The least integer which when subtracted from the antecedent and added to the consequent of the ratio ` 9 : 8` gives a ratio less than the ratio ` 15 : 26` is

Options

A. 2
B. 3
C. 4
D. 1

Explanation

To solve the problem step by step, we need to find the least integer \( x \) such that when \( x \) is subtracted from the antecedent (numerator) of the ratio \( 9:8 \) and added to the consequent (denominator), the resulting ratio is less than \( 15:26 \).

### Step 1: Set up the equation
The original ratio is \( \frac{9}{8} \). When we subtract \( x \) from the numerator and add \( x \) to the denominator, we have:

\[ \frac{9 - x}{8 + x} \]

We need this ratio to be less than \( \frac{15}{26} \):

\[ \frac{9 - x}{8 + x} < \frac{15}{26} \]

### Step 2: Cross-multiply to eliminate the fractions
To solve the inequality, we can cross-multiply:

\[ (9 - x) \cdot 26 < (8 + x) \cdot 15 \]

### Step 3: Expand both sides
Expanding both sides gives:

\[ 234 - 26x < 120 + 15x \]

### Step 4: Combine like terms
Now, we will move all terms involving \( x \) to one side and constant terms to the other side:

\[ 234 - 120 < 15x + 26x \]

This simplifies to:

\[ 114 < 41x \]

### Step 5: Solve for \( x \)
Now, divide both sides by 41 to find \( x \):

\[ x > \frac{114}{41} \]

Calculating \( \frac{114}{41} \):

\[ \frac{114}{41} \approx 2.78 \]

### Step 6: Find the least integer greater than \( 2.78 \)
The least integer greater than \( 2.78 \) is \( 3 \).

### Conclusion
Thus, the least integer \( x \) which satisfies the condition is:

\[ \boxed{3} \]