Mental math questions reward a handful of shortcuts more than raw speed: split numbers by place value, round and correct, multiply through easy factors, and know which operation comes first. Most wrong answers come from a skipped rule, such as adding before multiplying, from a dropped correction, or from a percent applied to the wrong base. The chapters cover the shortcuts for each kind of question and end with quick checks that catch a wrong answer before you choose it.
Each chapter opens with the short version. Tap one to read the detail.
Adding fast, in the right order
~2 min
Add by place value, round to a friendly number and correct, and pair values that make round totals. In a longer expression, multiply and divide before you add and subtract, and work out parentheses first.
Split numbers by place value and add each place: 358 + 427 is 700 + 70 + 15 = 785. When one number sits just below a round one, use the round number and correct: 638 + 299 is 638 + 300 − 1 = 937, and 721 − 396 is 721 − 400 + 4 = 325. Moving a little from one addend to the other works too: 593 + 248 becomes 600 + 241 = 841. In a list, pair the values that make round totals first: in 27 + 45 + 73 + 55, the pairs 27 + 73 and 45 + 55 each make 100.
In an expression with several operations, multiply and divide before you add and subtract, and work out anything in parentheses first. So 16 + 8 × 7 is 16 + 56 = 72, not 24 × 7 = 168, and (19 + 6) × 4 − 13 is 25 × 4 − 13 = 87. Within one level, work left to right: 72 ÷ 9 × 4 is 8 × 4 = 32, not 72 ÷ 36 = 2. Writing in the parentheses the rules imply, as in 16 + (8 × 7), prevents most slips.
Rule: add by place value or round and correct, and settle the order of operations before computing anything.
Multiplying and squaring
~2 min
Split a factor into easy parts, multiply by a nearby round number and correct, and use the doubling, halving, 5, 9, and 11 shortcuts. Square a number near a round one by expanding, and use a² − b² for two numbers spaced evenly around a round one.
Split one factor into parts that are easy to multiply: 6 × 347 = 1,800 + 240 + 42 = 2,082. When a factor is just below a round number, use the round number and subtract the extra: 36 bottles at $2.98 cost 36 × $3 − 36 × $0.02 = $108 − $0.72 = $107.28. Halving one factor and doubling the other keeps the product: 14 × 45 = 7 × 90 = 630. Multiplying by 5 is multiplying by 10 and halving; by 9, it is multiplying by 10 and subtracting one copy (9 × 136 = 1,360 − 136 = 1,224); and a two-digit number times 11 puts the sum of its digits between them (11 × 54 = 594).
Two numbers spaced evenly around a round number multiply to its square minus the spacing squared: 104 × 96 = 10,000 − 16 = 9,984. The same expansion squares a number near a round one: 98² = (100 − 2)² = 10,000 − 400 + 4 = 9,604. For a number ending in 5, multiply the tens part by one more than itself and write 25 after it: 115² starts with 11 × 12 = 132, so it is 13,225.
The last digit of a power repeats in a cycle no longer than four, so 7^20, whose exponent is divisible by 4, ends in 1, just as 7⁴ = 2,401 does.
Rule: turn a hard product into easy ones by splitting, rounding and correcting, or doubling and halving, and use the square identities for numbers near a round one.
Dividing, remainders, and fractions
~2 min
Treat division as a missing factor, split the dividend into easy parts, and halve repeatedly to divide by 4 or 8. The digit sum gives the remainder on division by 9 or 3, and a fraction of a quantity is a division followed by a multiplication.
A division asks for a missing factor: 182 ÷ 14 asks what times 14 makes 182, and 13 × 14 = 182. Split the dividend into parts the divisor divides evenly: 378 ÷ 9 = 360 ÷ 9 + 18 ÷ 9 = 40 + 2 = 42. Dividing by 4 or 8 is halving two or three times (424 ÷ 8 runs 212, 106, 53), and dividing by 5 is doubling and dividing by 10 (435 ÷ 5 = 870 ÷ 10 = 87).
A number leaves the same remainder on division by 9, or by 3, as the sum of its digits does. The digits of 6,247 add to 19, and 19 leaves 1 on division by 9, so 6,247 leaves 1 as well. The reason is that 10, 100, and 1,000 each leave remainder 1 on division by 9, so each digit contributes only its own value. Whatever the remainder, it must be smaller than the divisor, and the dividend always equals divisor × quotient + remainder.
A fraction of a quantity is a division followed by a multiplication: two fifths of 85 is 85 ÷ 5 = 17, times 2 = 34. To count how many fractional pieces fit in an amount, divide by the fraction, which means multiplying by its reciprocal: cups that each hold 3/4 of a liter fill 18 liters in 18 × 4/3 = 24 cups.
Rule: divide by finding the missing factor or splitting the dividend, use the digit sum for remainders on division by 9 or 3, and multiply by the reciprocal to divide by a fraction.
Percentages
~2 min
Build percents from 10% and from simple fractions, multiply the rates for a percent of a percent, and scale from a known percent. Successive changes multiply, and a change is undone by dividing by its factor.
Ten percent is a tenth, and most other percents build from it: for 620, 10% is 62, 5% is 31, and 15% is 93. Some percents are simple fractions, so 75% of 64 is three quarters of 64, or 48. A percent of a percent multiplies the rates: 45% of 20% of 300 is 0.09 of 300, or 27. When one percent of a number is known, scale it: if 45% of a number is 36, then 5% is 4 and the whole number is 80.
Successive changes multiply. A 30% rise followed by a 30% fall leaves 1.3 × 0.7 = 0.91 of the start, a 9% loss rather than no change, because the fall applies to the larger amount. To find a price before a change, divide by its factor: $126 after a 30% discount was 126 ÷ 0.7 = $180, and $138 after a 15% markup was 138 ÷ 1.15 = $120.
Rule: turn each percent into a fraction or a multiplier, multiply to apply changes in sequence, and divide to undo one.
Sums, averages, and clock angles
~2 min
Add evenly spaced numbers by pairing the ends, find a missing value through the total an average implies, and place both clock hands from 12 before subtracting.
A sum of evenly spaced numbers is the number of terms times the average of the first and last term: 5 + 10 + … + 100 has 20 terms averaging 52.5, so the sum is 1,050. The numbers 1 to n add to n(n + 1) ÷ 2, so 1 to 80 add to 3,240.
An average times its count gives a total, which turns a missing-value question into a subtraction. If 8 test scores average 15 and the first 7 add up to 103, the scores total 120 and the last is 17.
On a clock, the minute hand turns 6° a minute, and the hour hand 30° an hour plus 0.5° a minute. At 8:15 the minute hand is at 90° and the hour hand at 240° + 7.5° = 247.5°, so the hands are 157.5° apart. If a difference comes out above 180°, subtract it from 360° for the smaller angle.
Rule: pair the ends of an evenly spaced sum, work averages through their totals, and place both clock hands before subtracting.
Checking the answer
~2 min
Check the last digit, the digit sum, and the size of an answer before choosing it. Each check takes seconds and rules out most wrong choices.
The last digit of a product depends only on the last digits of the factors: 23 × 47 must end in 1, because 3 × 7 = 21. Any choice ending in another digit is wrong.
Digit sums check multiplication, a method long known as casting out nines. Reduce each factor and the answer to a single digit by adding digits repeatedly: for 23 × 47 = 1,081, the factors reduce to 5 and 2, whose product 10 reduces to 1, and 1,081 also reduces to 1. A mismatch proves an error. A match makes the answer very likely right, though not certain, because swapping two digits leaves the digit sum unchanged.
Size checks are faster still. Two two-digit numbers multiply to three or four digits, 15% of a number is well under a fifth of it, and a discounted price must be lower than the original. Rounding each factor gives a target: 23 × 47 is about 20 × 50 = 1,000, so 1,081 is plausible and 10,810 is not.
Rule: before choosing, check the last digit, the digit sum, and the size of the answer.
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