What number must be added to each of 4 10 12 and 24 so that the resulting numbers are in proportion

What least number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?

Let the number added be x.

∴ (6 + x) : (15 + x) :: (20 + x) (43 + x)

`=> (6+x)/(15 + x) = (20 + x)/(43 + x)`

⇒ (6 + x)(43 + x) (20 + x)(15 + x)

`=> 258 + 6x + 43x + x^2 = 300 + 20x + 15x + x^2`

`=> 49x - 35x = 300 - 258`

`=> 14x = 42`

`=> x = 3`

Thus, the required number which should be added is 3.

Concept: Concept of Proportion

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Question 10 Ratio and Proportion Exercise 7.2

Answer:

Consider x be added to each number

16 + x , 26 + x and 40 + x are in continued proportion

It can be written as

(16 + x)/ (26 + x) = (26 + x)/ (40 + x)

By cross multiplication

(16 + x) (40 + x) = (26 + x) (26 + x)

On further calculation

$\begin{array}{l} 640+16 x+40 x+x^{2}=676+26 x+26 x+x^{2} \\ 640+56 x+x^{2}=676+52 x+x^{2} \\ 56 x+x^{2}-52 x-x^{2}=676-640 \end{array}$

So we get

4x = 36

x = 36/4 = 9

Hence, 9 is the number to be added to each of the numbers.

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Hint: Product of extremes is equal to the product of mean. Proportional numbers are represented as \[a:b::c:d\] , where \[a,d\] are the extremes and \[b,c\] are known as the mean. In this question four proportional numbers are given so first we will add a common unknown number to them and then we will use means and extremes property to find the unknown number.

Complete step-by-step answer:

Given the four proportional numbers \[12:22::42:72\] Let \[x\] be the number added to each of these numbers, so the new numbers becomeFirst number \[ = 12 + x\] Second number \[ = 22 + x\] Third number \[ = 42 + x\] Fourth number \[ = 72 + x\] Now after adding an unknown numbers, since the resulting numbers need to be in proportion so we can write these numbers as \[12 + x:22 + x::42 + x:72 + x\] Now we know the means and extremes property of proportionality where the product of extremes is equal to the product of mean, hence we can write \[\left( {12 + x} \right) \times \left( {72 + x} \right) = \left( {22 + x} \right) \times \left( {42 + x} \right)\] Now by solving this \[\Rightarrow 12 \times 72 + 12x + 72x + {x^2} = 22 \times 42 + 22x + 42x + {x^2} \\ \Rightarrow 864 + 84x = 924 + 64x \\ \Rightarrow 84x - 64x = 924 - 864 \\ \Rightarrow 20x = 60 \\ \Rightarrow x = 3 \; \] Therefore we get the value of \[x = 3\] Hence we can say if we add 3 to the numbers 12, 22, 42 and 72 the resulting numbers will be in proportion.The resulting numbers are \[15:25::45:75\]

So, the correct answer is “3”.

Note: The numbers are proportional when the ratio of the LHS of the proportions is equal to the RHS of the proportion. To check if the numbers are in proportion we just find their ratio of both the sides.

The given numbers \[12:22::42:72\] are not in proportion since their ratios are not equal \[\dfrac{{12}}{{22}} = \dfrac{{42}}{{72}}\] Now if we add 3 to each numbers then they become proportional \[ \dfrac{{15}}{{25}} = \dfrac{{45}}{{75}} \\ \dfrac{3}{5} = \dfrac{3}{5} \\ \]