S can do a piece of work in 80 days and T in 60 days. With the help of U, they finish the work in 20 days. In how many days U alone can do the same work?

Options

A. 40
B. 60
C. 48
D. 80

Answer

To solve the problem step by step, we will first determine the work done by S, T, and U, and then find out how long U alone would take to complete the work.

### Step 1: Determine the work done by S and T

- S can complete the work in 80 days.
- T can complete the work in 60 days.

To find the work done by each in one day, we can use the formula:
\[ \text{Work done in one day} = \frac{1}{\text{Number of days to complete the work}} \]

- Work done by S in one day:
\[ \text{Work by S} = \frac{1}{80} \text{ of the work} \]

- Work done by T in one day:
\[ \text{Work by T} = \frac{1}{60} \text{ of the work} \]

### Step 2: Calculate the combined work done by S and T in one day

To find the combined work done by S and T in one day, we add the work done by both:
\[ \text{Combined work by S and T} = \frac{1}{80} + \frac{1}{60} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 80 and 60 is 240.

Converting the fractions:
\[ \frac{1}{80} = \frac{3}{240} \quad \text{and} \quad \frac{1}{60} = \frac{4}{240} \]

Now, adding these:
\[ \text{Combined work by S and T} = \frac{3}{240} + \frac{4}{240} = \frac{7}{240} \]

### Step 3: Determine the total work done by S, T, and U together

We know that S, T, and U together finish the work in 20 days. Therefore, the work done by all three in one day is:
\[ \text{Combined work by S, T, and U} = \frac{1}{20} \]

### Step 4: Set up the equation to find U's work

Now, we can set up the equation:
\[ \text{Work by S} + \text{Work by T} + \text{Work by U} = \text{Combined work by S, T, and U} \]
Substituting the values we found:
\[ \frac{7}{240} + \text{Work by U} = \frac{1}{20} \]

### Step 5: Convert \(\frac{1}{20}\) to a fraction with a denominator of 240

To solve for U's work, we convert \(\frac{1}{20}\) to have a denominator of 240:
\[ \frac{1}{20} = \frac{12}{240} \]

### Step 6: Solve for U's work

Now we substitute this back into the equation:
\[ \frac{7}{240} + \text{Work by U} = \frac{12}{240} \]
Subtract \(\frac{7}{240}\) from both sides:
\[ \text{Work by U} = \frac{12}{240} - \frac{7}{240} = \frac{5}{240} \]

### Step 7: Determine how many days U alone would take to complete the work

Since U does \(\frac{5}{240}\) of the work in one day, we can find out how many days U would take to complete the entire work:
\[ \text{Days taken by U} = \frac{240}{5} = 48 \text{ days} \]

### Final Answer
U alone can complete the work in **48 days**.

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