Maria can read 20 pages of economics in an hour. She can also read 50 pages of sociology in an hour. She spends 5 hours per day studying.
The correct answer and explanation is :
Let’s break down the information in the problem to figure out how Maria spends her study time.
Step 1: Understanding the Rates
- Maria can read 20 pages of economics per hour.
- Maria can read 50 pages of sociology per hour.
- She spends 5 hours per day studying.
Step 2: Maximizing Reading Time for Each Subject
Since Maria has 5 hours each day to study, we can calculate how much of each subject she might read depending on how she divides her study time.
Let’s say Maria decides to allocate some of her 5 hours to economics and some to sociology. We can represent the amount of time she spends on economics as ( t_E ) (in hours) and the amount of time spent on sociology as ( t_S ) (also in hours).
- Economics Pages Read: In ( t_E ) hours, she will read ( 20 \times t_E ) pages.
- Sociology Pages Read: In ( t_S ) hours, she will read ( 50 \times t_S ) pages.
Since she spends 5 hours per day studying, we know:
[
t_E + t_S = 5 \, \text{(total study time)}.
]
Step 3: Maximizing Pages Read
To find the optimal amount of time she should allocate to each subject, we aim to maximize the total number of pages she reads. The total pages read will be the sum of the pages read in economics and sociology:
[
\text{Total Pages Read} = 20 \times t_E + 50 \times t_S.
]
Substituting ( t_S = 5 – t_E ) into this equation (since ( t_E + t_S = 5 )):
[
\text{Total Pages Read} = 20 \times t_E + 50 \times (5 – t_E) = 20 \times t_E + 250 – 50 \times t_E = 250 – 30 \times t_E.
]
Step 4: Finding the Optimal Solution
To maximize the total pages read, we want to minimize ( 30 \times t_E ), which means we want to minimize the time spent on economics.
Thus, Maria should spend 0 hours on economics and 5 hours on sociology.
Step 5: Final Calculation
If she spends all 5 hours on sociology:
- Pages read in sociology = ( 50 \times 5 = 250 ) pages.
- Pages read in economics = ( 20 \times 0 = 0 ) pages.
Therefore, Maria reads 250 pages of sociology in 5 hours.
Conclusion:
Maria should spend all of her study time on sociology to maximize the number of pages she reads. This way, she will read 250 pages of sociology each day.