Find the smallest positive integer which must be subtracted from both the terms of ratio `6:7`, so that the result gives a ratio less than `16:21`.

Answer

To solve the problem step by step, we need to find the smallest positive integer \( x \) that must be subtracted from both terms of the ratio \( 6:7 \) so that the new ratio is less than \( 16:21 \).

### Step 1: Set up the inequality
We start with the original ratio \( 6:7 \). If we subtract \( x \) from both terms, the new ratio becomes \( (6 - x):(7 - x) \). We want this ratio to be less than \( 16:21 \).

This can be expressed as:
\[ \frac{6 - x}{7 - x} < \frac{16}{21} \]

### Step 2: Cross-multiply to eliminate the fraction
To eliminate the fractions, we cross-multiply:
\[ (6 - x) \cdot 21 < (7 - x) \cdot 16 \]

### Step 3: Expand both sides
Now we expand both sides:
\[ 126 - 21x < 112 - 16x \]

### Step 4: Rearrange the inequality
Next, we rearrange the terms to isolate \( x \):
\[ 126 - 112 < 21x - 16x \]
\[ 14 < 5x \]

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

### Step 6: Find the smallest integer greater than 2.8
Since \( x \) must be a positive integer, the smallest integer greater than \( 2.8 \) is \( 3 \).

### Conclusion
Thus, the smallest positive integer that must be subtracted from both terms of the ratio \( 6:7 \) so that the result gives a ratio less than \( 16:21 \) is:
\[ \boxed{3} \]