To find out when Miguel and Christina will have equal amounts of gas in their cars, we need to set up an equation based on their initial capacities and their rates of gas consumption.
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Initial Capacities:
- Miguel's car capacity: 20 gallons
- Christina's car capacity: 16 gallons
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Consumption Rates:
- Miguel uses 1.5 gallons per week.
- Christina uses 1 gallon per week.
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Gas Amount After \( t \) Weeks:
- After \( t \) weeks, the amount of gas remaining in Miguel's car will be: \[ 20 - 1.5t \]
- The amount of gas remaining in Christina's car will be: \[ 16 - 1t \]
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Setting the Equations Equal: To find out when they have equal amounts of gas, we set the two expressions equal to each other: \[ 20 - 1.5t = 16 - 1t \]
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Solving for \( t \): Rearranging the equation: \[ 20 - 16 = 1.5t - 1t \] \[ 4 = 0.5t \] \[ t = \frac{4}{0.5} = 8 \]
Thus, Miguel and Christina will have equal amounts of gas in 8 weeks.
- Finding the Amount of Gas: To find out how much gas they will have at that time, we can plug \( t = 8 \) back into either of the initial expressions.
For Miguel: \[ 20 - 1.5 \times 8 = 20 - 12 = 8 \text{ gallons} \]
For Christina: \[ 16 - 1 \times 8 = 16 - 8 = 8 \text{ gallons} \]
Both will have 8 gallons of gas in their tanks after 8 weeks.