Question

The sum of three numbers is 136. If the ratio between the first number and the second number is 2:3, and that between the second and the third number is 5:3, then the first number is:

• A. 42
• B. 40
• C. 36
• D. 32

Answer 1

The Correct Option is B

To solve the problem, we need to establish the relationship between the three numbers based on their given ratios and their sum. Let's denote the first number as \(x\), the second number as \(y\), and the third number as \(z\). We need to find \(x\). 

1. From the problem, we are given:

\(x + y + z = 136\)

The ratio of the first number to the second number is 2:3, meaning:

\( \frac{x}{y} = \frac{2}{3} \Rightarrow x = \frac{2}{3}y\)

1. Similarly, the ratio between the second and third number is 5:3, so:

\(\frac{y}{z} = \frac{5}{3} \Rightarrow z = \frac{3}{5}y\)

1. Substitute these values into the sum equation:

\(\frac{2}{3}y + y + \frac{3}{5}y = 136\)

1. Simplify and solve the equation:

\(\frac{2}{3}y + y + \frac{3}{5}y = 136\)

To clear the fractions, find a common denominator which is 15:

\(\frac{10}{15}y + \frac{15}{15}y + \frac{9}{15}y = 136\)

\(\frac{10y + 15y + 9y}{15} = 136\)

\(\frac{34y}{15} = 136\)

Multiply both sides by 15:

\(34y = 136 \times 15\)

\(34y = 2040\)

\(y = \frac{2040}{34}\)

\(y = 60\)

1. Now substitute \(y\) back to find \(x\):

\(x = \frac{2}{3}y = \frac{2}{3} \times 60 = 40\)

Thus, the first number is 40.

Answer 2

To find the first number, let's denote the three numbers as \( x \), \( y \), and \( z \). According to the problem, we have: 

• The sum of the numbers: \( x + y + z = 136 \).
• The ratio between the first and second number: \( \frac{x}{y} = \frac{2}{3} \) implies \( x = \frac{2}{3}y \).
• The ratio between the second and third number: \( \frac{y}{z} = \frac{5}{3} \) implies \( z = \frac{3}{5}y \).

We need to solve these equations simultaneously. Express \( x \), \( y \), and \( z \) in terms of \( y \):

• \( x = \frac{2}{3}y \)
• \( z = \frac{3}{5}y \)

Substitute these into the sum equation:

\( \frac{2}{3}y + y + \frac{3}{5}y = 136 \)

To combine these terms, find a common denominator. The least common multiple of 3, 1, and 5 is 15:

\( \frac{10}{15}y + \frac{15}{15}y + \frac{9}{15}y = 136 \)

\( \frac{34}{15}y = 136 \)

Multiply both sides by 15 to eliminate the fraction:

\( 34y = 136 \times 15 \)

\( 34y = 2040 \)

Divide by 34 to solve for \( y \):

\( y = \frac{2040}{34} \)

\( y = 60 \)

Now, find \( x \):

\( x = \frac{2}{3}y = \frac{2}{3} \times 60 = 40 \)

Thus, the first number is 40.