Question

The ratio of two-digit natural numbers to the number obtained on reversing its digits is \[7:4\]. Find the sum of all possible numbers of such pairs.
A.310
B.330
C.328
D.332
E. none of these

Answer 1

Hint: In order to find the sum of all possible numbers, we must consider a two digits number at first. And then we must be considering the reversed number of the considered number and we must be representing it in the form of ratio and equate it to \[7:4\]. After equating, we should solve the obtained equation and obtain the values of the considered two-digit number. Then we must be checking out randomly by substituting the values accordingly in the number such that the condition satisfies. The sum of obtained numbers would be our required answer.

Complete step-by-step solution:
Let us have a brief regarding the ratios. A ratio is generally used to compare values between things. Ratios can also be written in the form of fraction and vice versa. Ratios can also be shown pictorially. When ratios of two different aspects are considered and the change between them is proportionate, then they are considered to be in proportion. Generally, ratios are written as shown- \[a:b\]. Two things can only be compared only when the units of the aspects being compared are identical or same.
Now let us start solving the problem.
Let us consider \[10a+b\] as our two-digit number.
Upon reversing the digits, we get \[10b+a\].
We are given the ratio as \[7:4\]
Upon expressing it numerically, we get \[\dfrac{10a+b}{10b+a}=\dfrac{7}{4}\]
Upon solving it, we get
\[\begin{align}
  & \Rightarrow \dfrac{10a+b}{10b+a}=\dfrac{7}{4} \\
 & \Rightarrow 4\left( 10a+b \right)=7\left( 10b+a \right) \\
 & \Rightarrow 40a+4b=70b+7a \\
 & \Rightarrow 33a=66b \\
 & \Rightarrow a=2b \\
\end{align}\]
So, we can see that if \[a=1\], then \[b\] would be \[2\].
Now let us check out the possible numbers.
If \[a=1\], then \[b=2\]. The number would be \[12\] and the reversed number would be \[21\].
If \[a=2\], then \[b=4\]. The number would be \[24\] and the reversed number would be \[42\].
If \[a=3\], then \[b=6\]. The number would be \[36\] and the reversed number would be \[63\].
If \[a=4\], then \[b=8\]. The number would be \[48\] and the reversed number would be \[84\].
After this the number forming would be a three digit number, so we will be stopping here.
Now let us find the sum of the numbers obtained, we get
\[12+21+24+36+42+48+63+84=330\]
\[\therefore \] Option B is the correct answer.

Note: The most common error while considering an unknown two-digit number could occur if we don’t multiply the number of tens placed with ten. We must note that even upon reversing, only the digits change but the place values remain the same.