Ratio, Proportion, Indices, LogarithmPYQ June 24Question 893 of 220
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If 2x=4y=8z\displaystyle 2^x = 4^y = 8^z and 12x+14y+16z=247thenthevalueof\displaystyle \frac{1}{2x} + \frac{1}{4y} + \frac{1}{6z} = \frac{24}{7} then the value ofz$ is:

Options

A716\displaystyle \frac{7}{16}
B732\displaystyle \frac{7}{32}
C748\displaystyle \frac{7}{48}
D764\displaystyle \frac{7}{64}
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Correct Answer

✅ Option c — 748\displaystyle \frac{7}{48}

All Options:

  • A716\displaystyle \frac{7}{16}
  • B732\displaystyle \frac{7}{32}
  • C748\displaystyle \frac{7}{48}
  • D764\displaystyle \frac{7}{64}

Detailed Solution & Explanation

We start with the given equality:
2x=4y=8z2^x = 4^y = 8^z
Since 4=22\displaystyle 4 = 2^2 and 8=23\displaystyle 8 = 2^3, we can rewrite the equation as:
2x=22y=23z2^x = 2^{2y} = 2^{3z}
Since the bases are identical, we equate the exponents:
x=2y=3zx = 2y = 3z
This allows us to express x\displaystyle x and y\displaystyle y in terms of z\displaystyle z:
- x=3z\displaystyle x = 3z
- 2y=3z  ⟹  y=3z2\displaystyle 2y = 3z \implies y = \frac{3z}{2}
Substitute these expressions into the given sum:
12x+14y+16z=247\frac{1}{2x} + \frac{1}{4y} + \frac{1}{6z} = \frac{24}{7}
12(3z)+14(3z2)+16z=247\frac{1}{2(3z)} + \frac{1}{4\left(\frac{3z}{2}\right)} + \frac{1}{6z} = \frac{24}{7}
16z+16z+16z=247\frac{1}{6z} + \frac{1}{6z} + \frac{1}{6z} = \frac{24}{7}
36z=247  ⟹  12z=247\frac{3}{6z} = \frac{24}{7} \implies \frac{1}{2z} = \frac{24}{7}
Solve for z\displaystyle z by cross-multiplying:
2z×24=7  ⟹  48z=7  ⟹  z=7482z \times 24 = 7 \implies 48z = 7 \implies z = \frac{7}{48}
This mathematically matches **Option C**.
Note: The textbook answer key incorrectly specifies **Option B** (732\displaystyle \frac{7}{32}) as correct. However, our rigorous algebraic solution clearly proves that z=748\displaystyle z = \frac{7}{48} (Option C).
Hence, **Option C** 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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