Discussion
What is the value of \( 99\dfrac{98}{99} \times 99 \)?
Answer: A
Correct answer: 9899
Rewrite the mixed number as a round figure minus a small fraction, which is what makes the arithmetic collapse: \[ 99\frac{98}{99} = 100 - \frac{1}{99}, \] since \( \frac{98}{99} = 1 - \frac{1}{99} \). Now multiply: \[ \left(100 - \frac{1}{99}\right) \times 99 = 9900 - 1 = 9899. \] The alternative is to convert to an improper fraction: \( \frac{99 \times 99 + 98}{99} = \frac{9801 + 98}{99} = \frac{9899}{99} \), and multiplying by 99 gives 9899 - the same answer by a longer route. The technique generalises to any mixed number just below a round figure. \( 999\frac{998}{999} \times 999 = 999000 - 1 = 998999 \), and \( 49\frac{48}{49} \times 49 = 2450 - 1 = 2449 \). The distractor 9900 is the product before subtracting the 1, and 9801 is \( 99^{2} \) alone.
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