Question

Three numbers are in the ratio 2:5:7. If the first number, the resulting number on the subtraction of 7 from the second number and the third number form an arithmetic sequence, then find the numbers?

Answer 1

Hint: Assume the three numbers as 2x, 5x and 7x respectively. Now, use the information given in the question and subtract 7 from the second number to get the three numbers in A.P. Use the formula for the arithmetic mean of three term in A.P given as $b=\dfrac{a+c}{2}$ where a, b and c are three numbers in A.P. Solve the linear equation in x and substitute in the expressions 2x, 5x and 7x to get the answers.

Complete step by step answer:
Here we have been provided with the three numbers in ratio of 2:5:7 and we are asked to find these numbers with the help of the given information.
Now, the three numbers in the ratio 2:5:7 can be assumed as 2x, 5x and 7x respectively. It is given that the first number, 7 subtracted from the second number and the third number forms and A.P, so we have 7 subtracted from the second number equal to 5x – 7. Therefore, 2x, (5x – 7) and 7x are in A.P.
We know that when three numbers a, b and c are in A.P then the formula of arithmetic mean of a and c is given as $b=\dfrac{a+c}{2}$, so in the above case 5x – 7 will be the arithmetic mean between 2x and 7x. So we get,
$\begin{align}
  & \Rightarrow 5x-7=\dfrac{2x+7x}{2} \\
 & \Rightarrow 5x-7=\dfrac{9x}{2} \\
\end{align}$
By cross multiplication we get,
$\begin{align}
  & \Rightarrow 10x-14=9x \\
 & \Rightarrow 10x-9x=14 \\
 & \Rightarrow x=14 \\
\end{align}$
Therefore, the three numbers can be obtained by substituting the value of x in 2x, 5x and 7x respectively, so we get,
$\therefore 2x=28$, $5x=70$ and $7x=98$
Hence, the three numbers are 28, 70 and 98.

Note: You must remember the formulas of the arithmetic mean, geometric mean and the harmonic mean between the two numbers. Note that you have to be careful while selecting the middle number to calculate the arithmetic mean in the above question. You may see that the middle number mentioned is 5x – 7 and therefore it is the mean of two boundary numbers 2x and 7x.