The common quantity that must be added to each term of `a^(2):b^(2)` to make it equal to a:b is

Options

A. ab
B. `a+b`
C. a-b
D. `(a)/(b)`

Answer

To solve the problem, we need to find the common quantity that must be added to each term of the ratio \( a^2 : b^2 \) to make it equal to the ratio \( a : b \).

### Step-by-Step Solution:

1. **Set Up the Problem**:
We start with the ratio \( a^2 : b^2 \). We want to add a common quantity \( x \) to both \( a^2 \) and \( b^2 \) such that the new ratio becomes \( a : b \).

\[ \frac{a^2 + x}{b^2 + x} = \frac{a}{b} \]

2. **Cross-Multiply**:
To eliminate the fraction, we cross-multiply:

\[ (a^2 + x) \cdot b = (b^2 + x) \cdot a \]

This leads to:

\[ a^2b + xb = b^2a + ax \]

3. **Rearranging the Equation**:
We can rearrange the equation to isolate terms involving \( x \):

\[ a^2b - b^2a = ax - xb \]

Factoring out \( x \) from the right side gives us:

\[ a^2b - b^2a = x(a - b) \]

4. **Solve for \( x \)**:
Now, we can solve for \( x \):

\[ x = \frac{a^2b - b^2a}{a - b} \]

5. **Factor the Numerator**:
The numerator can be factored:

\[ a^2b - b^2a = ab(a - b) \]

Thus, we can simplify \( x \):

\[ x = \frac{ab(a - b)}{a - b} \]

Since \( a \neq b \), we can cancel \( a - b \):

\[ x = ab \]

### Final Answer:
The common quantity that must be added to each term of \( a^2 : b^2 \) to make it equal to \( a : b \) is \( ab \).