Section B Practice Problems

1.  
   1. $\frac{7}{6}$ (or equivalent). Sample response: I just added the number of sixths, $5 +2$.
   2. $\frac{9}{6}$ (or equivalent). Sample response: I know that there are 2 sixths in each third, so $\frac{2}{3}$ is $\frac{4}{6}$, and then I have $5 + 4$ sixths. 
2. Sample response: Both are sums, and both have a $\frac{5}{6}$. Both sums can be written as sixths. For the second sum, I need to rewrite the thirds as sixths.

Sample responses:

1. There are 4 $\frac{1}{12}$s in each $\frac{1}{3}$ and $2 \times 4=8,$ so $\frac{2}{3}$ is the same as $\frac{8}{12}$.
2. I can just subtract twelfths and find $\frac{8}{12} - \frac{7}{12} = \frac{1}{12}$.

1. $\frac{9}{20}$ (or equivalent). Sample response: $\frac{1}{4} = \frac{5}{20}$, $\frac{1}{5} = \frac{4}{20}$, and $\frac{5}{20} + \frac{4}{20} = \frac{9}{20}$.
2. $\frac{13}{36}$ (or equivalent). Sample response: $\frac{10}{9} = \frac{40}{36}$, $\frac{3}{4}= \frac{27}{36}$, and $\frac{40}{36} - \frac{27}{36} = \frac{13}{36}$.

1. $2\frac{5}{12}$. Sample response: $\frac{3}{4} = \frac{9}{12}$ and $\frac{1}{3} = \frac{4}{12}$ so   $2\frac{9}{12} - \frac{4}{12} = 2\frac{5}{12}$.
2. $2\frac{17}{35}$. Sample response: $\frac{4}{5} + \frac{1}{5} = 1$, and then I need to add $2 \frac{2}{7}$ more to get $3\frac{2}{7}$. Then $\frac{1}{5} = \frac{7}{35}$ and $\frac{2}{7} = \frac{10}{35}$, so $\frac{7}{35} + \frac{10}{35} = \frac{17}{35}$.

1. $8 \frac{7}{24}$ cups (or equivalent). Sample response: I know $\frac{2}{3} + \frac{5}{8} =\frac{16}{24} + \frac{15}{24}$, so that’s $\frac{31}{24}$ or $1 \frac{7}{24}$, and I put that together with the $3 + 4$ or 7 whole cups.
2. $1\frac{1}{24}$ cups (or equivalent). Sample response: I found $4\frac{16}{24} - 3\frac{15}{24}$ by subtracting the whole numbers and subtracting the fractions.

Sample response:

1. $\frac{123}{104}$. Sample response: $\frac{7}{8} = \frac{7}{8}\times \frac{13}{13}$ and $\frac{4}{13} = \frac{4}{13}\times \frac{8}{8}$. The answer is $\frac{91}{104}+\frac{32}{104}$ or $\frac{123}{104}$.
2. $\frac{29}{40}$. Sample response: $\frac{7}{8} = \frac{35}{40}$ and $\frac{3}{20} = \frac{6}{40}$, so the difference is $\frac{29}{40}$.

1. 

2. $12 \frac{3}{8}$ inches. Sample response: $3 \times 4 = 12$ and $3 \times \frac{1}{8} = \frac{3}{8}$.

Sample response: There are 13 measurements on the line plot, so I am just missing 2. One measurement is another seedling of the shortest height, $\frac{1}{2}$ inch, and the other measurement is the tallest seedling, with a height of $2\frac{5}{8}$ inches. 

Sample response:

1. $\frac{2}{4} + \frac{3}{5} = \frac{22}{20}$. I need to use the 4 and the 5 for the denominators, or else my answer will be too big. The other possibility, $\frac{3}{4} + \frac{2}{5}$ is $\frac{1}{20}$ more.
2. $\frac{4}{3} - \frac{2}{5} = \frac{14}{15}$. I tried a lot of possibilities, and this difference is the closest. I can’t get a bigger denominator in my difference, except 20, using $4 \times 5$, but using 2 and 3 as the numerators does not get close to 1. The only way to do better would be to get 1 exactly.

Sample response:

I first chose my greatest and least measurements so that they added up to a whole number (in this case, 3) with $\frac{3}{8}$ more inches. That meant the rest of my 10 measurements needed to add up to 15 inches. I made half of those 1 inch and the other half 2 inches.