What is the least fraction which when subtracted from 4 3/7 − 3 5/9 + 11 5/21…

Concept

To turn a mixed-number expression into a whole number, first combine every fractional part over one common denominator (the LCM of the denominators) to find the exact fractional remainder sitting on top of the whole-number part. Subtracting a fraction equal to that remainder is exactly what cancels it and leaves a whole number; subtracting any other fraction leaves a non-zero fraction behind.

Application

1. Separate the whole-number and fractional parts of 4 3/7 − 3 5/9 + 11 5/21: whole numbers give 4 − 3 + 11 = 12; fractional parts give 3/7 − 5/9 + 5/21.

2. Find the LCM of the denominators 7, 9 and 21. Since 21 = 3 × 7 and 9 = 3², the least common multiple is 63.

3. Convert every fraction to 63rds: 3/7 = 27/63, 5/9 = 35/63, 5/21 = 15/63.

4. Combine the numerators: 27/63 − 35/63 + 15/63 = 7/63, which simplifies to 1/9.

5. So the whole expression equals 12 and a leftover fraction of 1/9 (i.e. 12 1/9).

6. The least fraction that, when subtracted, removes this leftover and leaves a whole number is exactly that leftover amount, 1/9.

Cross-check

Convert every mixed number to an improper fraction and recompute directly: 4 3/7 = 31/7, 3 5/9 = 32/9, 11 5/21 = 236/21. Over the common denominator 63, these are 279/63, 224/63 and 708/63. So the expression equals (279 − 224 + 708)/63 = 763/63 = 12 remainder 7/63, i.e. 12 1/9 — the same leftover fraction found above, confirming the result.

Result

The least fraction that must be subtracted to leave a whole number is 1/9.