Problem Statement:
Nolan can eat a bag of popcorn in 2 minutes, and Madison can eat a bag of popcorn in 3 minutes. If Nolan and Madison share a bag of popcorn, how long will it take them to finish the bag together?
---
Solution:
#### Step 1: Understand the problem
-
Nolan's rate: Nolan can eat a bag of popcorn in 2 minutes. Therefore, his eating rate is:
$$
\text{Rate of Nolan} = \frac{1}{2} \text{ bags per minute}
$$
-
Madison's rate: Madison can eat a bag of popcorn in 3 minutes. Therefore, her eating rate is:
$$
\text{Rate of Madison} = \frac{1}{3} \text{ bags per minute}
$$
When Nolan and Madison work together, their combined rate is the sum of their individual rates. Let \( x \) be the time (in minutes) it takes for them to finish one bag of popcorn together.
#### Step 2: Set up the equation
The combined rate of Nolan and Madison is:
$$
\text{Combined rate} = \frac{1}{2} + \frac{1}{3}
$$
Since they finish one bag of popcorn together in \( x \) minutes, their combined rate can also be expressed as:
$$
\text{Combined rate} = \frac{1}{x}
$$
Equating the two expressions for the combined rate:
$$
\frac{1}{2} + \frac{1}{3} = \frac{1}{x}
$$
#### Step 3: Solve for \( x \)
First, find a common denominator for the fractions on the left-hand side. The least common denominator of 2 and 3 is 6:
$$
\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}
$$
Add the fractions:
$$
\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}
$$
So the equation becomes:
$$
\frac{5}{6} = \frac{1}{x}
$$
To solve for \( x \), cross-multiply:
$$
5x = 6
$$
Divide both sides by 5:
$$
x = \frac{6}{5}
$$
#### Step 4: Interpret the result
The value \( x = \frac{6}{5} \) represents the time in minutes it takes for Nolan and Madison to finish the bag of popcorn together. Converting this to a mixed number:
$$
\frac{6}{5} = 1 \frac{1}{5} \text{ minutes}
$$
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Final Answer:
$$
\boxed{\frac{6}{5}}
$$
This means it will take Nolan and Madison \( \frac{6}{5} \) minutes, or 1 minute and 12 seconds, to finish the bag of popcorn together.
Parent Tip: Review the logic above to help your child master the concept of rational equation word problems worksheet.
