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

Answer

To find the percentage error in the product of three quantities \( A \), \( B \), and \( C \), we can follow these steps:

### Step 1: Understand the Concept of Percentage Error
The percentage error in a measurement is given by the formula:
\[ \text{Percentage Error} = \left( \frac{\Delta X}{X} \right) \times 100 \]
where \( \Delta X \) is the absolute error in the measurement of \( X \).

### Step 2: Define the Variables
Let:
- \( a \) be the percentage error in the measurement of \( A \)
- \( b \) be the percentage error in the measurement of \( B \)
- \( c \) be the percentage error in the measurement of \( C \)

### Step 3: Relate the Errors in the Product
For the product \( ABC \), the absolute error \( \Delta X \) can be expressed in terms of the individual errors:
\[ \Delta X = X \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C} \right) \]
where \( X = ABC \).

### Step 4: Substitute the Errors
Substituting the definitions of \( \Delta A \), \( \Delta B \), and \( \Delta C \) in terms of their respective percentage errors:
\[ \frac{\Delta A}{A} = \frac{a}{100}, \quad \frac{\Delta B}{B} = \frac{b}{100}, \quad \frac{\Delta C}{C} = \frac{c}{100} \]

### Step 5: Combine the Errors
Now, substituting these into the equation for \( \Delta X \):
\[ \frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C} = \frac{a}{100} + \frac{b}{100} + \frac{c}{100} \]

### Step 6: Calculate the Percentage Error in \( ABC \)
Now, multiplying the entire equation by 100 gives us the percentage error in \( ABC \):
\[ \text{Percentage Error in } ABC = a + b + c \]

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