Chapter 2.4, Problem 20P

Suppose you start with a proper fraction and you add 1 to both the numerator and the denominator.

For example, if you started with $\frac{2}{3}$ , then you’d get a new fraction $\frac{2+1}{3+1}=\frac{3}{4}.$

a. Give at least 5 examples of proper fractions $\frac{A}{B}$ . In each example, compare the sizes of $\frac{A}{B}$ and $\frac{A+1}{B+1}$ What do you notice? Make sure you are working with proper fractions (where the numerator is less than the denominator).

b. Frank says that if $\frac{A}{B}$ B is a proper fraction, then $\frac{A+1}{B+1}$ is always greater than $\frac{A}{B}$ because $\frac{A+1}{B+1}$ has more parts.

Regardless of whether Frank’s conclusion is correct, discuss whether Frank’s reasoning is valid. Did Frank give a convincing explanation that $\frac{A+1}{B+1}$ is greater than $\frac{A}{B}?$ If not, what objections could you make to Frank’s reasoning?

c. Explain the phenomenon you discovered in part (a). Compare the sizes of $\frac{A}{B}$ and $\frac{A+1}{B+1}$ Explain why the fraction you say is larger really is larger.