To solve the problem, we need to set up equations that represent the amount of ice cream left in each machine after a certain number of days.
Let \( x \) be the number of days that pass.
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The vanilla ice cream machine starts with a capacity of 180 cups and sells 20 cups per day. The amount of vanilla ice cream left after \( x \) days can be represented as: \[ V = 180 - 20x \]
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The chocolate ice cream machine starts with a capacity of 140 cups and sells 16 cups per day. The amount of chocolate ice cream left after \( x \) days can be represented as: \[ C = 140 - 16x \]
To find when the two machines have the same amount of ice cream left, we set these two equations equal to each other: \[ 180 - 20x = 140 - 16x \]
Now, we will solve for \( x \).
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Start by adding \( 20x \) to both sides: \[ 180 = 140 + 4x \]
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Next, subtract 140 from both sides: \[ 40 = 4x \]
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Finally, divide both sides by 4: \[ x = 10 \]
The two machines will have the same amount of ice cream left after 10 days.
To verify, let's calculate the amounts left after 10 days:
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For vanilla: \[ V = 180 - 20(10) = 180 - 200 = -20 , \text{(this indicates it is sold out before 10 days)} \]
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For chocolate: \[ C = 140 - 16(10) = 140 - 160 = -20 , \text{(this indicates it is also sold out before 10 days)} \]
Since both machines will run out of ice cream before reaching 10 days, we can conclude they will never have the same amount left because the vanilla machine will run out in 9 days (180/20 = 9) and the chocolate in 8.75 days (140/16 = 8.75).
Thus, the result from \( x = 10 \) indicates a point in time after both machines are emptied, confirming they can't have the same amount left due to their selling capacities and initial amounts.
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