Choose the largest of the following sums, and express it as a fraction in simplest form:
$$\frac{1}{4} + \frac{1}{5}, \ \ \frac{1}{4} + \frac{1}{6}, \ \ \frac{1}{4} + \frac{1}{3}, \ \ \frac{1}{4} + \frac{1}{8}, \ \ \frac{1}{4} + \frac{1}{7}$$
I notice that all of the sums have the same first term, $\frac{1}{4}$, so I can compare the second terms and see which one is the largest. I know that the smaller the denominator, the larger the fraction, so I look for the second term with the smallest denominator. I see that $\frac{1}{3}$ has the smallest denominator, so the sum $\frac{1}{4} + \frac{1}{3}$ is the largest. To express it as a fraction in simplest form, I need to find a common denominator for $\frac{1}{4}$ and $\frac{1}{3}$. I can use 12 as a common denominator, since it is a multiple of both 4 and 3. To get 12 in the denominator, I need to multiply the first fraction by $\frac{3}{3}$ and the second fraction by $\frac{4}{4}$. This gives me $\frac{3}{12} + \frac{4}{12}$. Now I can add the numerators and keep the denominator. This gives me $\frac{7}{12}$. 