Mental Math Tricks for the Daily Quizzes

Habit 1: round, then adjust

Add 47 and 38? Add 50 and 40 (90), then take away the 3 and the 2 you added (85). Subtract 29 from 73? Subtract 30 (43), then add 1 back (44). Multiply 19 by 6? Multiply 20 by 6 (120), then subtract one 6 (114). Rounding to the nearest ten or hundred and correcting afterward is the single most useful trick in arithmetic, and it works for every operation.

Habit 2: work left to right

School arithmetic goes right to left (units first, carry the one). Mental arithmetic goes the other way: big parts first. 356 + 278: three hundred and two hundred is 500; fifty and seventy is 120, total 620; six and eight is 14, total 634. Working left to right means you always have an approximate answer early, and a slip in the last step costs little.

Doubling and halving

• To multiply by 4, double twice. To multiply by 8, double three times. 37 × 4: 74, 148.
• To multiply by 5, multiply by 10 and halve. 86 × 5: 860, 430.
• To multiply by 25, multiply by 100 and halve twice. 48 × 25: 4800, 2400, 1200.
• To multiply by 50, multiply by 100 and halve.
• To multiply two even numbers, halve one and double the other to make it easier: 14 × 15 = 7 × 30 = 210; 16 × 35 = 8 × 70 = 560.

Multiplying by 9 and 11

• By 9: multiply by 10 and subtract the number. 9 × 47 = 470 − 47 = 423. For single digits, the fingers trick works: hold up ten fingers, fold the nth finger down, and read the tens to the left and the units to the right (9 × 7: fold the seventh finger; six fingers left, three right: 63).
• By 11, two-digit numbers: write the digits apart and put their sum in the middle. 11 × 34 = 3 (3 + 4) 4 = 374. If the sum is ten or more, carry: 11 × 68 = 6 (14) 8 = 748.

Squaring numbers that end in 5

Take the tens digit, multiply it by the next number up, and write 25 after it. 35²: 3 × 4 = 12, so 1225. 65²: 6 × 7 = 42, so 4225. 105²: 10 × 11 = 110, so 11025. It always works, and it comes up in quizzes more than you would expect.

Squares and near squares

Know the squares to 15² (225) and the easy ones beyond: 20² = 400, 25² = 625, 30² = 900. Then use (a + b)(a − b) = a² − b² for near squares: 29 × 31 = 30² − 1 = 899; 48 × 52 = 50² − 4 = 2496; 19 × 21 = 399.

Division and divisibility

• A number is divisible by 3 if its digits add to a multiple of 3, and by 9 if they add to a multiple of 9. 738: 7 + 3 + 8 = 18, so it divides by 9 (738 ÷ 9 = 82).
• Divisible by 4 if its last two digits are; by 8 if its last three digits are; by 6 if it is even and divisible by 3.
• To divide by 5, double and divide by 10: 345 ÷ 5 = 690 ÷ 10 = 69.
• To divide by 25, multiply by 4 and divide by 100.

Percentages

• 10% is a tenth: move the decimal point. 10% of 340 is 34.
• 5% is half of 10%; 1% is a tenth of 10%; 15% is 10% plus 5%; 20% is double 10%.
• Percentages commute: 8% of 25 is the same as 25% of 8, which is 2. Whenever one side is an easy fraction (25% is a quarter, 50% a half, 12.5% an eighth), swap.
• To add 20% to a price, multiply by 1.2; to take 20% off, multiply by 0.8.

Times tables under pressure

The Times Tables Quiz is pure recall, and recall is built by repetition in both directions. The tables that trip people up are 6, 7 and 8 multiplied by each other: 6 × 7 = 42, 6 × 8 = 48, 7 × 8 = 56 ("five, six, seven, eight" for 56 = 7 × 8 is a well-known hook), 7 × 7 = 49, 8 × 8 = 64. Drill those nine facts and the rest of the tables feel easy. For anything above 10, split: 12 × 7 = 70 + 14 = 84.

Estimation first

Before computing exactly, estimate: 38 × 52 is about 40 × 50 = 2000, so an answer of 1976 is plausible and 19760 is not. In a multiple-choice quiz, estimation alone often rules out three of the four options, and it catches the slips that time pressure causes.

Common mistakes

1. Working right to left and losing track of carries.
2. Not rounding. 99 × 7 is 700 − 7, not a long multiplication.
3. Forgetting percentages commute. 4% of 75 is 75% of 4, which is 3.
4. Guessing instead of estimating. An estimate takes two seconds and removes wrong options.
5. Rushing the units digit. Check the last digit of a product against the last digits of its factors (7 × 8 must end in 6).

Glossary

• Rounding and adjusting: computing with a nearby round number and correcting afterward.
• Difference of squares: (a + b)(a − b) = a² − b², used for products of numbers equally spaced around a round number.
• Divisibility rule: a quick test for whether one number divides another.
• Commuting percentages: a% of b equals b% of a.