A does 80% of a work in 20 days. He then calls in B and they together finish the remaining work in 3 days. How long B alone would take to do the whole work? 

A does 80% of a work in 20 days. He then calls in B and they together finish the remaining work in 3 days. How long B alone would take to do the whole work?

B alone would take 37 1/2 days to do the whole work.

Here's how to find how long B alone would take to do the whole work:

Step 1: Calculate the remaining work after A finishes.

• A does 80% of the work, so the remaining work is 100% - 80% = 20%.

Step 2: Find the combined rate of A and B.

• We know they finish the remaining 20% of the work in 3 days.
• To find their combined rate, divide the work completed (20%) by the time taken (3 days):
  • Combined rate = (20% / 3 days) = 1/15 per day

Step 3: Find A's rate.

• We know A completes 80% of the work in 20 days.
• To find his rate, divide the work completed (80%) by the time taken (20 days):
  • A's rate = (80% / 20 days) = 1/25 per day

Step 4: Find B's rate.

• Now, subtract A's rate from the combined rate to find B's rate:
  • B's rate = Combined rate - A's rate = (1/15 per day) - (1/25 per day) = 2/75 per day

Step 5: Calculate the time B takes to do the whole work.

• Since B's rate is the amount of work he can do per day, divide 1 (representing the whole work) by B's rate to find the time he takes:
  • Time taken by B = 1 / (2/75 per day) = 75/2 days = 37.5 days

Therefore, B alone would take 37 1/2 days to do the whole work.

Work and Time in Mathematics

In mathematics, work and time problems often deal with the relationship between the amount of work done, the time taken to do that work, and the rate at which the work is completed. These problems can involve scenarios such as multiple workers collaborating to complete a task, a single worker completing a task over time, or machines performing work at different rates.

Here are some common concepts and formulas used in work and time problems:

1. Work Formula: Work (W) = Rate (R) × Time (T)

2. Rate Formula: Rate (R) = Work (W) ÷ Time (T)

3. Time Formula: Time (T) = Work (W) ÷ Rate (R)

4. Combined Work: When multiple workers are involved, their combined work rate is the sum of their individual work rates.

5. Inverse Proportionality: Sometimes, the work done is inversely proportional to the time taken, meaning that as one increases, the other decreases. This relationship is expressed mathematically as RT = k, where R is the rate, T is the time, and k is a constant.

6. Examples:

   • If it takes 6 hours for a single worker to complete a task, their rate is 1/6 of the job per hour.
   • If two workers working together can complete a task in 4 hours, their combined rate is 1/4 of the job per hour.

When solving work and time problems, it's important to clearly define variables, such as the rate at which work is done, the time taken to complete the task, and the total amount of work to be done. Substituting these values into the appropriate formulas and solving for the unknown variables can help find solutions to various work and time problems.