Question

Which of the following arrangements of the numbers $ 75,4,3,100 $ forms a proportion?
(A) $ 100,3,75,4 $
(B) $ 3,4,75,100 $
(C) $ 3,100,4,75 $
(D) $ 3,75,100,4 $

Answer 1

Hint: A proportion $ a:b::c:d $ means $ \dfrac{a}{b} = \dfrac{c}{d}. $ Write the given numbers in such a way that they follow the given condition of proportion and the option that follows a proportion is the required.

Complete step-by-step answer:
A proportion $ a:b::c:d $ means, $ \dfrac{a}{b} = \dfrac{c}{d}. $
Which after cross multiplication can be written as $ ad = bc. $
Now, we have to arrange $ 75,4,3,100 $ by using the above property of proportion.
We would use the method of proof in reverse order to solve this question.
That is, we would start from the last step
 $ ad = bc $ ……….. (1)
And move up to the first step $ a:b::c:d. $
To get the required arrangement
From equation (1) we can observe that
We need to separate the $ 4 $ given numbers into two pairs such that their product is equal clearly.
 $ 75 \times 4 = 300 $ and $ 100 \times 3 = 300 $
Therefore, we can write
 $ 75 \times 4 = 100 \times 3 $
Which can now be written as
 $ \dfrac{{75}}{3} = \dfrac{{100}}{4} $ . . . (2)
Now, we can write this ratio in terms of proportion as
 $ 75:3::100:4 $
Which is not in any of the options.
So we would rearrange equation (2) as
 $ \dfrac{{75}}{{100}} = \dfrac{3}{4} $
We can further re-arrange it as
 $ \dfrac{3}{4} = \dfrac{{75}}{{100}} $
Which now can be written in proportion as
 $ 3:4::75:100 $ . . . (3)
Therefore, from the above explanation the correct option is (B) $ 3,4,75,100 $ .
So, the correct answer is “Option B”.

Note: We can have multiple arrangements for the same proportion. As you can see that equation (2) and equation (3) are the different arrangements of the same proportion.