If `a,b,c` are the percentage errors in the measurement of `A`, `B` and `C` , then percentage error in `ABC` would be approximately :

Options

A. (a)`abc`
B. (b)`a+b+c`
C. (c)`ab+bc+ac`
D. (d)`(a)/(b)+(b)/(c )+(c )/(a)`

Answer

To solve the problem of finding the percentage error in the product \( ABC \) given the percentage errors \( a, b, c \) in the measurements of \( A, B, C \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Percentage Error**:
The percentage error in a measurement is defined as:
\[ \text{Percentage Error} = \left( \frac{\Delta X}{X} \right) \times 100 \]
where \( \Delta X \) is the absolute error in the measurement of \( X \).

2. **Setting Up the Equations**:
For the measurements \( A, B, C \), we can express the percentage errors as:
- For \( A \):
\[ a = \left( \frac{\Delta A}{A} \right) \times 100 \]
- For \( B \):
\[ b = \left( \frac{\Delta B}{B} \right) \times 100 \]
- For \( C \):
\[ c = \left( \frac{\Delta C}{C} \right) \times 100 \]

3. **Finding the Absolute Errors**:
Rearranging the equations gives us the absolute errors:
- \( \Delta A = \frac{a}{100} A \)
- \( \Delta B = \frac{b}{100} B \)
- \( \Delta C = \frac{c}{100} C \)

4. **Calculating the Product**:
The product \( ABC \) has an absolute error given by:
\[ \Delta (ABC) = \Delta A \cdot B \cdot C + A \cdot \Delta B \cdot C + A \cdot B \cdot \Delta C \]

5. **Substituting the Absolute Errors**:
Substituting the expressions for \( \Delta A, \Delta B, \Delta C \):
\[ \Delta (ABC) = \left( \frac{a}{100} A \right) BC + A \left( \frac{b}{100} B \right) C + AB \left( \frac{c}{100} C \right) \]

6. **Factoring Out \( ABC \)**:
Factoring out \( ABC \) from the equation:
\[ \Delta (ABC) = ABC \left( \frac{a}{100} + \frac{b}{100} + \frac{c}{100} \right) \]

7. **Calculating the Percentage Error in \( ABC \)**:
Now, we can express the percentage error in \( ABC \):
\[ \text{Percentage Error in } ABC = \left( \frac{\Delta (ABC)}{ABC} \right) \times 100 = a + b + c \]

### Final Result:
Thus, the percentage error in the product \( ABC \) is approximately:
\[ \text{Percentage Error in } ABC \approx a + b + c \]