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

Introduction / Context:
This problem tests understanding of repeating decimals and their conversion to fractions. We are given two repeating decimals whose repeating parts have length three, and we must compute the sum of their reciprocals. This requires converting each repeating decimal to a rational number, taking reciprocals, and then simplifying the resulting expression. Such problems help strengthen links between decimal representations and rational numbers.

Given Data / Assumptions:

• N = 0.369369369..., where 369 repeats indefinitely.
• M = 0.531531531..., where 531 repeats indefinitely.
• We must calculate (1 / N) + (1 / M) as a single simplified fraction.

Concept / Approach:
A standard technique for converting a repeating decimal with a block of three digits into a fraction is to use the fact that 0.abcabcabc... equals abc / 999, where abc is the three digit integer formed by the repeating block. After converting N and M into fractions, we take their reciprocals, add them, and simplify. Working in fraction form avoids rounding errors and ensures an exact answer.

Step-by-Step Solution:
First convert N = 0.369369... to a fraction.The repeating block is 369, so N = 369 / 999.Simplify 369 / 999 by dividing numerator and denominator by 9: 369 / 9 = 41 and 999 / 9 = 111.So N = 41 / 111.Next convert M = 0.531531... to a fraction.The repeating block is 531, so M = 531 / 999.Simplify 531 / 999 by dividing numerator and denominator by 9: 531 / 9 = 59 and 999 / 9 = 111.So M = 59 / 111.Now compute 1 / N and 1 / M.1 / N = 1 / (41 / 111) = 111 / 41.1 / M = 1 / (59 / 111) = 111 / 59.We need (1 / N) + (1 / M) = 111 / 41 + 111 / 59.Take the common denominator 41 * 59.Numerator becomes 111 * 59 + 111 * 41 = 111 * (59 + 41) = 111 * 100 = 11100.Denominator is 41 * 59 = 2419.So (1 / N) + (1 / M) = 11100 / 2419.

Verification / Alternative check:
We can approximate the decimals to check plausibility.N is approximately 0.369 and M is approximately 0.531.Then 1 / N is about 1 / 0.369 ≈ 2.71 and 1 / M is about 1 / 0.531 ≈ 1.88.Their sum is roughly 4.59.Now evaluate 11100 / 2419 numerically: 2419 * 4 = 9676 and 2419 * 4.5 is slightly larger than 10885, so 11100 / 2419 is close to 4.59, which matches the rough decimal estimate.

Why Other Options Are Wrong:
111 / 100 equals 1.11, which is far too small compared to the approximate sum of reciprocals near 4.6.1897 / 3162 and 2419 / 11100 are also far from the correct magnitude and do not arise from correct arithmetic with the exact fractions.Only 11100 / 2419 matches the exact fractional computation.

Common Pitfalls:
Some learners misapply the conversion rule and write 0.abcabc... as abc / 99 instead of abc / 999.Another common error is simplifying 369 / 999 or 531 / 999 incorrectly.Careless mistakes in adding the fractions 111 / 41 and 111 / 59 can also produce wrong numerators or denominators.

Final Answer:
The value of (1 / N) + (1 / M) is 11100/2419.