Ratio, Proportion, Indices, LogarithmMCQMTP June 22Question 913 of 302
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If (25)150=(25x)50\displaystyle (25)^{150} = (25x)^{50}, then the value of x\displaystyle x will be:

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — 54\displaystyle 5^4

All Options:

  • A53\displaystyle 5^3
  • B54\displaystyle 5^4
  • C52\displaystyle 5^2
  • D5\displaystyle 5

Detailed Solution & Explanation

We are given the equation:
(25)150=(25x)50\left( 25 \right)^{150} = \left( 25x \right)^{50}

To isolate the term with x\displaystyle x, let us raise both sides of the equation to the power of 150\displaystyle 150\displaystyle \frac{1}{50}:
[(25)150]150=[(25x)50]150\left[ \left( 25 \right)^{150} \right]^{150\displaystyle \frac{1}{50}} = \left[ \left( 25x \right)^{50} \right]^{150\displaystyle \frac{1}{50}}
Applying the exponent law (ab)c=abc\displaystyle \left(a^b\right)^c = a^{bc}:
2515050=25x25^{15050\displaystyle \frac{150}{50}} = 25x
253=25x25^3 = 25x

Dividing both sides of the equation by $25\displaystyle :<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><msup><mn>25</mn><mn>3</mn></msup><mn>25</mn></mfrac><mo>=</mo><msup><mn>25</mn><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x = 25325=252\displaystyle \frac{25^3}{25} = 25^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">25</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord">2</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><br><br>We know that &#36;25 = 5^2. Let us express x\displaystyle x with a base of 5\displaystyle 5:
x=(52)2=52×2=54x = \left( 5^2 \right)^2 = 5^{2 ×\displaystyle \times 2} = 5^4

Hence, **Option B** is the correct answer.

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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