What is the least integer which when added to both terms of the ratio 5 : 9 will make a ratio greater than 7 : 10?

Options

A. 6
B. 8
C. 5
D. 7

Answer

To solve the problem, we need to find the least integer \( x \) such that when added to both terms of the ratio \( 5:9 \), the new ratio is greater than \( 7:10 \).

### Step-by-Step Solution:

1. **Set up the new ratio**:
We start with the ratio \( 5:9 \). If we add \( x \) to both terms, the new ratio becomes:
\[ \frac{5 + x}{9 + x} \]

2. **Set up the inequality**:
We want this new ratio to be greater than \( \frac{7}{10} \):
\[ \frac{5 + x}{9 + x} > \frac{7}{10} \]

3. **Cross-multiply to eliminate the fraction**:
Cross-multiplying gives us:
\[ 10(5 + x) > 7(9 + x) \]

4. **Distribute both sides**:
Expanding both sides results in:
\[ 50 + 10x > 63 + 7x \]

5. **Rearranging the inequality**:
To isolate \( x \), we subtract \( 7x \) from both sides:
\[ 10x - 7x > 63 - 50 \]
This simplifies to:
\[ 3x > 13 \]

6. **Solve for \( x \)**:
Dividing both sides by 3 gives:
\[ x > \frac{13}{3} \]
Calculating \( \frac{13}{3} \) gives approximately \( 4.33 \).

7. **Finding the least integer**:
Since \( x \) must be an integer greater than \( 4.33 \), the least integer satisfying this condition is:
\[ x = 5 \]

### Conclusion:
The least integer which when added to both terms of the ratio \( 5:9 \) will make a ratio greater than \( 7:10 \) is \( \boxed{5} \).