P can do a work 3 times faster than Q and therefore takes 40 days less than Q. Find the time in whichPandQcan complete the work individually.

Answer

To solve the problem step by step, let's denote the time taken by Q to complete the work as \( x \) days. Since P can do the work 3 times faster than Q, the time taken by P to complete the work will be \( \frac{x}{3} \) days.

According to the problem, P takes 40 days less than Q. Therefore, we can set up the following equation:

\[ x - \frac{x}{3} = 40 \]

Now, let's solve this equation step by step.

### Step 1: Simplify the equation
To simplify the left side, we need a common denominator. The common denominator of 1 and 3 is 3. Thus, we can rewrite \( x \) as \( \frac{3x}{3} \):

\[ \frac{3x}{3} - \frac{x}{3} = 40 \]

### Step 2: Combine the fractions
Now, we can combine the fractions on the left side:

\[ \frac{3x - x}{3} = 40 \]

This simplifies to:

\[ \frac{2x}{3} = 40 \]

### Step 3: Eliminate the fraction
To eliminate the fraction, multiply both sides of the equation by 3:

\[ 2x = 120 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 2 to find \( x \):

\[ x = 60 \]

### Step 5: Find the time taken by P
Since \( x \) represents the time taken by Q, we can now find the time taken by P:

\[ \text{Time taken by P} = \frac{x}{3} = \frac{60}{3} = 20 \text{ days} \]

### Summary of Results
- Time taken by Q: \( 60 \) days
- Time taken by P: \( 20 \) days

### Final Answer
P can complete the work in **20 days**, and Q can complete the work in **60 days**.

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