(a + c)^{2}}{(b + c)^{2}} = \frac{a}{b}\)

⟹ b(a + c)\(^{2}\) = a(b + c)\(^{2}\)

⟹ b (a\(^{2}\) + c\(^{2}\) + 2ac)