Food for thought

## Suggestion

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?