Name: Proportion
Uploaded: 2019-07-19
Duration: 5 min 17 s
Description: A T-Rex looks strange because it has disproportionately sized arms. But, what are proportions? To understand their significance in mathematics, here are some...

Question

Find four proportionals such that the sum of the extremes is

21

, the sum of the means

19

, and sum of the squares of all the four numbers is

442 .

Accepted Answer

Let the four proportionals are $a, b, c, d$ .

$\Rightarrow \frac{a}{b} = \frac{c}{d}$

$\Rightarrow$ $a d = b c$

Given,

Sum of the extremes is $21$

$∴ a + d = 21 \dots (i)$

Given,

Sum of the means is $19$

$∴ b + c = 19 \dots (ii)$

Given,

Sum of the squares of all four numbers is $442$

$∴ a^{2} + b^{2} + c^{2} + d^{2} = 442 \dots (iii)$

Squaring $(i)$ and $(ii)$ and adding together, we get,

${(a + d)}^{2} + {(b + c)}^{2} = 19^{2} + 21^{2}$

$\Rightarrow a^{2} + b^{2} + c^{2} + d^{2} + 2 (a d + b c) = 441 + 361$

$\Rightarrow 442 + 2 (2 a d) = 441 + 361$ $(∵ a d = b c)$

$\Rightarrow 4 a d = 360$

$\Rightarrow$ $a d = 90 . . . (iv)$

$\Rightarrow b c = 90 . . . (v)$

So, from the equation $(i) & (iv)$

$a - d = \sqrt{{(a + d)}^{2} - 4 a d}$

$\Rightarrow a - d = \sqrt{{(21)}^{2} - 4 \times 90}$

$\Rightarrow a - d = 9 . . . (vi)$

From equation $(i) & (vi)$

$a = 15, d = 6$

So, from the equation $(ii) & (v)$

$b - c = \sqrt{{(b + c)}^{2} - 4 b c}$

$\Rightarrow b - c = \sqrt{{(19)}^{2} - 4 \times 90}$

$\Rightarrow b - c = 1 . . . (vii)$

From equation $(ii) & (vii)$

$b = 10, c = 9$

So, the value of proportionals are

$a = 15, d = 6, b = 10, c = 9 .$

Find four proportionals such that the sum of the extremes is $21$ , the sum of the means $19$ , and sum of the squares of all the four numbers is $442 .$

Important Questions on Proportion

Learn and Prepare for any exam you want !

Find four proportionals such that the sum of the extremes is 21, the sum of the means 19, and sum of the squares of all the four numbers is 442.

Important Questions on Proportion

Learn and Prepare for any exam you want !

Find four proportionals such that the sum of the extremes is $21$ , the sum of the means $19$ , and sum of the squares of all the four numbers is $442 .$