Question

What number must be added to each of the numbers $5,11,19$ and $37$ so that they are in proportion?

Answer 1

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

Complete step by step answer:
Since from the given that we have numbers $5,11,19$ and $37$ which are the four proportional numbers.
Let fix $x$ as the number added to each of these numbers, so that the new number become
The first number is $5 + x$, the second number $11 + x$, the third number $19 + x$ , and the fourth number is $37 + x$
Now after adding an unknown’s numbers, since the resulting numbers need to be in proportion so we write these numbers as $5 + x:11 + x::19 + x:37 + x$
Now we know that the means and extremes property of the proportionality where the product of the extreme is equal to the product of the mean.
Hence, we write it as \[(5 + x) \times (37 + x) = (11 + x) \times (19 + x)\]
Now solving this equation, we get $185 + 42x + {x^2} = 209 + 30x + {x^2}$
Canceling the common terms, we get $185 + 42x = 209 + 30x \Rightarrow 12x = 24$
By the division operation, we get $x = 2$ and hence which is the least number that needs to be added in the given question and thus the resulting is $5 + x:11 + x::19 + x:37 + x \Rightarrow 7:13::21:39$
Therefore, the answer is $x = 2$

Note:
The number is proportional when the ratio of the LHS of the proportional is equal to the RHS of the proportional. To check the numbers are proportional we just find their ratio of both sides. Since the given numbers are $5,11,19$ and $37$. Then we found it proportional values a$7:13::21:39$ and hence we get $\dfrac{7}{{13}} = \dfrac{{21}}{{39}} \Rightarrow \dfrac{7}{{13}} = \dfrac{7}{{13}}$