Mean and Third Proportional

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) = a(b$^{2}$ + c$^{2}$ + 2bc)

⟹ b (a$^{2}$ + c$^{2}$) = a(b$^{2}$ + c$^{2}$)

⟹ ba$^{2}$ + bc$^{2}$ = ab$^{2}$ + ac$^{2}$

⟹ ba$^{2}$ - ab$^{2}$ = ac$^{2}$ - bc$^{2}$

⟹ ab(a - b) = c$^{2}$(a - b)

⟹ ab = c$^{2}$, [Since, a ≠ b, cancelling a - b]

Therefore, c is mean proportional of a and b.

8. Find the third proportional of 2x^2, 3xy

Solution:

Let the third proportional be k

Therefore, 2x^2, 3xy and k are in continued proportion

Therefore,

\frac{2x^{2}}{3xy} = \frac{3xy}{k}

⟹ 2x$^{2}$k = 9x$^{2}$y$^{2}$

⟹ 2k = 9y$^{2}$

⟹ k = $\frac{9y^{2}}{2}$

Therefore, the third proportional is $\frac{9y^{2}}{2}$.

● Ratio and proportion

• Basic Concept of Ratios
• Important Properties of Ratios
• Ratio in Lowest Term
  
• Types of Ratios
• Comparing Ratios
• Arranging Ratios
  
• Dividing into a Given Ratio
  
• Divide a Number into Three Parts in a Given Ratio
• Dividing a Quantity into Three Parts in a Given Ratio
  
• Problems on Ratio
  
• Worksheet on Ratio in Lowest Term
  
• Worksheet on Types of Ratios
  
• Worksheet on Comparison on Ratios
• Worksheet on Ratio of Two or More Quantities
  
• Worksheet on Dividing a Quantity in a Given Ratio
• Word Problems on Ratio
  
• Proportion
  
• Definition of Continued Proportion
  
• Mean and Third Proportional
  
• Word Problems on Proportion
  
• Worksheet on Proportion and Continued Proportion
  
• Worksheet on Mean Proportional
  
• Properties of Ratio and Proportion

10th Grade Math

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