Multiply Near a Round Number: Puzzle and Step-by-Step Solution
Mental Math
Easy
Reported at Optiver, SIG, Akuna, IMC, DRW
The puzzle
Without paper, compute 47 × 53, then 97 × 103, then 48². You have about six seconds each, as on an 80-in-8 style speed arithmetic screen.
Short answer: 47 × 53 = 2491, 97 × 103 = 9991 and 48² = 2304, all by expanding around a round number.
Step 1: Straddling a round number: difference of squares
47 and 53 sit 3 either side of 50. Write them as (50 - 3)(50 + 3). The identity (a - b)(a + b) = a² - b² turns the product into 50² - 3² = 2500 - 9 = 2491.
The same move gives 97 × 103 = 100² - 3² = 10000 - 9 = 9991, and 38 × 42 = 40² - 2² = 1600 - 4 = 1596. Whenever two factors are equally spaced around a number whose square you know, the multiplication becomes one subtraction.
Step 2: Squares near 50 and 100
Expand around the round number: (50 + d)² = 2500 + 100d + d². For 48, d = -2: 2500 - 200 + 4 = 2304. For 53, d = 3: 2500 + 300 + 9 = 2809.
Around 100: (100 + d)² = 10000 + 200d + d². So 97² = 10000 - 600 + 9 = 9409, and 104² = 10000 + 800 + 16 = 10816. The rule of thumb: near 100, the answer is (100 + 2d) hundreds plus d², padded to two digits.
Step 3: Squares ending in 5
For a number written as n5, its square is n × (n + 1) followed by 25. So 65² is 6 × 7 = 42 followed by 25: 4225. And 115² is 11 × 12 = 132 followed by 25: 13225.
It works because (10n + 5)² = 100n² + 100n + 25 = 100 n(n + 1) + 25. Knowing why it works is what lets you trust it under time pressure.
Step 4: Turning tricks into speed
On a timed screen with about six seconds an item and negative marking, the gain is not that each trick is faster than long multiplication, although it is. It is that recognising the pattern in under a second tells you immediately whether an item is fast or should be skipped.
Other patterns worth drilling: percentages can be swapped (15% of 240 equals 240% of 15, which is 36), multiplying by 25 is dividing by 4 and multiplying by 100, and dividing by 0.25 is multiplying by 4. Each one turns a hard-looking item into a two-step one.
Why interviewers ask it
Interviewers and online screens reward recognising structure, not raw multiplication. Practise the patterns under a clock until you see them without looking.
Knowing the proof is half of it. Work Multiply Near a Round Number interactively, then take it under a clock with the rest of the Outcry simulators.