To find when the two ice cream machines will have the same amount of ice cream left, we need to set up equations based on their initial capacities and their rates of sales over time.
Let's denote \(x\) as the number of days after which we want to find the remaining ice cream amounts.
-
Vanilla Ice Cream:
- Initial capacity: 180 cups
- Sells 20 cups per day
- Remaining after \(x\) days: \[ 180 - 20x \]
-
Chocolate Ice Cream:
- Initial capacity: 150 cups
- Sells 16 cups per day
- Remaining after \(x\) days: \[ 150 - 16x \]
Now set the remaining amounts equal to each other: \[ 180 - 20x = 150 - 16x \]
Now, we'll solve for \(x\).
Step 1: Rearrange the equation
Add \(20x\) to both sides and subtract \(150\) from both sides: \[ 180 - 150 = 20x - 16x \] \[ 30 = 4x \]
Step 2: Solve for \(x\)
Divide both sides by 4: \[ x = \frac{30}{4} = 7.5 \]
Conclusion
After 7.5 days, the two machines will have the same amount of ice cream left. Thus, the correct option would be: \[ 180 - 20(7.5) = 150 - 16(7.5) \]
Calculating both sides:
- Vanilla: \(180 - 150 = 30\) cups
- Chocolate: \(150 - 120 = 30\) cups
Both machines will have 30 cups remaining after 7.5 days.
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