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The problem states:

**Given that \( b \) is a positive number and \( a < b \), show that**
\[
\frac{a}{b} < \frac{a+1}{b+1}.
\]

### To prove this inequality:

1. Start with the given inequality that we need to prove:
   \[
   \frac{a}{b} < \frac{a+1}{b+1}.
   \]

2. Cross-multiply to eliminate the denominators, but be careful with the direction of the inequality since all terms are positive (due to the condition \( b > 0 \)):
   \[
   a(b+1) < (a+1)b.
   \]

3. Expand both sides:
   \[
   ab + a < ab + b.
   \]

4. Cancel out \( ab \) from both sides:
   \[
   a < b.
   \]

5. This is the original condition given in the problem, \( a < b \), so the inequality holds.

### Therefore, the inequality \( \frac{a}{b} < \frac{a+1}{b+1} \) is true whenever \( a < b \) and \( b > 0 \).

Would you like any further details or have any questions?

Here are 5 related questions:
1. How does the condition \( b > 0 \) ensure that cross-multiplication does not change the inequality?
2. Can the inequality be generalized for cases where \( a = b \)?
3. How would the inequality change if \( a > b \)?
4. Can this type of inequality be extended to more complex fractions?
5. What happens to the inequality when \( a \) and \( b \) are both negative?

**Tip**: Always consider the sign of variables when working with inequalities, as it affects the direction of the inequality when multiplying or dividing both sides.

Show that if b is a positive number and a < b, then a/b < (a + 1)/(b + 1).

Learn how to prove the inequality a/b < (a + 1)/(b + 1) given that a < b and b > 0. This step-by-step guide explores key algebraic concepts, including cross-multiplication and properties of inequalities. Suitable for Grades 9-11.

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