If a, b, and c are the distinct positive integers, and 10% of abc is 5, then which of the following is a possible value of a+b?

Options

A. `3`
B. `5`
C. `8`
D. `25`

Answer

To solve the problem step by step, we need to find distinct positive integers \( a, b, \) and \( c \) such that \( 10\% \) of \( abc = 5 \).

### Step 1: Set up the equation from the percentage.
We know that \( 10\% \) of \( abc \) can be expressed mathematically as:
\[ \frac{10}{100} \times abc = 5 \]

### Step 2: Simplify the equation.
This simplifies to:
\[ \frac{1}{10} \times abc = 5 \]
Multiplying both sides by \( 10 \) gives:
\[ abc = 5 \times 10 = 50 \]

### Step 3: Factorize 50 into distinct positive integers.
Next, we need to find distinct positive integers \( a, b, \) and \( c \) such that their product is \( 50 \). We can start by listing the factor pairs of \( 50 \):
- \( 1 \times 50 \)
- \( 2 \times 25 \)
- \( 5 \times 10 \)

### Step 4: Identify combinations of factors.
We need to find combinations of three distinct integers. Let's consider the factorization of \( 50 \):
1. \( 1, 2, 25 \) (distinct)
2. \( 1, 5, 10 \) (distinct)

### Step 5: Calculate possible values of \( a + b \).
Now we can calculate \( a + b \) for the distinct combinations:
1. For \( 1, 2, 25 \):
- If we take \( a = 1 \) and \( b = 2 \), then \( a + b = 1 + 2 = 3 \).
- If we take \( a = 1 \) and \( b = 25 \), then \( a + b = 1 + 25 = 26 \).
- If we take \( a = 2 \) and \( b = 25 \), then \( a + b = 2 + 25 = 27 \).

2. For \( 1, 5, 10 \):
- If we take \( a = 1 \) and \( b = 5 \), then \( a + b = 1 + 5 = 6 \).
- If we take \( a = 1 \) and \( b = 10 \), then \( a + b = 1 + 10 = 11 \).
- If we take \( a = 5 \) and \( b = 10 \), then \( a + b = 5 + 10 = 15 \).

### Step 6: Determine the possible values of \( a + b \).
From the combinations we calculated:
- From \( 1, 2, 25 \): Possible values are \( 3, 26, 27 \).
- From \( 1, 5, 10 \): Possible values are \( 6, 11, 15 \).

### Conclusion:
The only possible value of \( a + b \) that matches the options provided in the question is \( 3 \).

Thus, the answer is:
\[ \boxed{3} \]