Ratio, Proportion, Indices, LogarithmMCQMTP June 2023 Series IIQuestion 846 of 302
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If (x−9):(3x+6)\displaystyle (x-9):(3x+6) is the duplicate ratio of $4:9$, find the value of $x$.

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

✅ Option d — x=25\displaystyle x=25

All Options:

  • Ax=9\displaystyle x=9
  • Bx=16\displaystyle x=16
  • Cx=36\displaystyle x=36
  • Dx=25\displaystyle x=25

Detailed Solution & Explanation

The problem states that the ratio (x−9):(3x+6)\displaystyle (x-9):(3x+6) is the duplicate ratio of $4:9$. First, let us understand the definition of a duplicate ratio. The duplicate ratio of a:b\displaystyle a:b is defined as a2:b2\displaystyle a^2:b^2.
Applying this definition to $4:9$, the duplicate ratio of $4:9$ is 42:92\displaystyle 4^2:9^2. 42:92=16:814^2:9^2 = 16:81
According to the problem statement, (x−9):(3x+6)\displaystyle (x-9):(3x+6) is equal to this duplicate ratio. Therefore, we can set up the following equality: (x−9):(3x+6)=16:81(x-9):(3x+6) = 16:81
To solve for x\displaystyle x, we convert this ratio equality into a fractional equation. A ratio a:b\displaystyle a:b can be expressed as the fraction ab\displaystyle ab\displaystyle \frac{a}{b}. x−93x+6=1681x−93x+6=1681\displaystyle \frac{x-9}{3x+6} = \frac{16}{81}
Now, we can solve this equation by cross-multiplication. Multiply the numerator of the left side by the denominator of the right side, and vice versa. 81(x−9)=16(3x+6)81(x-9) = 16(3x+6)
Next, distribute the numbers on both sides of the equation: 81x−81×9=16×3x+16×681x - 81 ×\displaystyle \times 9 = 16 ×\displaystyle \times 3x + 16 ×\displaystyle \times 6 81x−729=48x+9681x - 729 = 48x + 96
Now, we need to gather all terms involving x\displaystyle x on one side of the equation and all constant terms on the other side. Subtract $48x$ from both sides: 81x−48x−729=9681x - 48x - 729 = 96 33x−729=9633x - 729 = 96
Add $729$ to both sides of the equation: 33x=96+72933x = 96 + 729 33x=82533x = 825
Finally, to find the value of x\displaystyle x, divide both sides by $33$: x=82533x = 82533\displaystyle \frac{825}{33}
To simplify the fraction, we can perform the division. Both $825\displaystyle and $33 are divisible by 3\displaystyle 3: x=825÷333÷3=27511x = 825÷333÷3=27511\displaystyle \frac{825 \div 3}{33 \div 3} = \frac{275}{11}
Now, divide $275\displaystyle by $11: 275÷11=25275 ÷\displaystyle \div 11 = 25 So, x=25x = 25

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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