If 2x = 4y = 8z and 1/2x + 1/4y + 1/6z = 24/7 , then the value of z is:

Option c: 7/48. Solution: We start with the given equality: 2x = 4y = 8z Since 4 = 22 and 8 = 23 , we can rewrite the equation as: 2x = 22y = 23z Since the bases are identical, we equate the exponents: x = 2y = 3z This allows us to express x and y in terms of z : - x = 3z - $2y = 3z y = 3z/2 Substitute these expressions into the given sum: 1/2x + 1/4y + 1/6z = 24/7 1/2(3z) + 1/4(3z2) + 1/6z = 24/7 1/6z + 1/6z + 1/6z = 24/7 3/6z = 24/7 1/2z = 24/7 Solve for z by cross-multiplying: 2z × 24 = 7 48z = 7 z = 7/48 This mathematically matches Option C. Note: The textbook answer key incorrectly specifies Option A ( 7/16 ) as correct. However, our rigorous algebraic solution clearly proves that z = 7/48$ (Option C). Hence, Option C is the correct answer.

If 2x = 4y = 8z and 1/2x + 1/4y + 1/6z = 24/7 , then the value of z is:

The correct answer is Option c: 7/48. We start with the given equality: 2x = 4y = 8z Since 4 = 22 and 8 = 23 , we can rewrite the equation as: 2x = 22y = 23z Since the bases are identical, we equate the exponents: x = 2y = 3z This allows us to express x and y in terms of z : - x = 3z - $2y = 3z y = 3z/2 Substitute these expressions into the given sum: 1/2x + 1/4y + 1/6z = 24/7 1/2(3z) + 1/4(3z2) + 1/6z = 24/7 1/6z + 1/6z + 1/6z = 24/7 3/6z = 24/7 1/2z = 24/7 Solve for z by cross-multiplying: 2z × 24 = 7 48z = 7 z = 7/48 This mathematically matches Option C. Note: The textbook answer key incorrectly specifies Option A ( 7/16 ) as correct. However, our rigorous algebraic solution clearly proves that z = 7/48$ (Option C). Hence, Option C is the correct answer.

Ratio, Proportion, Indices, LogarithmMCQPYQ Dec 23Question 890 of 302

All Questions

If $\displaystyle 2^x = 4^y = 8^z$ and $\displaystyle$ , then the value of $\displaystyle z$ is:

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

For any discrepancies in this question, email contact@cadada.in

Correct Answer

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

All Options:

A $\displaystyle \frac{7}{16}$
B $\displaystyle \frac{7}{32}$
C $\displaystyle \frac{7}{48}$
D $\displaystyle \frac{7}{64}$

Detailed Solution & Explanation

We start with the given equality:

2^x = 4^y = 8^z

Since

\displaystyle 4 = 2^2

and

\displaystyle 8 = 2^3

, we can rewrite the equation as:

2^x = 2^{2y} = 2^{3z}

Since the bases are identical, we equate the exponents:

x = 2y = 3z

This allows us to express

\displaystyle x

and

\displaystyle y

in terms of

\displaystyle z

:
-

\displaystyle x = 3z

- $2y = 3z \implies y =

\displaystyle \frac{3z}{2}<span

class="katex">

\displaystyle <br>Substitute these expressions into the given sum:<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><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>6</mn><mi>z</mi></mrow></mfrac><mo>=</mo><mfrac><mn>24</mn><mn>7</mn></mfrac></mrow><annotation encoding="application/x-tex">

\displaystyle by cross-multiplying:<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><mn>2</mn><mi>z</mi><mo>×</mo><mn>24</mn><mo>=</mo><mn>7</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mn>48</mn><mi>z</mi><mo>=</mo><mn>7</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>z</mi><mo>=</mo><mfrac><mn>7</mn><mn>48</mn></mfrac></mrow><annotation encoding="application/x-tex">2z

\displaystyle \frac{7}{16}

$) as correct. However, our rigorous algebraic solution clearly proves that $z =

\displaystyle \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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If 2x=4y=8z\displaystyle 2^x = 4^y = 8^z2x=4y=8z and 12x+14y+16z=247\displaystyle 12x+14y+16z=247\displaystyle \frac{1}{2x} + \frac{1}{4y} + \frac{1}{6z} = \frac{24}{7}2x1​+4y1​+6z1​=724​2x1​+4y1​+6z1​=724​, then the value of z\displaystyle zz is: