Interactive laboratory · Algebra
Strategies, shortcuts and algebraic identities for faster calculation
A difficult calculation becomes simpler when we represent it differently. For example, 48×52 becomes (50−2)(50+2), a difference of squares. Every shortcut here follows from the distributive law, algebraic identities, compensation, doubling and halving, or digit structure—not magic.
48×52=(50−2)(50+2)=50²−2²=2496
In each card, look at the trick and example first; then open the mathematical explanation and try the tool when available.
Technique 1 / 18
The trick: Insert the sum of the digits between them, carrying if it exceeds 9.
Example: 53×11: 5+3=8, giving 583. 85×11: 8+5=13, giving 935.
(10a+b)11=100a+10(a+b)+b
Because 11=10+1. The formula produces adjacent sums; any sum above 9 carries to the left.
Technique 2 / 18
The trick: Multiply the part before 5 by the next number; finish with 25.
Example: 85²: 8×9=72, giving 7225.
(10a+5)²=100a(a+1)+25
Expanding gives 100a²+100a+25, so the leading part is a(a+1).
Technique 3 / 18
The trick: Find the center, then subtract the square of the offset.
Example: 48×52=50²−2²=2496. The same idea works around 10, 100 or 1000.
(m−d)(m+d)=m²−d²
The mixed terms −md and +md cancel. A half-integer center also works.
Technique 4 / 18
The trick: Compute a(a+1), followed by the two-digit product of the units.
Example: 83×87: 8×9=72 and 3×7=21, giving 7221. 61×69: 42 and 09, giving 4209.
(10a+b)(10a+10−b)=100a(a+1)+b(10−b)
The leading part is a multiple of 100. The unit product is below 100 and occupies two positions, even if it starts with zero.
Technique 5 / 18
The trick: Use offsets from 100 and normalize the two-digit tail.
Example: 97×96: 93 | 12 = 9312. 103×107: 110 | 21 = 11021.
(100+a)(100+b)=100(100+a+b)+ab
The result is the left part times 100 plus the offset product. A negative or oversized tail needs borrowing or carrying.
Technique 6 / 18
The trick: Use the same idea with a three-digit tail after normalization.
Example: 997×994: 991 | 018 = 991018. 998×997: 995 | 006 = 995006.
(1000+a)(1000+b)=1000(1000+a+b)+ab
Divide ab by 1000 with a nonnegative remainder. The quotient adjusts the left part; the remainder always has three digits.
Technique 7 / 18
The trick: Multiply by 100 and subtract the number.
Example: 47×99=4700−47=4653; for 999 use 1000n−n.
n(10^k−1)=10^k n−n
99, 999 and 9999 are one below powers of ten.
Technique 8 / 18
The trick: Multiply by 100 and add the number.
Example: 53×101=5300+53=5353.
101n=100n+n
Any apparent concatenation follows from addition; it is not an unconditional rule.
Technique 9 / 18
The trick: Multiply by 100 and divide by 4.
Example: 84×25=(84/4)×100=2100.
25n=100n/4
25 is one quarter of 100. If n is not divisible by 4, multiply by 100 before dividing.
Technique 10 / 18
The trick: Multiply by 1000 and divide by 8.
Example: 48×125=(48/8)×1000=6000.
125n=1000n/8
125 is one eighth of 1000.
Technique 11 / 18
The trick: Double one factor while halving the other.
Example: 25×48=50×24=100×12=1200; 16×75=8×150=4×300=1200.
ab=(2a)(b/2)
The factors 2 and 1/2 cancel. Choose the direction that creates easier numbers.
Technique 12 / 18
The trick: Round one factor and make a correction.
Example: 49×37=50×37−37=1813; 102×46=100×46+2×46=4692.
(A+d)b=Ab+db
This is the distributive law applied to a number near a round one.
Technique 13 / 18
The trick: Square the base, add twice the base times the offset, then square the offset.
Example: 104²=10000+800+16=10816; 96²=10000−800+16=9216.
(100+d)²=10000+200d+d²
It follows from (a+b)²; d may be negative.
Technique 14 / 18
The trick: Multiply the difference by the sum.
Example: 51²−49²=(51−49)(51+49)=2×100=200.
A²−B²=(A−B)(A+B)
Expanding the product cancels the mixed terms.
Technique 15 / 18
The trick: Factor or split into a sum, choosing the easier route.
Example: 37×24=37×(20+4)=740+148=888; or 37×(6×4).
a(b+c)=ab+ac; a(bc)=(ab)c
Associativity and distributivity change the form without changing the value.
Technique 16 / 18
The trick: Multiply by 10, 100 or 1000 and divide by 2.
Example: 68×50=6800/2=3400.
5n=10n/2; 50n=100n/2; 500n=1000n/2
Each is half a power of ten.
Technique 17 / 18
The trick: Multiply by the next multiple of ten, then subtract the number.
Example: 47×19=47×20−47=893.
n(10k−1)=10kn−n
9, 19, 29 and 39 are each one below a round multiple of ten.
Technique 18 / 18
The trick: Recognize the structure before calculating. Several methods can work.
Example: 48×52 favors squares; 25×48 favors doubling and halving.
same product, different representations
A convenient method reduces mental steps; the best choice also depends on the person doing the calculation.
Enter whole numbers. The lab displays each step and verifies the product using exact arithmetic.
Look at the product and choose a convenient method. More than one may work.
Answer before opening the hint or solution. Steps stay hidden until you request them.
A new problem is generated from twelve categories, and its answer is computed on the spot.
Ten questions, one at a time. After a mistake you can try again or reveal the solution; a point is awarded only for a correct first attempt. Progress stays on this device.
Fast mental calculation is not about memorizing hundreds of rules. It is about recognizing a structure and turning a difficult calculation into an equivalent, simpler one. The identities below explain many of the techniques in this laboratory.
(a+b)(a−b)=a²−b²
(a+b)²=a²+2ab+b²
(a−b)²=a²−2ab+b²
a(b+c)=ab+ac