Question

Which‌ ‌of‌ ‌the‌ ‌following‌ ‌are‌ ‌in‌ ‌proportion?‌ ‌
(A)‌ ‌‌\[2,3,20,30\]‌ ‌
(B)‌ ‌‌\[3,4,15,18\]‌ ‌
(C)‌ ‌‌\[1,3,11,22\]‌ ‌
(D)‌ ‌‌\[2,5,40,80\]‌

Answer 1

Hint: We use the concepts of ratios, relative ratios and proportions to solve this problem. A ratio is generally defined as comparison between two or more quantities of the same units. It also tells us how much the first quantity is present in the second quantity. And proportion is the comparison between two ratios.

Complete step-by-step solution:
Generally, proportion is referred to as part, share, or number considered in comparative relation to a whole. When two ratios are equivalent, then they are said to be in proportion. Its symbol is “ \[::\] ” or “ \[ = \] ”.
Consider four numbers \[a,b,c{\text{ and }}d\] .
So, if \[a:b\] and \[c:d\] are equivalent, then these are said to be in proportion.
\[ \Rightarrow a:b::c:d\]
Now, in option (A), the numbers given are \[2,3,20,30\] .
So, here, \[a:b = 2:3\] and \[c:d = 20:30\] which in simplification is equal to \[2:3\] .
So, the two ratios are equivalent. So, they are in proportion.
In option (B), the ratios are \[3:4\] and \[15:18\] which is equal to \[5:6\]
So, here the ratios are not equivalent. So, they are not in proportion.
In option (C), the ratios are \[1:3\] and \[11:22\] which on simplification is equal to \[1:2\]
And here also, the ratios are not equivalent, so, they are not proportional.
In option (D), the ratios are \[2:5\] and \[1:2\] (on simplification of \[40:80\] )
So, here the ratios are not equivalent, so they are not in proportion.
So, option (A) is the correct option.

Note: A ratio can also be written in fraction form i.e. \[a:b = \dfrac{a}{b}\] , so, if two fractions are equal, then they are said to be proportional. Also make a note that, a ratio is always simplified to its least.
If \[\dfrac{a}{b} = \dfrac{c}{d}\] , then we can conclude that those are in proportion and we can further extend it as \[ad = bc\] , by cross multiplying the fractions.