The quantity that must be added to each term of a : b,so as to make it c:d,is

Options

A. `(ab-cd)/(a-b)`
B. `(ac + bd)/(c +a)`
C. `(ad + bc)/(c +d)`
D. `(ad - bc)/(c-d)`

Answer

To solve the problem of finding the quantity that must be added to each term of the ratio \( A : B \) to make it equal to the ratio \( C : D \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Quantity to be Added**:
Let the quantity that must be added to each term of the ratio \( A : B \) be denoted as \( X \).

2. **Set Up the New Ratios**:
After adding \( X \) to each term, the new ratio becomes \( (A + X) : (B + X) \).

3. **Set Up the Equation**:
We want this new ratio to be equal to \( C : D \). Therefore, we can set up the equation:
\[ \frac{A + X}{B + X} = \frac{C}{D} \]

4. **Cross-Multiply**:
To eliminate the fraction, we cross-multiply:
\[ (A + X) \cdot D = (B + X) \cdot C \]

5. **Expand Both Sides**:
Expanding both sides gives us:
\[ AD + XD = BC + XC \]

6. **Rearrange the Equation**:
We want to isolate \( X \). Rearranging the equation, we get:
\[ AD - BC = XC - XD \]

7. **Factor Out \( X \)**:
Now, factor \( X \) out from the right side:
\[ AD - BC = X(C - D) \]

8. **Solve for \( X \)**:
Finally, divide both sides by \( (C - D) \) to find \( X \):
\[ X = \frac{AD - BC}{C - D} \]

### Final Answer:
The quantity that must be added to each term of the ratio \( A : B \) to make it equal to \( C : D \) is:
\[ X = \frac{AD - BC}{C - D} \]