A and B completed a work together in 5 days. had A worked at twice the speed and B at half the speed, It would have taken them four days to complete the job. How much time would it take for A alone to do the work?

Options

A. 10 days
B. 20 days
C. 25 days
D. 15 days

Answer

To solve the problem, we need to determine how much time A alone would take to complete the work. Let's break down the steps:

### Step 1: Define the total work
Let's assume the total work is 100 units.

### Step 2: Calculate the combined work rate of A and B
Since A and B complete the work together in 5 days, their combined work rate is:
\[ \text{Work rate of A and B} = \frac{100 \text{ units}}{5 \text{ days}} = 20 \text{ units per day} \]

### Step 3: Set up the equations for the modified speeds
If A works at twice his speed and B works at half his speed, let's denote the work rates of A and B as \( a \) and \( b \) respectively. Therefore, we have:
\[ a + b = 20 \quad \text{(1)} \]
When A's speed is doubled and B's speed is halved, their new work rates become \( 2a \) and \( \frac{b}{2} \). They complete the work in 4 days, so:
\[ 2a + \frac{b}{2} = \frac{100 \text{ units}}{4 \text{ days}} = 25 \text{ units per day} \]
This gives us the second equation:
\[ 2a + \frac{b}{2} = 25 \quad \text{(2)} \]

### Step 4: Solve the equations
From equation (1), we can express \( b \) in terms of \( a \):
\[ b = 20 - a \quad \text{(3)} \]
Substituting equation (3) into equation (2):
\[ 2a + \frac{20 - a}{2} = 25 \]
Multiplying through by 2 to eliminate the fraction:
\[ 4a + 20 - a = 50 \]
Combining like terms:
\[ 3a + 20 = 50 \]
Subtracting 20 from both sides:
\[ 3a = 30 \]
Dividing by 3:
\[ a = 10 \]

### Step 5: Find the value of b
Using equation (3):
\[ b = 20 - a = 20 - 10 = 10 \]

### Step 6: Calculate the time taken by A alone
Now that we have \( a = 10 \) (the work rate of A), we can find the time taken by A to complete the work alone:
\[ \text{Time taken by A} = \frac{100 \text{ units}}{a} = \frac{100}{10} = 10 \text{ days} \]

### Final Answer
A alone would take **10 days** to complete the work.

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