Question

What number shall be added to each of the number 4, 7, 14, 22 to form the terms in a proportion?
A. 1
B. 2
C. 3
D. 4

Answer 1

Hint: The product of extremes equals the mean product. Proportion number can be represented as \[a:b::c:d\]where a, d are the extremes and b, c are known as the mean. Because there are four proportional numbers in this question, we must first add a common number to them before using the means and extremes principle to discover the unknown means.

Complete step by step answer:
Given the four proportional numbers \[4:7::14:22\]
Consider the x be the number which is added to each of these number, so the new numbers become
First number\[=4+x\]
Second number\[=7+x\]
Third number\[=14+x\]
Fourth number\[=22+x\]
Now after adding an unknown number, the resulting numbers need to be in proportion so we can write these numbers as.
\[4+x:7+x::14+x:22+x\]
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.
\[\left( 4+x \right)\times \left( 22+x \right)=\left( 7+x \right)\times \left( 14+x \right)\]
By solving this we get:
\[4\times 22+4x+22x+{{x}^{2}}=7\times 14+7x+14x+{{x}^{2}}\]
By simplifying further we get:
\[88+26x=98+21x\]
By further solving this we get:
\[26x-21x=98-88\]
By simplifying further we get:
\[5x=10\]
Therefore, we get:
\[\therefore x=2\]
Hence we can say that if we add 3 to the number 4, 7, 14 and 22 the resulting numbers will be in proportion.
The resulting numbers are \[6:9::16:24\]

So, the correct answer is “Option B”.

Note:
When the ratio of the LHS of the proportions equals the RHS of the proportion, the numbers are proportionate. We simply find the ratios on both sides to see if the numbers are in proportion.
The given numbers \[4:7::14:22\]are not in proportion since their ratios are not equal
\[\dfrac{4}{7}\ne \dfrac{14}{22}\]. Even after reducing it is not equal, that means\[\dfrac{4}{7}\ne \dfrac{7}{11}\].