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

Hint:

To quickly merge two ratios \( A : B = a : b \) and \( B : C = c : d \), you can write:
\[ A : B : C = (a \times c) : (b \times c) : (b \times d) \]
Applying this directly here:
\[ A : B : C = (2 \times 5) : (3 \times 5) : (3 \times 3) = 10 : 15 : 9 \]
This saves time during exams by skipping intermediate steps.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given two separate ratios of three numbers and their sum, which is 136.
Our objective is to combine these separate ratios into a single continuous ratio for the three numbers and then find the value of the first number.

Step 2: Key Formula or Approach:
Let the three numbers be \( A \), \( B \), and \( C \).
We are given the ratios \( A : B = 2 : 3 \) and \( B : C = 5 : 3 \).
To combine these into a single ratio \( A : B : C \), we make the terms corresponding to the common variable \( B \) equal in both ratios by finding their Least Common Multiple (LCM).

Step 3: Detailed Explanation:
1. Let the three numbers be \( A \), \( B \), and \( C \).
2. The ratio of the first number to the second number is:
\[ A : B = 2 : 3 \]
3. The ratio of the second number to the third number is:
\[ B : C = 5 : 3 \]
4. The term for \( B \) is 3 in the first ratio and 5 in the second ratio. The LCM of 3 and 5 is 15.
5. To make the \( B \) term equal to 15 in both ratios:
- Multiply both terms of the ratio \( A : B \) by 5:
\[ A : B = (2 \times 5) : (3 \times 5) = 10 : 15 \]
- Multiply both terms of the ratio \( B : C \) by 3:
\[ B : C = (5 \times 3) : (3 \times 3) = 15 : 9 \]
6. Combining these, we obtain the unified ratio:
\[ A : B : C = 10 : 15 : 9 \]
7. Let the actual values of the three numbers be \( 10x \), \( 15x \), and \( 9x \), where \( x \) is a constant multiplier.
8. We are given that the sum of these three numbers is 136:
\[ 10x + 15x + 9x = 136 \]
9. Combining the like terms:
\[ 34x = 136 \]
10. Solving for \( x \):
\[ x = \frac{136}{34} = 4 \]
11. Now, we compute the first number \( A \):
\[ A = 10x = 10 \times 4 = 40 \]

Step 4: Final Answer:
The value of the first number is 40, which corresponds to option (B).