A and B can do a piece of work in 16 and 15 days. If they were ill they could do 80 percent and 60% of their efficiency respectively. How many days will they take to completely the w work

To solve this problem, let's first determine the efficiency of A and B under normal conditions and then calculate their reduced efficiency when they are ill. Finally, we'll find the combined time they would take to complete the work together under reduced efficiency.

### Step 1: Determine the normal work rates of A and B

- A can complete the work in 16 days.
- B can complete the work in 15 days.

The work rate (efficiency) of a person is defined as the fraction of work they can complete in one day.

\[
\text{Rate of A} = \frac{1}{16} \quad (\text{work per day})
\]
\[
\text{Rate of B} = \frac{1}{15} \quad (\text{work per day})
\]

### Step 2: Calculate their reduced efficiencies due to illness

When ill, their efficiencies are reduced to 80% and 60% of their normal efficiencies respectively:

\[
\text{Reduced Rate of A} = 0.8 \times \frac{1}{16} = \frac{0.8}{16} = \frac{8}{160} = \frac{1}{20} \quad (\text{work per day})
\]

\[
\text{Reduced Rate of B} = 0.6 \times \frac{1}{15} = \frac{0.6}{15} = \frac{6}{150} = \frac{2}{50} = \frac{1}{25} \quad (\text{work per day})
\]

### Step 3: Calculate the combined reduced rate

The combined work rate of A and B when they are ill is the sum of their individual reduced rates:

\[
\text{Combined Rate} = \frac{1}{20} + \frac{1}{25}
\]

To add these fractions, find a common denominator. The least common multiple (LCM) of 20 and 25 is 100:

\[
\text{Combined Rate} = \frac{5}{100} + \frac{4}{100} = \frac{9}{100}
\]

### Step 4: Calculate the time taken to complete the work

Since \(\frac{9}{100}\) of the work is completed per day, the number of days required to complete the entire work is the reciprocal of the combined rate:

\[
\text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{9}{100}} = \frac{100}{9} \text{ days}
\]

\[
\text{Time} \approx 11.11 \text{ days}
\]

### Conclusion

A and B, working together at their reduced efficiency levels due to illness, will take approximately **11.11 days** (or \(11 \frac{1}{9}\) days) to complete the work.

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How long would A or B take to complete the work alone if they were ill?
2. What is the percentage decrease in their combined efficiency due to illness?
3. How many days will A and B take to complete half of the work if they are not ill?
4. If only B was ill, how many days would they take to complete the work together?
5. How would the time change if A's illness reduced their efficiency to 70% instead of 80%?

**Tip:** When dealing with work and efficiency problems, always remember to convert percentages into fractions or decimals to calculate the actual work rates.

Learn how to calculate the efficiency and work rates of A and B to solve a work completion problem. This problem involves determining work rates under normal and reduced efficiency conditions, suitable for students in Grades 7-9.

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