Clever additions and subtractions

Break numbers apart to calculate accurately, then check the result.

Look for a round number

Mental arithmetic is easier when you choose an intermediate step that is easy to remember. To add 38 and 27, first give two units to 38 so it becomes 40. There are 25 units left to add. Now 40 + 25 = 65. This is still the value of 38 + 27: the two units were moved from one addend to the other, so the total did not change. This small rearrangement avoids carrying several digits in your head at the same time.

Before moving anything, look for a nearby multiple of ten. For 59 + 18, moving one unit gives 60 + 17 = 77. For 24 + 39, moving one unit the other way gives 23 + 40 = 63. The direction may change, but the sum stays the same. Write both versions on paper while learning the method. Later, try saying the transformed sum aloud before computing it. Accuracy matters more than speed.

Compensate when subtracting

For 72 - 29, you could subtract 20 and then 9. Another route is to subtract 30 and give one back: 72 - 30 = 42, then 42 + 1 = 43. You give one back because 30 is one greater than 29. Without that correction, your answer is one too small. Likewise, 103 - 48 can become 103 - 50 + 2, which is 55. Pay attention to the sign of the correction, not just the first subtraction.

You can also add the same amount to both numbers in a subtraction. The difference between them stays fixed: 72 - 29 becomes 73 - 30, again giving 43. Choose the route you can explain clearly. An elegant shortcut is only helpful when it reduces mistakes. If it feels confusing, split the second number into tens and units and subtract those parts in turn.

Check before answering

A rough estimate can catch an implausible answer. Since 38 is near 40 and 27 is near 30, their sum should be near 70. An exact result of 65 makes sense; 95 does not. Subtracting about 30 from 72 should leave about 42, so the exact answer 43 is plausible. Estimation is a check, not a replacement for the exact calculation.

Try three sums and three differences with a number ending in 8 or 9. Turn each one into a calculation using a round ten. Then use the opposite operation to check it. If you found 56 + 19 = 75, verify that 75 - 19 = 56. If you found 81 - 28 = 53, verify that 53 + 28 = 81. Say what quantity you moved or compensated. If you cannot explain where a final one or two came from, redo that step slowly.

In a sum, moving a quantity from one addend to the other keeps the total fixed. In a difference, subtracting too much means you must add the excess back. A quick estimate helps catch an impossible answer.