Which is the decimal expansion of ( \frac{29}{42} )?

Correct answer: (0.690476190476\ldots)

Direct answer: option A, 0.690476190476... . Divide 29 by 42. Since 29<42, write 0 and a decimal point, then use 290: 42 goes 6 times, remainder 38. Bring down 0 to get 380: 42 goes 9 times, remainder 2. Bring down 0 to get 20: digit 0, remainder 20. Bring down 0 to get 200: digit 4, remainder 32. Bring down 0 to get 320: digit 7, remainder 26. Bring down 0 to get 260: digit 6, remainder 8. Bring down 0 to get 80: digit 1, remainder 38. The remainder 38 has appeared before, so the block 904761... repeats after the initial 6, giving 0.690476190476... . Option A matches this division. Option B, 0.692929..., has different digits. Option C is too large; option D is far too small and does not represent 29/42. A repeated remainder is the reason the decimal is recurring.