Seven groups of twenty-three
7 × 20 = 1407 × 3 = 21140 + 21 = 161Check: 7 × 23 must be between 7 × 20 and 7 × 25, so between 140 and 175.
Method 03 • exact arithmetic
Arithmetic with fewer hurdles
Make the numbers friendlier without changing the answer. A small adjustment can turn an awkward calculation into two steps you can see clearly.
Try one adjustment
Add a friendly number, then undo the extra
Suppose you need 48 + 27. Forty-eight is close to 50, so calculate 50 + 27 = 77. You added two more than the original question allowed, so subtract two: 77 − 2 = 75. This is often called compensation. The useful part is not just moving to a round number; it is keeping track of the correction.
You can also transfer an amount between the two numbers. Move two from 27 to 48, giving 50 + 25 = 75. The total stays the same because one part rises by exactly what the other loses. Choose whichever version is easier to hold in mind. You do not need to learn both before using one.
Subtract a nearly round number
83 − 29 = 83 − 30 + 153 + 1 = 54Check: 54 + 29 = 83. Subtracting 30 removed one too many, so add one back.
For subtraction, another exact method is to add the same amount to both numbers. In 83 − 29, adding one to each gives 84 − 30 = 54. The gap remains unchanged. This can be useful when the second number becomes a convenient multiple of ten.
Split multiplication into useful chunks
Multiplication works over a sum: 7 × 23 is the same as 7 × 20 plus 7 × 3. This is the distributive property, but you do not need the name to use it. Think of seven groups, each containing twenty items and three more. Count the twenties first, then count the extra threes.
Six groups of nineteen
6 × 19 = 6 × 20 − 6 × 1120 − 6 = 114Check: adding back one group of six gives 120.
Double one factor and halve the other
For 16 × 25, halve 16 and double 25 to get 8 × 50. Repeat to get 4 × 100 = 400. The product stays the same because one factor is multiplied by two while the other is divided by two. This is especially convenient when an even number sits beside 25, 50 or another factor that becomes simpler when doubled.
This method is exact, including with decimals, but the easiest route depends on the numbers. Halving 15 produces 7.5, which may not feel friendlier. For 15 × 24, it is more convenient to double 15 and halve 24, giving 30 × 12 = 360. You are allowed to stop adjusting as soon as the calculation becomes clear.
Use twenty-five as a quarter of one hundred
Multiplying by 25 can be done by multiplying by 100 and dividing by four. For 36 × 25, calculate 3,600 ÷ 4 = 900. You could instead divide 36 by four first, then multiply by 100. Both routes give 900. The second route uses smaller intermediate numbers when the original number divides neatly by four.
Similarly, multiplying by five is multiplying by ten and halving. For 38 × 5, use 380 ÷ 2 = 190. These shortcuts work because 25 = 100 ÷ 4 and 5 = 10 ÷ 2. They are not special exceptions; they are ordinary multiplication rearranged.
Divide by splitting into known multiples
To calculate 156 ÷ 12, split 156 into 120 and 36. Since 120 ÷ 12 = 10 and 36 ÷ 12 = 3, the total is 13. Check by multiplying: 13 × 12 = 156. Choose chunks that divide cleanly by the divisor, rather than splitting digits in a way that ignores their place value.
For equal shares of money, an exact mathematical share may contain a fraction of the smallest currency unit. If 10 is shared among three people, each ideal share is 3⅓. Three rounded shares of 3.33 total 9.99. Agree how to allocate the remaining 0.01; rounding does not make the remainder disappear.
Choose clarity over a clever trick
If a shortcut requires more mental bookkeeping than the original calculation, use paper or a calculator. Keep one intermediate value per line and label the unit. Microsoft Excel, Google Sheets and LibreOffice Calc can help with repeated calculations, but entering the wrong expression still gives the wrong result. Estimate first and check a sample calculation independently.
A foundation for these methods
OpenStax Prealgebra 2e explains arithmetic properties. Desmos Scientific Calculator can be used to check calculations. Neither resource is affiliated with this site.