Two fractions are shown below.

\[\textrm{Fraction} \ A: \frac{10^a +1}{10^a +2} \qquad \textrm{Fraction} \ B: \frac{10^b + 1}{10^b + 2}\]

For what values of \(a\) and \(b\) will fraction A be greater than fraction B?

What happens when you substitute values in place of \(a\) and \(b\)?

Think about \(a=0\) and \(b=1\), giving the fractions \[A: \frac{10^0 +1}{10^0 +2} \qquad B: \frac{10^1 +1}{10^1 +2}.\]

Which fraction has the bigger result in this situation?

What happens when you substitute different values in place of \(a\) and \(b\)?

- Which fraction has the bigger result in each case?

How do the numerators and denominators of the two fractions compare?

What is the same and what’s different?

How could you use any similarities to help you to simplify the problem?

Could you take an approach based on inequalities?