sophia runs $2\frac{3}{10}$ miles before school and $1\frac{9}{10}$ miles after school. select all of the equations that you can use to find how far sophia runs. $\square\ 1\frac{9}{10}+2\frac{3}{10}=3\frac{2}{10}$ $\square\ \frac{19}{10}+\frac{23}{10}=\frac{42}{10}$ $\square\ \frac{19}{10}+\frac{23}{10}=\frac{42}{20}$ $\square\ 1\frac{9}{10}+\frac{4}{10}=2\frac{3}{10}$ $\square\ 1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$

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# Brief Explanations:
To find the total distance Sophia runs, we need to add the distance she runs before school ($2\frac{3}{10}$ miles) and after school ($1\frac{9}{10}$ miles).

1. First, convert the mixed numbers to improper fractions:
   - $2\frac{3}{10}=\frac{2\times10 + 3}{10}=\frac{23}{10}$
   - $1\frac{9}{10}=\frac{1\times10+9}{10}=\frac{19}{10}$
2. Now, add the improper fractions: $\frac{19}{10}+\frac{23}{10}=\frac{19 + 23}{10}=\frac{42}{10}$
3. We can also add the mixed numbers directly:
   - Add the whole numbers: $2 + 1 = 3$
   - Add the fractions: $\frac{3}{10}+\frac{9}{10}=\frac{12}{10}=1\frac{2}{10}$
   - Combine the whole numbers and fractions: $3+1\frac{2}{10}=4\frac{2}{10}$? Wait, no, wait. Wait, $2\frac{3}{10}+1\frac{9}{10}$:
     - Whole numbers: $2 + 1 = 3$
     - Fractions: $\frac{3}{10}+\frac{9}{10}=\frac{12}{10}=1\frac{2}{10}$
     - Then total is $3 + 1\frac{2}{10}=4\frac{2}{10}$. Wait, but let's check the improper fractions:
       - $2\frac{3}{10}=\frac{23}{10}$, $1\frac{9}{10}=\frac{19}{10}$
       - $\frac{23}{10}+\frac{19}{10}=\frac{42}{10}=4\frac{2}{10}$ (since $42\div10 = 4$ with remainder $2$).
       - Wait, earlier when I added the mixed numbers, I made a mistake. Let's correct that:
         - $2\frac{3}{10}+1\frac{9}{10}=(2 + 1)+(\frac{3}{10}+\frac{9}{10})=3+\frac{12}{10}=3 + 1\frac{2}{10}=4\frac{2}{10}$. Yes, that's correct.
       - And $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$, which simplifies to $4\frac{2}{10}$.
       - Now let's check each option:
         - Option 1: $1\frac{9}{10}+2\frac{3}{10}=3\frac{2}{10}$ → Incorrect, because we saw it's $4\frac{2}{10}$.
         - Option 2: $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$ → Correct, because $\frac{19 + 23}{10}=\frac{42}{10}$.
         - Option 3: $\frac{19}{10}+\frac{23}{10}=\frac{42}{20}$ → Incorrect, because when adding fractions with the same denominator, we add the numerators, not the denominators. So denominator remains 10, not 20.
         - Option 4: $1\frac{9}{10}+\frac{4}{10}=2\frac{3}{10}$ → Let's check: $1\frac{9}{10}+\frac{4}{10}=1+\frac{9}{10}+\frac{4}{10}=1+\frac{13}{10}=1 + 1\frac{3}{10}=2\frac{3}{10}$. Wait, but this is not related to finding the total distance Sophia runs (which is before + after), but this equation is correct in itself (since $1\frac{9}{10}+\frac{4}{10}=2\frac{3}{10}$), but is it an equation to find how far she runs? No, because she runs $2\frac{3}{10}$ before and $1\frac{9}{10}$ after, so total is before + after, not after + some fraction = before. So this equation is mathematically correct but not for finding the total distance. But the question is "Select all of the equations that you can use to find how far Sophia runs." So we need equations that represent total distance = before + after.
         - Option 5: $1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$ → Correct, because that's the sum of the two distances.
         - Wait, let's re - evaluate:
           - The total distance is $2\frac{3}{10}+1\frac{9}{10}$ (or $1\frac{9}{10}+2\frac{3}{10}$).
           - Converting to improper fractions: $\frac{23}{10}+\frac{19}{10}=\frac{42}{10}$ (Option 2 is correct).
           - And $1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$ (Option 5 is correct), because $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}=4\frac{2}{10}$.
           - Wait, and what about Option 4: $1\frac{9}{10}+\frac{4}{10}=2\frac{3}{10}$. Let's check: $1\frac{9}{10}+\frac{4}{10}=1+\frac{9 + 4}{10}=1+\frac{13}{10}=2\frac{3}{10}$. This is a correct equation, but is it used to find the total distance? No, because the total distance is $2\frac{3}{10}+1\frac{9}{10}$, not $1\frac{9}{10}+\frac{4}{10}$. So Option 4 is a correct equation but not for finding the total distance.
           - Wait, maybe I made a mistake earlier. Let's re - express all:
             - Sophia's total distance $D = 2\frac{3}{10}+1\frac{9}{10}$.
             - Let's compute $2\frac{3}{10}+1\frac{9}{10}$:
               - As improper fractions: $\frac{23}{10}+\frac{19}{10}=\frac{42}{10}$ (Option 2: $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$ is correct, since addition is commutative).
               - As mixed numbers: $2\frac{3}{10}+1\frac{9}{10}=(2 + 1)+(\frac{3}{10}+\frac{9}{10})=3+\frac{12}{10}=3 + 1\frac{2}{10}=4\frac{2}{10}$ (Option 5: $1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$ is correct).
               - Option 4: $1\frac{9}{10}+\frac{4}{10}=2\frac{3}{10}$. Let's check: $1\frac{9}{10}+\frac{4}{10}=1+\frac{9 + 4}{10}=1+\frac{13}{10}=2\frac{3}{10}$. This is a correct equation, but it's not for finding the total distance. It's an equation that relates $1\frac{9}{10}$, $\frac{4}{10}$, and $2\frac{3}{10}$, but not for finding the total (which is $2\frac{3}{10}+1\frac{9}{10}$).
             - Wait, maybe the diagram is helpful. The first diagram has two red bars (each labeled 1) and three yellow $\frac{1}{10}$ bars, which is $2+\frac{3}{10}=2\frac{3}{10}$. The second diagram has one red bar (labeled 1) and nine yellow $\frac{1}{10}$ bars, which is $1+\frac{9}{10}=1\frac{9}{10}$. To find the total, we add these two: $2\frac{3}{10}+1\frac{9}{10}$.
             - Now, let's re - check the options:
               - Option 1: $1\frac{9}{10}+2\frac{3}{10}=3\frac{2}{10}$ → Incorrect, because $3\frac{2}{10}$ is not equal to $4\frac{2}{10}$.
               - Option 2: $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$ → Correct, since $\frac{19 + 23}{10}=\frac{42}{10}$.
               - Option 3: $\frac{19}{10}+\frac{23}{10}=\frac{42}{20}$ → Incorrect, denominator should be 10, not 20.
               - Option 4: $1\frac{9}{10}+\frac{4}{10}=2\frac{3}{10}$ → Let's see, $1\frac{9}{10}+\frac{4}{10}=1+\frac{9 + 4}{10}=1+\frac{13}{10}=2\frac{3}{10}$. This is a correct equation, but is it used to find the total? No, but maybe the question is "select all equations that you can use to find how far Sophia runs". Maybe this equation is part of a different approach? Wait, no, the total is $2\frac{3}{10}+1\frac{9}{10}$. But if we know that $2\frac{3}{10}=1\frac{9}{10}+\frac{4}{10}$, then substituting into the total: $D=(1\frac{9}{10}+\frac{4}{10})+1\frac{9}{10}=1\frac{9}{10}+1\frac{9}{10}+\frac{4}{10}$, which is not helpful. So probably Option 4 is not for finding the total.
               - Option 5: $1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$ → Correct, as we saw.
             - Wait, I think I made a mistake in the first addition of mixed numbers. Let's do it again:
               - $2\frac{3}{10}+1\frac{9}{10}$:
                 - Add the whole numbers: $2 + 1 = 3$
                 - Add the fractions: $\frac{3}{10}+\frac{9}{10}=\frac{12}{10}=1\frac{2}{10}$
                 - Now add the results: $3+1\frac{2}{10}=4\frac{2}{10}$. Yes, that's correct. And $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}=4\frac{2}{10}$, so Option 2 and Option 5 are correct? Wait, no, Option 2 is $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$, which is correct, and Option 5 is $1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$, which is also correct. Wait, but let's check the arithmetic:
                   - $1\frac{9}{10}=\frac{19}{10}$, $2\frac{3}{10}=\frac{23}{10}$
                   - $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$, and $42\div10 = 4$ with remainder $2$, so $\frac{42}{10}=4\frac{2}{10}$. So $1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$ (Option 5) and $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$ (Option 2) are both correct.
                   - Wait, but earlier I thought Option 1 was incorrect, which it is, because $3\frac{2}{10}
eq4\frac{2}{10}$. Option 3 is incorrect because $\frac{42}{10}
eq\frac{42}{20}$. Option 4: $1\frac{9}{10}+\frac{4}{10}=2\frac{3}{10}$ is correct as an equation, but is it used to find the total distance? The total distance is $2\frac{3}{10}+1\frac{9}{10}$, so if we use $2\frac{3}{10}=1\frac{9}{10}+\frac{4}{10}$, then we can write the total as $1\frac{9}{10}+(1\frac{9}{10}+\frac{4}{10})$, but that's more complicated. So probably the intended correct options are Option 2 ($\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$) and Option 5 ($1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$)? Wait, but let's check the original problem again.

# Answer:
The correct options are:
- $\boldsymbol{\frac{19}{10}+\frac{23}{10}=\frac{42}{10}}$ (second option)
- $\boldsymbol{1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}}$ (fifth option)

Wait, but when I first calculated, I thought Option 5 was correct, and Option 2 is also correct. Let's verify with the diagram:

- The first diagram represents $2\frac{3}{10}$ (2 whole units + 3 tenths)
- The second diagram represents $1\frac{9}{10}$ (1 whole unit + 9 tenths)
- To find the total, we add them: $2\frac{3}{10}+1\frac{9}{10}$

Converting to improper fractions: $2\frac{3}{10}=\frac{23}{10}$, $1\frac{9}{10}=\frac{19}{10}$, so $\frac{23}{10}+\frac{19}{10}=\frac{42}{10}$ (Option 2 is $\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$, which is the same due to commutativity of addition).

As mixed numbers: $2\frac{3}{10}+1\frac{9}{10}=(2 + 1)+(\frac{3}{10}+\frac{9}{10})=3+\frac{12}{10}=3 + 1\frac{2}{10}=4\frac{2}{10}$, so $1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$ (Option 5) is correct.

Option 4: $1\frac{9}{10}+\frac{4}{10}=2\frac{3}{10}$ is a correct equation, but it's not used to find the total distance (it's a relation between $1\frac{9}{10}$, $\frac{4}{10}$, and $2\frac{3}{10}$, not the sum of the two distances Sophia runs).

So the correct options are the second one ($\frac{19}{10}+\frac{23}{10}=\frac{42}{10}$) and the fifth one ($1\frac{9}{10}+2\frac{3}{10}=4\frac{2}{10}$).

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