Math Problem Statement
A and B started working on a project and finished it in 5 days. If A had worked twice as efficiently and B worked at only half of his efficiency then the same job could have been competed in 4 days. A alone can could do it in how many days?
Solution
Let the work done by A in 1 day be and by B in 1 day be .
Scenario 1: They finish the project in 5 days
In 1 day, A and B together complete work. Since they finish the project in 5 days, the total work done is: This gives the equation:
Scenario 2: A works twice as efficiently, and B works at half efficiency
In this case, A does work per day, and B does work per day. They complete the project in 4 days, so: Simplifying:
Solving the equations
From Equation 1: From Equation 2: We now solve these equations simultaneously.
Step 1: Multiply Equation 1 by 2 to align terms with Equation 2:
Step 2: Subtract Equation 2 from Equation 3:
Simplifying: Multiply through by 2 to eliminate the fraction:
Step 3: Substitute into Equation 1:
Final Step: Find the time A alone takes to complete the work
If A does of the work per day, the time A alone would take to finish the project is:
Thus, A alone can complete the project in days.
Would you like more details, or have any questions?
Here are 5 related questions to expand on this:
- How many days would B alone take to complete the project?
- What if both A and B improved their efficiency by 50%? How long would it take them to finish the job?
- If A had worked thrice as efficiently, how would the total time to complete the project change?
- What is the work rate of A and B combined in the second scenario where efficiencies are altered?
- How would you model this situation if A and B’s efficiencies changed continuously over time?
Tip: To solve work problems, always relate efficiency to the total job, and try using simultaneous equations to break down complex scenarios.
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Math Problem Analysis
Mathematical Concepts
Work and Time
Efficiency
Simultaneous Equations
Formulas
Work done per day by A and B: a + b = Total work / Time
New efficiency: 2a + b/2 = Total work / New time
Time to complete work: Time = Total work / Rate
Theorems
Simultaneous Equations
Suitable Grade Level
Grades 9-12
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