Question

Shyam ran in a race in his club. If one more runner had finished before him there would have been as many people in front of him as behind him. But if there had been one more runner finishing behind him there would have been three times as many behind him as in front of him. How many people took part in the race and at what position did Shyam finish?

• A. 7, 4th
• B. 9, 4th
• C. 3, 5th
• D. 4, 9th

Answer 1

The correct option is B

9, 4th

Step 1: Find the linear equation to obtain the required solution

Let $x$ people finish in front of Shyam.

Let $y$ people finish behind Shyam.

It is given that if one more runner had finished before him, there would have been as many people in front of him as behind him.

$\begin{array}{rcl}x+1& =& y-1\\ x-y+2& =& 0...\left(1\right)\end{array}$

Also, if there had been one more runner finishing behind him, there would have been three times as many behind him as in front of him.

$\begin{array}{rcl}& \Rightarrow & \left(y+1\right)=3\left(x-1\right)\\ & \Rightarrow & y+1=3x-3\\ 3x-y-4& =& 0...\left(2\right)\end{array}$

Step 2: Solve the above system to obtain the required solution.

Subtracting equation (1) from equation (2), we get

$\begin{array}{rcl}\left(3x-y\right)-\left(x-y\right)& =& 4-\left(-2\right)\\ & \Rightarrow & 2x=6\\ & \Rightarrow & x=3\end{array}$

Substituting the value of $x$ in equation (1), we get

$\begin{array}{rcl}x-y+2& =& 0\\ & \Rightarrow & 3-y+2=0\\ & \Rightarrow & y=5\end{array}$

Hence, there are 3 people finishing in front of Shyam and 5 people behind him.

$\therefore$There is a total of 9 people in the race, and Shyam completes the race in the 4th position.

Hence, the correct answer is option B.