Question

1. the model below can be used to find the sum of two mixed numbers. what is the sum? what is the difference? show your work.

Explanation:

Step1: Identify the mixed numbers

First mixed number: The red bar is 1, and the blue parts are 8 of $\frac{1}{10}$, so it's $1 + \frac{8}{10}=1\frac{4}{5}$ (simplifying $\frac{8}{10}$ to $\frac{4}{5}$).
Second mixed number: Two red bars (each 1) and three green parts of $\frac{1}{3}$, so it's $2 + \frac{3}{3}=3$? Wait, no, looking again: Wait, the second model: red bars are two 1s? Wait, no, maybe the first model: red is 1, blue is 8 tenths, so $1+\frac{8}{10}=1\frac{4}{5}$. The second model: red is two 1s? Wait, no, the green parts: three $\frac{1}{3}$? Wait, $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1$, and then two red 1s? Wait, no, maybe the second mixed number is $2 + \frac{3}{3}$? No, $\frac{3}{3}=1$, so $2 + 1 = 3$? Wait, no, maybe I misread. Wait, first model: red bar (1) plus 8 tenths: $1+\frac{8}{10}=1\frac{4}{5}$. Second model: two red bars (each 1) and three thirds: $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1$, so total is $2 + 1 = 3$? Wait, no, maybe the second mixed number is $2\frac{3}{3}$? But $\frac{3}{3}=1$, so $2 + 1 = 3$. Wait, maybe the second mixed number is $2\frac{1}{3}+\frac{1}{3}+\frac{1}{3}$? No, $\frac{1}{3}\times3 = 1$, so $2 + 1 = 3$. Now, let's check again.

Wait, maybe the first mixed number is $1\frac{8}{10}=1\frac{4}{5}$, and the second is $2\frac{3}{3}$? No, $\frac{3}{3}=1$, so $2 + 1 = 3$. Wait, maybe the second mixed number is $2\frac{1}{3}\times3$? No, let's do it correctly.

Wait, first model: 1 (red) + 8*(1/10) = 1 + 8/10 = 1 + 4/5 = 1 4/5.

Second model: 2 (red bars? Wait, two red bars, each 1, so 2, and green parts: three 1/3, which is 1, so total 2 + 1 = 3. Wait, but 3 is a whole number.

Step2: Calculate the sum

Sum: $1\frac{4}{5}+3$. Convert $1\frac{4}{5}$ to improper fraction: $\frac{9}{5}$, 3 is $\frac{15}{5}$, so $\frac{9}{5}+\frac{15}{5}=\frac{24}{5}=4\frac{4}{5}$. Wait, no, maybe I misidentified the second number. Wait, maybe the second mixed number is $2\frac{3}{3}$? No, $\frac{3}{3}=1$, so 2 + 1 = 3. Alternatively, maybe the second mixed number is $2\frac{1}{3}\times3$? No, let's re - examine the models.

Wait, maybe the first mixed number is $1\frac{8}{10}=1.8$ and the second is $2\frac{3}{3}=3$? No, $\frac{3}{3}=1$, so 2 + 1 = 3. Then sum is $1.8 + 3 = 4.8 = 4\frac{4}{5}$.

Difference: $3 - 1\frac{4}{5}$. Convert 3 to $\frac{15}{5}$ and $1\frac{4}{5}$ to $\frac{9}{5}$, so $\frac{15}{5}-\frac{9}{5}=\frac{6}{5}=1\frac{1}{5}$.

Wait, maybe the second mixed number is $2\frac{3}{3}$ is wrong. Wait, maybe the second model: red is two 1s, and green is three $\frac{1}{3}$, so $2+\frac{3}{3}=3$. The first is $1+\frac{8}{10}=1\frac{4}{5}$.

Step3: Recalculate if needed

Wait, maybe the second mixed number is $2\frac{3}{3}$ is incorrect. Let's see: the green parts are three $\frac{1}{3}$, so $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1$, and the red parts: two 1s? So total is $2 + 1 = 3$. The first number: red 1, blue 8 tenths: $1+\frac{8}{10}=1\frac{4}{5}$.

Step4: Sum calculation

$1\frac{4}{5}+3$. Convert $1\frac{4}{5}$ to improper fraction: $1\frac{4}{5}=\frac{5 + 4}{5}=\frac{9}{5}$. 3 is $\frac{15}{5}$. So $\frac{9}{5}+\frac{15}{5}=\frac{24}{5}=4\frac{4}{5}$.

Step5: Difference calculation

$3 - 1\frac{4}{5}$. Convert 3 to $\frac{15}{5}$ and $1\frac{4}{5}$ to $\frac{9}{5}$. Then $\frac{15}{5}-\frac{9}{5}=\frac{6}{5}=1\frac{1}{5}$.

Answer:

Sum: $4\frac{4}{5}$
Difference: $1\frac{1}{5}$