$3x-2:5x+6$ the duplicate ratio of $2:3$ then find the value $x$.

The correct answer is Option b: $6$. • The problem states that the ratio $3x-2:5x+6$ is the duplicate ratio of $2:3$. • First, let's understand what a duplicate ratio is. The duplicate ratio of $a:b$ is $a^2:b^2$. • So, the duplicate ratio of $2:3$ is $2^2:3^2$. • Calculating this, we get $2^2 = 4$ and $3^2 = 9$. So, the duplicate ratio is $4:9$. • Now, we are given that $3x-2:5x+6$ is equal to this duplicate ratio. Therefore, we can write the equation: $\frac{3x-2}{5x+6} = \frac{4}{9}$. • To solve for $x$, we cross-multiply: $9(3x-2) = 4(5x+6)$. • Distribute the numbers on both sides of the equation: $27x - 18 = 20x + 24$. • Now, we need to gather the $x$ terms on one side and the constant terms on the other side. Subtract $20x$ from both sides: $27x - 20x - 18 = 24$. $7x - 18 = 24$. • Add $18$ to both sides: $7x = 24 + 18$. $7x = 42$. • Finally, divide by $7$ to find the value of $x$: $x = \frac{42}{7}$. $x = 6$. • The correct answer is $6$. • Option (B) is $6$, which matches our calculated value for $x$. This is why it is the correct answer. • Option (A) is $2$. If $x=2$, then $\frac{3(2)-2}{5(2)+6} = \frac{6-2}{10+6} = \frac{4}{16} = \frac{1}{4}$. This is not equal to $\frac{4}{9}$. So, $2$ is incorrect. • Option (C) is $5$. If $x=5$, then $\frac{3(5)-2}{5(5)+6} = \frac{15-2}{25+6} = \frac{13}{31}$. This is not equal to $\frac{4}{9}$. So, $5$ is incorrect.

Ratio, Proportion, Indices, LogarithmPYQ Nov 18Question 788 of 211

All Questions

$\displaystyle 3x-2:5x+6$ the duplicate ratio of $\displaystyle 2:3$ then find the value $\displaystyle x$ .

Options

\displaystyle 2

\displaystyle 6

\displaystyle 5

\displaystyle 9

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Correct Answer

✅ Option b — $\displaystyle 6$

All Options:

A $\displaystyle 2$
B $\displaystyle 6$
C $\displaystyle 5$
D $\displaystyle 9$

Detailed Solution & Explanation

• The problem states that the ratio

\displaystyle 3x-2:5x+6

is the duplicate ratio of

\displaystyle 2:3

. • First, let's understand what a duplicate ratio is. The duplicate ratio of

\displaystyle a:b

\displaystyle a^2:b^2

. • So, the duplicate ratio of

\displaystyle 2:3

\displaystyle 2^2:3^2

. • Calculating this, we get

\displaystyle 2^2 = 4

and

\displaystyle 3^2 = 9

. So, the duplicate ratio is

\displaystyle 4:9

. • Now, we are given that

\displaystyle 3x-2:5x+6

is equal to this duplicate ratio. Therefore, we can write the equation:

\displaystyle \frac{3x-2}{5x+6} = \frac{4}{9}

. • To solve for

\displaystyle x

, we cross-multiply:

\displaystyle 9(3x-2) = 4(5x+6)

. • Distribute the numbers on both sides of the equation:

\displaystyle 27x - 18 = 20x + 24

. • Now, we need to gather the

\displaystyle x

terms on one side and the constant terms on the other side. Subtract

\displaystyle 20x

from both sides:

\displaystyle 27x - 20x - 18 = 24

\displaystyle 7x - 18 = 24

. • Add

\displaystyle 18

to both sides:

\displaystyle 7x = 24 + 18

\displaystyle 7x = 42

. • Finally, divide by

\displaystyle 7

to find the value of

\displaystyle x

\displaystyle x = \frac{42}{7}

\displaystyle x = 6

. • The correct answer is

\displaystyle 6

. • Option (B) is

\displaystyle 6

, which matches our calculated value for

\displaystyle x

. This is why it is the correct answer. • Option (A) is

\displaystyle 2

. If

\displaystyle x=2

, then

\displaystyle \frac{3(2)-2}{5(2)+6} = \frac{6-2}{10+6} = \frac{4}{16} = \frac{1}{4}

. This is not equal to

\displaystyle \frac{4}{9}

. So,

\displaystyle 2

is incorrect. • Option (C) is

\displaystyle 5

. If

\displaystyle x=5

, then

\displaystyle \frac{3(5)-2}{5(5)+6} = \frac{15-2}{25+6} = \frac{13}{31}

. This is not equal to

\displaystyle \frac{4}{9}

. So,

\displaystyle 5

is incorrect.

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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3x−2:5x+6\displaystyle 3x-2:5x+63x−2:5x+6 the duplicate ratio of 2:3\displaystyle 2:32:3 then find the value x\displaystyle xx.