The smallest integer, which subtracted from both the terms of `6 : 7` gives a ratio less than `16 : 21`. is :

Options

A. 5
B. 4
C. 3
D. 2

Answer

To solve the problem of finding the smallest integer that, when subtracted from both terms of the ratio \(6:7\), results in a ratio less than \(16:21\), we can follow these steps:

### Step 1: Understand the Ratios
First, we need to convert the ratios into fractions to make comparisons easier. The ratio \(6:7\) can be expressed as:
\[ \frac{6}{7} \]
And the ratio \(16:21\) can be expressed as:
\[ \frac{16}{21} \]

### Step 2: Set Up the Inequality
We need to find an integer \(x\) such that:
\[ \frac{6 - x}{7 - x} < \frac{16}{21} \]

### Step 3: Cross Multiply to Eliminate the Fractions
To solve the inequality, we cross-multiply:
\[ (6 - x) \cdot 21 < (7 - x) \cdot 16 \]

### Step 4: Expand Both Sides
Now, we will expand both sides of the inequality:
\[ 126 - 21x < 112 - 16x \]

### Step 5: Rearrange the Inequality
Next, we will rearrange the inequality to isolate \(x\):
\[ 126 - 112 < 21x - 16x \]
\[ 14 < 5x \]

### Step 6: Solve for \(x\)
Now, divide both sides by 5:
\[ \frac{14}{5} < x \]
This simplifies to:
\[ x > 2.8 \]

### Step 7: Find the Smallest Integer
Since \(x\) must be an integer, the smallest integer greater than \(2.8\) is \(3\).

### Conclusion
Thus, the smallest integer that can be subtracted from both terms of the ratio \(6:7\) to make it less than \(16:21\) is:
\[ \boxed{3} \]