Question 7: Working Together vs. Working Alone: Finding Individual Times

Objective:

To learn how to calculate the time taken by an individual (A in this case) to complete a task when working with others.

Key Concepts:

1. Work Rate Formula: The rate at which a person or group works is the reciprocal of the time taken to complete the task. If an individual takes T days to finish the work, their rate is:

   Rate of work=1T\text{Rate of work} = \frac{1}{T}Rate of work=T1​

   When multiple people work together, their combined work rate is the sum of their individual work rates.

2. Working Together: When two people, A and B, work together, their combined rate is the sum of their individual rates:

   $$\text{Rate of A + B}=\text{Rate of A}+\text{Rate of B}$$

   From this combined rate, we can find the individual time required for A or B to complete the work alone.

Given Information:

• A and B together can complete the work in 15 days, so their combined rate is: Rate of A + B=115\text{Rate of A + B} = \frac{1}{15}Rate of A + B=151​
• B alone can complete the work in 20 days, so B’s rate is: Rate of B=120\text{Rate of B} = \frac{1}{20}Rate of B=201​

Step-by-Step Solution:

1. Find A’s Rate: To find A’s rate, we subtract B’s rate from the combined rate of A + B:

   $$\text{Rate of A}=\text{Rate of A + B}-\text{Rate of B}$$

   Substitute the given values:

   Rate of A=115−120\text{Rate of A} = \frac{1}{15} – \frac{1}{20}Rate of A=151​−201​

2. Subtract the Fractions: To subtract these fractions, we need to find a common denominator. The least common denominator of 15 and 20 is 60. Rewrite the fractions:

   Rate of A=460−360=160\text{Rate of A} = \frac{4}{60} – \frac{3}{60} = \frac{1}{60}Rate of A=604​−603​=601​

   Therefore, A’s rate is 160\frac{1}{60}601​.

3. Find A’s Time to Complete the Work: Since A’s rate is 160\frac{1}{60}601​, it means A will take 60 days to complete the work alone.

Final Answer:

A alone will complete the work in 60 days.