Question

The sum of three numbers is \[123\]. If the ratio between first and second number is \[2:5\] and that of between second and third is \[3:4\] then find the difference between second and third number.

Answer 1

Hint: To find the difference between second and third number, we need to assume some variables as the three numbers i.e., \[a,b\]and \[c\], and then solve with respect to the given ratios to get the value of \[a,b\] and \[c\], next we need to find the difference between the second and the third number.

Complete step by step answer:
Let the three numbers be \[a,b\]and \[c\].
Given that
\[a:b = 2:5\]
This can be written as:
\[ \Rightarrow a:b = 6:15\]
As per the second given condition i.e.,
\[b:c = 3:4\]
This can be written as:
\[ \Rightarrow b:c = 15:20\]
Hence from above equations
\[ \Rightarrow a:b:c = 6:15:20\]
Let us assume that:
\[a = 6x,b = 15x\] and \[c = 20x\]
Given, sum of all the three numbers \[ = 123\]
Hence, on adding all the three numbers we get,
\[a + b + c = 123\]
\[ \Rightarrow 6x + 15x + 20x = 123\]
\[ \Rightarrow 41x = 123\]
Simplifying the terms, we get:
\[ \Rightarrow x = \dfrac{{123}}{{41}}\]
\[ \Rightarrow x = 3\]
Hence, now substituting the values for \[a,b\] and \[c\] we get:
\[
  a = 6x \\
   \Rightarrow a = 6\left( 3 \right) \\
   \Rightarrow a = 18 \\
\]
\[
  b = 15x \\
   \Rightarrow b = 15\left( 3 \right) \\
   \Rightarrow b = 45 \\
\]
\[
  c = 20x \\
   \Rightarrow c = 20\left( 3 \right) \\
   \Rightarrow c = 60 \\
\]
 Therefore, the values of three numbers \[a,b\] and \[c\] are:
 \[a = 18\], \[b = 45\] and \[c = 60\].
The difference between the second and the third number is:
\[60 - 45 = 15\].

Note: To find any values for these types of statements we need to consider any random variables as the unknown term, next solving as per the statements stated we can find out the values of any number asked. If they are asked to find three numbers then consider \[a,b,c\]as the unknown variables and solve the sum.