A student finishes the first half of an exam in [tex]$\frac{2}{3}$[/tex] the time it takes him to finish the second half. If the entire exam takes him an hour, how many minutes does he spend on the first half of the exam?
(A) 20
(B) 24
(C) 27
Answer (2)
Define t 1 as the time for the first half and t 2 as the time for the second half.
Write the equations: t 1 = 3 2 t 2 and t 1 + t 2 = 60 .
Substitute t 1 in the second equation: 3 2 t 2 + t 2 = 60 .
Solve for t 2 and then t 1 : t 1 = 24 minutes.
The student spends 24 minutes on the first half of the exam.
Explanation
Define variables and write the given equation Let t 1 be the time spent on the first half of the exam and t 2 be the time spent on the second half of the exam. We are given that the student finishes the first half of the exam in 3 2 the time it takes him to finish the second half. This can be written as: t 1 = 3 2 t 2
Write the equation for the total time We also know that the entire exam takes the student 1 hour, which is 60 minutes. Therefore, the sum of the time spent on the first half and the second half is 60 minutes: t 1 + t 2 = 60
Substitute the first equation into the second equation Now we have a system of two equations with two variables. We can substitute the first equation into the second equation to solve for t 2 :
3 2 t 2 + t 2 = 60
Combine the terms Combine the terms with t 2 :
3 2 t 2 + 3 3 t 2 = 3 5 t 2 = 60
Solve for t_2 Multiply both sides of the equation by 5 3 to solve for t 2 :
t 2 = 60 × 5 3 = 5 180 = 36
Solve for t_1 Now that we have the value of t 2 , we can find the value of t 1 using the first equation: t 1 = 3 2 t 2 = 3 2 × 36 = 2 × 12 = 24
Final Answer Therefore, the student spends 24 minutes on the first half of the exam.
Examples
Understanding how to manage time effectively is crucial in many real-life situations, such as project management, cooking, or even planning a road trip. This problem demonstrates how to break down a task into smaller parts and allocate time accordingly. For example, if you know that one part of a project takes twice as long as another, you can use similar equations to plan your schedule and ensure you complete everything on time. By understanding these concepts, you can improve your time management skills and become more efficient in your daily life.
The student spends 24 minutes on the first half of the exam, which is determined by setting up equations based on the time relationship between the two halves of the exam. By solving these simultaneous equations, we find that t 1 = 24 minutes. Therefore, the answer is option (B) 24.
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