If $\sqrt{x}$% of $x$ is 80, then what is $x$?
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Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC CISF-AC-EXE – 2024
“$\sqrt{x}$%” means $\frac{\sqrt{x}}{100}$.
“of $x$” means multiply by $x$.
“is 80” means equals 80.
So the equation is:
$\frac{\sqrt{x}}{100} \times x = 80$.
Rewrite $\sqrt{x}$ as $x^{1/2}$ and $x$ as $x^1$:
$\frac{x^{1/2} \times x^1}{100} = 80$.
Using the rule of exponents $a^m \times a^n = a^{m+n}$, combine the terms with x:
$\frac{x^{1/2 + 1}}{100} = 80$
$\frac{x^{3/2}}{100} = 80$.
Multiply both sides by 100:
$x^{3/2} = 80 \times 100 = 8000$.
To solve for x, raise both sides of the equation to the power of (2/3), which is the reciprocal of 3/2:
$(x^{3/2})^{2/3} = (8000)^{2/3}$.
$x^1 = (8000)^{2/3}$.
$(8000)^{2/3}$ can be calculated as $(8000^{1/3})^2$ or $(8000^2)^{1/3}$. It’s usually easier to find the cube root first.
We need to find the cube root of 8000. $8000 = 8 \times 1000 = 2^3 \times 10^3 = (2 \times 10)^3 = 20^3$.
So, $8000^{1/3} = 20$.
Now, square the result:
$x = (20)^2 = 400$.
Let’s verify the answer: $\sqrt{400} = 20$. 20% of 400 = $\frac{20}{100} \times 400 = 0.20 \times 400 = 80$. The result matches the given condition.