Question

The sum of three numbers is \[98.\] The ratio of the first to the second is $ 2:3 $ and the ratio of the second to the third is $ 5:8. $ Find the second number.
A. $ 15 $
B. $ 20 $
C. $ 30 $
D. $ 32 $

Answer 1

Hint: First of all any three variables for the three numbers, then convert the word statements in the form of mathematical expressions. Find the correlation between them and find the required number.

Complete step-by-step answer:
Complete step by step solution:
Let us assume that three numbers are
First number be $ = x, $
Second number be $ = y $ and
Third number be $ = z $
Given that sum of three numbers is $ 98 $
 $ \therefore x + y + z = 98 $ .... (A)
Also, given that the ratio of the first to the second number is $ 2:3 $
 $ \therefore \dfrac{x}{y} = \dfrac{2}{3} $
Cross multiplication implies –
 $ \Rightarrow x = \dfrac{{2y}}{3} $ .... (B)
Also, given that the ratio of the second number to the third is $ 5:8. $
 $ \therefore \dfrac{y}{z} = \dfrac{5}{8} $
The above expression can be re-written as –
 $ \Rightarrow \dfrac{z}{y} = \dfrac{8}{5} $
Cross multiplication implies –
 $ \Rightarrow z = \dfrac{{8y}}{5} $ .... (C)
Place the values of equation (B) and (C) in (A)
 $ \therefore x + y + z = 98 $
 \[ \Rightarrow \dfrac{{2y}}{3} + y + \dfrac{{8y}}{5} = 98\]
Take LCM (Least common multiple) on the left hand side of the equation –
 \[ \Rightarrow \dfrac{{2(5)y}}{{3(5)}} + \dfrac{{15y}}{{15}} + \dfrac{{8(3)y}}{{5(3)}} = 98\]
Simplify the above expression –
 \[ \Rightarrow \dfrac{{10y}}{{15}} + \dfrac{{15y}}{{15}} + \dfrac{{24y}}{{15}} = 98\]
When the denominators are equal we can add the numerator part of the fraction –
 \[ \Rightarrow \dfrac{{10y + 15y + 24y}}{{15}} = 98\]
Simplify the above equation, and do cross-multiplication –
 \[
   \Rightarrow \dfrac{{49y}}{{15}} = 98 \\
   \Rightarrow 49y = 98 \times 15 \;
 \]
Make the required variable the subject, take the multiplicative term on the opposite side. Remember when any term in multiplicative at one side changes its side then it goes in the denominator.
 \[ \Rightarrow y = \dfrac{{98 \times 15}}{{49}}\]
Find the factors of the terms in the numerator.
 \[ \Rightarrow y = \dfrac{{49 \times 2 \times 15}}{{49}}\]
Common factors from the numerator and the denominator cancel each other.
 \[
   \Rightarrow y = 2 \times 15 \\
   \Rightarrow y = 30 \;
 \]
Therefore, the second number is $ 30 $
So, the correct answer is “Option C”.

Note: Be careful while you are simplifying the equations. Always remember that when any term multiplicative at one side is moved to the opposite side, then it goes in the division and vice-versa. Be clear with the least common multiple and highest common factor and apply accordingly.