If N= 0.369369369369.. and M= 0.531531531531…, then what is the value of (1/N) + (1/M)?

Options

A. 11100/2419
B. 111/100
C. 1897/3162
D. 2419/11100

Answer

To solve for the value of \( \frac{1}{N} + \frac{1}{M} \) where \( N = 0.369369369... \) and \( M = 0.531531531... \), we can follow these steps:

### Step 1: Convert N and M to fractions

1. **Convert \( N \) to a fraction:**
- Let \( N = 0.369369369... \)
- This can be expressed as \( N = 0.369 + 0.000369369369... \)
- Let \( x = 0.369369369... \)
- Then, \( 1000x = 369.369369369... \)
- Subtracting these, we get:
\[ 1000x - x = 369.369369369... - 0.369369369... \]
\[ 999x = 369 \]
\[ x = \frac{369}{999} \]
- Simplifying \( \frac{369}{999} \):
\[ \frac{369 \div 9}{999 \div 9} = \frac{41}{111} \]
- Thus, \( N = \frac{41}{111} \).

2. **Convert \( M \) to a fraction:**
- Let \( M = 0.531531531... \)
- Similarly, let \( y = 0.531531531... \)
- Then, \( 1000y = 531.531531531... \)
- Subtracting these, we get:
\[ 1000y - y = 531.531531531... - 0.531531531... \]
\[ 999y = 531 \]
\[ y = \frac{531}{999} \]
- Simplifying \( \frac{531}{999} \):
\[ \frac{531 \div 9}{999 \div 9} = \frac{59}{111} \]
- Thus, \( M = \frac{59}{111} \).

### Step 2: Calculate \( \frac{1}{N} + \frac{1}{M} \)

3. **Calculate \( \frac{1}{N} \) and \( \frac{1}{M} \):**
- \( \frac{1}{N} = \frac{111}{41} \)
- \( \frac{1}{M} = \frac{111}{59} \)

4. **Add \( \frac{1}{N} + \frac{1}{M} \):**
- To add these fractions, we need a common denominator:
\[ \frac{1}{N} + \frac{1}{M} = \frac{111}{41} + \frac{111}{59} \]
- The common denominator is \( 41 \times 59 \):
\[ = \frac{111 \times 59}{41 \times 59} + \frac{111 \times 41}{59 \times 41} \]
\[ = \frac{111 \times 59 + 111 \times 41}{41 \times 59} \]
\[ = \frac{111(59 + 41)}{41 \times 59} \]
\[ = \frac{111 \times 100}{41 \times 59} \]

5. **Final calculation:**
- \( 41 \times 59 = 2419 \)
- Thus, the final result is:
\[ \frac{11100}{2419} \]

### Final Answer:
The value of \( \frac{1}{N} + \frac{1}{M} = \frac{11100}{2419} \).