Question

The angry Arjun carried some arrows for fighting with Bheeshm. With half the arrows, he cut down the arrows thrown by Bheeshm on him, and with six other arrows, he killed the rath of the driver of Bheeshm. With one arrow each, he knocked down respectively the rath, flag, and the bow of Bhishma. Finally, with one more than four times the square root of arrows, he laid Bheeshm unconscious on an arrow bed. Find the total number of arrows Arjun had.

Answer 1

Hint: First, let the number of total arrows as $x$. Then, form the equations according to the given conditions. Solve the equation and eliminate the value which contradicts the given statement.

Complete step-by-step answer:

First of all, let the number of arrows carried by Arjun be $x$.
Then, according to the question,
It is given that Arjun uses half the number of total arrows to cut down the arrows thrown by Bheeshm, hence it is given by $\dfrac{x}{2}$.
And we know that he has used 6 arrows to kill the rath of the driver of Bheeshm.
Next, he also used 1 arrow to knock down the rath,
And, 1 arrow is used to knock down the flag and 1 arrow for the bow of Bheeshm.
Next, he used one more than 4 times the square root of arrows he laid Bheeshm on an arrow bed, given by $1 + 4\sqrt x $
Therefore, when we add all the arrows used by Arjun, it will be equal to $x$
Hence, we can write the sum of all the arrows as,
$
  x = \dfrac{x}{2} + 6 + 1 + 1 + 1 + 1 + 4\sqrt x \\
  \Rightarrow x = \dfrac{x}{2} + 10 + 4\sqrt x \\
$
On solving the above equation, we get,
$
  x = \dfrac{x}{2} + 10 + 4\sqrt x \\
  \Rightarrow 2x = x + 20 + 8\sqrt x \\
 \Rightarrow x - 20 = 8\sqrt x \\
 $
On squaring both sides we get,
$
 \Rightarrow x - 20 = 8\sqrt x \\
 \Rightarrow {x^2} + 400 - 40x = 64x \\
 \Rightarrow {x^2} - 104x + 400 = 0 \\
$
Now, factorize the equation and solve for the value of \[x\].
$
 \Rightarrow {x^2} - 104x + 400 = 0 \\
 \Rightarrow {x^2} - 100x - 4x + 400 = 0 \\
 \Rightarrow x\left( {x - 100} \right) - 4\left( {x - 100} \right) = 0 \\
 \Rightarrow \left( {x - 4} \right)\left( {x - 100} \right) = 0 \\
 \Rightarrow x = 4,100 \\
$
One answer is 4 and the other is 100.
But, the total number of arrows cannot be 4 as Arjun used 6 arrows to kill the rath of the driver of Bheeshm, so the number will be more than 6.
Hence, Arjun had 100 arrows.

Note: While solving the value of $x$, squaring both sides should be done by keeping only the square-root term on one side to avoid difficult calculations. Also, the value which satisfies all the conditions should be considered.