Question

todd makes \\(\frac{1}{5}\\) of a liter of lemonade. he splits it equally between himself and his sister. his sister already had \\(3\frac{1}{10}\\) liters of lemonade. how many liters of lemonade does she have now? liters

Explanation:

Step1: Find the amount Todd's sister gets from the new lemonade.

Todd has $\frac{1}{5}$ liter and splits it equally between 2 people (himself and his sister), so we divide $\frac{1}{5}$ by 2. The formula for dividing a fraction by a whole number is $\frac{a}{b}\div c=\frac{a}{b\times c}$, so $\frac{1}{5}\div2 = \frac{1}{5\times2}=\frac{1}{10}$.

Step2: Add the amount she got to what she already had.

She already had $3\frac{1}{10}$ liters. To add $3\frac{1}{10}$ and $\frac{1}{10}$, we add the fractional parts first. The fractional part of $3\frac{1}{10}$ is $\frac{1}{10}$, so $\frac{1}{10}+\frac{1}{10}=\frac{2}{10}=\frac{1}{5}$. Then we add the whole number part, which is 3, so $3+\frac{1}{5}=3\frac{1}{5}$? Wait, no, wait. Wait, $3\frac{1}{10}+\frac{1}{10}=3\frac{2}{10}=3\frac{1}{5}$? Wait, no, $1/10 + 1/10 = 2/10 = 1/5$, so $3 + 1/5 = 3\frac{1}{5}$? Wait, no, let's do it correctly. $3\frac{1}{10}$ is $3 + \frac{1}{10}$, and we add $\frac{1}{10}$, so $3 + (\frac{1}{10}+\frac{1}{10}) = 3 + \frac{2}{10} = 3 + \frac{1}{5} = 3\frac{1}{5}$? Wait, no, $\frac{2}{10}$ simplifies to $\frac{1}{5}$, so yes. Wait, but let's check again. Todd makes $\frac{1}{5}$ liter, splits between 2, so each gets $\frac{1}{5}\times\frac{1}{2}=\frac{1}{10}$ liter. Then sister had $3\frac{1}{10}$, so adding $\frac{1}{10}$: $3\frac{1}{10}+\frac{1}{10}=3\frac{2}{10}=3\frac{1}{5}$. Wait, but $3\frac{2}{10}$ is also $3\frac{1}{5}$? Wait, no, $2/10$ is $1/5$, so yes. Wait, but maybe I made a mistake. Wait, $\frac{1}{5}$ divided by 2 is $\frac{1}{10}$, correct. Then $3\frac{1}{10}+\frac{1}{10}=3\frac{2}{10}=3\frac{1}{5}$. Wait, but let's convert to improper fractions. $3\frac{1}{10}=\frac{31}{10}$, and $\frac{1}{10}$ is $\frac{1}{10}$. So $\frac{31}{10}+\frac{1}{10}=\frac{32}{10}=\frac{16}{5}=3\frac{1}{5}$. Yes, that's correct.

Wait, but the problem's diagram shows a mixed number, so we need to present it as a mixed number. So $3\frac{1}{5}$? Wait, no, wait, $3\frac{1}{10}+\frac{1}{10}=3\frac{2}{10}=3\frac{1}{5}$? Wait, $2/10$ is $1/5$, so yes. So the total is $3\frac{1}{5}$? Wait, no, wait, $3\frac{1}{10} + \frac{1}{10} = 3\frac{2}{10} = 3\frac{1}{5}$. Yes.

Answer:

$3\frac{1}{5}$ (or in the boxed form as per the diagram, but since the diagram has a mixed number, it's $3\frac{1}{5}$ liters. Wait, but let's check again. Maybe I messed up the division. $\frac{1}{5}$ liter split between 2 people: each gets $\frac{1}{5} \div 2 = \frac{1}{10}$ liter. Then sister had $3\frac{1}{10}$, so adding $\frac{1}{10}$: $3\frac{1}{10} + \frac{1}{10} = 3\frac{2}{10} = 3\frac{1}{5}$. Yes, that's correct.