What quantity must be added to each term of the ratio x : y so that it may become equal to `c : d`?

Answer

To solve the problem of what quantity must be added to each term of the ratio \( x : y \) so that it becomes equal to \( c : d \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Quantity to be Added**:
Let the quantity to be added to both \( x \) and \( y \) be \( p \).

2. **Set Up the Equation**:
After adding \( p \) to both terms, the new ratio becomes:
\[ \frac{x + p}{y + p} = \frac{c}{d} \]

3. **Cross Multiply**:
To eliminate the fraction, we cross multiply:
\[ d(x + p) = c(y + p) \]

4. **Expand Both Sides**:
Distributing \( d \) and \( c \) gives:
\[ dx + dp = cy + cp \]

5. **Rearrange the Equation**:
To isolate terms involving \( p \), we rearrange the equation:
\[ dp - cp = cy - dx \]

6. **Factor Out \( p \)**:
Factor \( p \) from the left side:
\[ p(d - c) = cy - dx \]

7. **Solve for \( p \)**:
Finally, divide both sides by \( (d - c) \) to find \( p \):
\[ p = \frac{cy - dx}{d - c} \]

### Final Answer:
The quantity that must be added to each term of the ratio \( x : y \) to make it equal to \( c : d \) is:
\[ p = \frac{cy - dx}{d - c} \]