The reciprocal of `2 (1)/(7) + ((-3)/( 14)) + ((-1)/( 28)) + 1 (1)/(4)` is

Options

A. `(7)/(22)`
B. `(22)/(7)`
C. `(20)/(21)`
D. `(17)/(21)`

Answer

To find the reciprocal of the expression \(2 \frac{1}{7} + \left(-\frac{3}{14}\right) + \left(-\frac{1}{28}\right) + 1 \frac{1}{4}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(2 \frac{1}{7}\) to an improper fraction:
\[ 2 \frac{1}{7} = \frac{(2 \times 7) + 1}{7} = \frac{14 + 1}{7} = \frac{15}{7} \]
2. Convert \(1 \frac{1}{4}\) to an improper fraction:
\[ 1 \frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4} \]

### Step 2: Rewrite the Expression
Now rewrite the expression using the improper fractions:
\[ \frac{15}{7} + \left(-\frac{3}{14}\right) + \left(-\frac{1}{28}\right) + \frac{5}{4} \]

### Step 3: Find a Common Denominator
The denominators are 7, 14, 28, and 4. The least common multiple (LCM) of these denominators is 28.

### Step 4: Convert Each Fraction to Have the Common Denominator
1. Convert \(\frac{15}{7}\) to have a denominator of 28:
\[ \frac{15}{7} = \frac{15 \times 4}{7 \times 4} = \frac{60}{28} \]
2. Convert \(-\frac{3}{14}\) to have a denominator of 28:
\[ -\frac{3}{14} = -\frac{3 \times 2}{14 \times 2} = -\frac{6}{28} \]
3. \(-\frac{1}{28}\) is already in the correct form.
4. Convert \(\frac{5}{4}\) to have a denominator of 28:
\[ \frac{5}{4} = \frac{5 \times 7}{4 \times 7} = \frac{35}{28} \]

### Step 5: Combine the Fractions
Now combine all the fractions:
\[ \frac{60}{28} - \frac{6}{28} - \frac{1}{28} + \frac{35}{28} = \frac{60 - 6 - 1 + 35}{28} = \frac{88}{28} \]

### Step 6: Simplify the Fraction
Now simplify \(\frac{88}{28}\):
\[ \frac{88 \div 4}{28 \div 4} = \frac{22}{7} \]

### Step 7: Find the Reciprocal
The reciprocal of \(\frac{22}{7}\) is:
\[ \frac{7}{22} \]

### Final Answer
The reciprocal of \(2 \frac{1}{7} + \left(-\frac{3}{14}\right) + \left(-\frac{1}{28}\right) + 1 \frac{1}{4}\) is \(\frac{7}{22}\).

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