For what value of P, the given equation is correct?
`(1- P)/(6) + (2P)/(3)- (1- 7P)/(4)= 2 (1)/(6)`

Options

A. `-1`
B. 2
C. `-2`
D. 1

Answer

To solve the equation \((1 - P)/6 + (2P)/3 - (1 - 7P)/4 = 2/6\), we will follow these steps:

### Step 1: Write down the equation
The equation given is:
\[ \frac{1 - P}{6} + \frac{2P}{3} - \frac{1 - 7P}{4} = \frac{2}{6} \]

### Step 2: Find the Least Common Multiple (LCM)
The denominators are 6, 3, and 4. The LCM of these numbers is 12.

### Step 3: Rewrite each term with the common denominator
Now, we will rewrite each fraction with the LCM (12) as the denominator:
\[ \frac{1 - P}{6} = \frac{2(1 - P)}{12}, \quad \frac{2P}{3} = \frac{4P}{12}, \quad \frac{1 - 7P}{4} = \frac{3(1 - 7P)}{12} \]
So, the equation becomes:
\[ \frac{2(1 - P) + 4P - 3(1 - 7P)}{12} = \frac{2}{6} \]

### Step 4: Simplify the left side
Now, we will simplify the left side:
\[ 2(1 - P) + 4P - 3(1 - 7P) = 2 - 2P + 4P - 3 + 21P \]
Combine like terms:
\[ (2 - 3) + (-2P + 4P + 21P) = -1 + 23P \]
So, the equation now looks like:
\[ \frac{-1 + 23P}{12} = \frac{2}{6} \]

### Step 5: Simplify the right side
The right side \(\frac{2}{6}\) simplifies to \(\frac{1}{3}\).

### Step 6: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ -1 + 23P = 12 \cdot \frac{1}{3} \]
Calculating the right side:
\[ -1 + 23P = 4 \]

### Step 7: Solve for P
Now, add 1 to both sides:
\[ 23P = 4 + 1 \]
\[ 23P = 5 \]
Now, divide both sides by 23:
\[ P = \frac{5}{23} \]

### Conclusion
The value of \(P\) that satisfies the equation is:
\[ P = \frac{5}{23} \]