ArithMind Mental Arithmetic Lessons
Stage 9 · Phase 2: Intermediate intermediate all

The Compensation Technique

Simplify mental math by rounding numbers and adjusting the result.

Adjust, Calculate, Correct

Compensation is a mental strategy that temporarily replaces an awkward number with a nearby friendly value, performs the easier calculation, and then corrects the result by the same amount. The previous stage focused on multiplying by 10, 100, and 1000, which strengthened recognition of place-value landmarks. Those landmarks now become convenient targets: 198 can become 200, 597 can become 600, and 1,002 can become 1,000.

The replacement is not an estimate because the final correction restores the exact original value. Think of borrowing an easier number for one line of work and then settling the difference.

Why It Helps in an Everyday Decision

Imagine that a community group has collected $503 and expects to spend $197 on equipment. Subtracting 200 is immediate, so you see $303 first. The planned cost is actually $3 less than $200, meaning the group keeps $3 more: $306 remains. This mental result can guide a quick purchasing decision while still being exact.

Compensation is useful when a value sits only a small distance from a multiple of ten or hundred. It is not automatically best for every calculation. Turning 243 into 300 creates a correction of 57, which may be harder than simply adding or subtracting in chunks.

Named Method: Round and Reverse

The round-and-reverse method has three questions:

  1. What friendly number is nearby?
  2. How did I change the original number?
  3. What action reverses that change in the result?

For addition, if an addend is increased, the temporary sum is too high. In 298 + 156, replacing 298 with 300 adds an unwanted 2. The easy sum is 456, then reversing that increase gives 456 - 2 = 454.

Rounding down works too. For 603 + 249, use 250 for the second addend. That temporary addend is 1 too high, so 603 + 250 = 853, followed by 853 - 1 = 852.

Named Method: Subtract and Restore

Subtraction needs closer attention because changing the amount removed has an opposite effect on what remains. The subtract-and-restore method uses a simple story:

  • If you subtract too much, restore the excess.
  • If you subtract too little, remove the shortage.

For 503 - 197, using 200 removes 3 too much. The easy difference is 303, and restoring 3 gives 306. By contrast, for 84 - 31, using 30 removes 1 too little. After 84 - 30 = 54, remove the missing 1 to get 53.

Naming “too much” or “too little” is more dependable than memorising a bare plus-or-minus rule.

Worked Examples With Reasoning

  1. 397 + 228: Lift 397 to 400. 400 + 228 = 628; the lift added 3, so reverse it: 628 - 3 = 625.
  2. 742 - 298: Treat 298 as 300. 742 - 300 = 442; 2 too many were removed, so restore 2 for 444.
  3. 615 + 99: Replace 99 with 100. 615 + 100 = 715; remove the extra 1 to reach 714.
  4. 936 - 402: Use 400 instead of 402. 936 - 400 = 536; because 2 too few were taken away, remove 2 more. The answer is 534.

A quick estimate protects the correction. Since 742 - 298 is close to 740 - 300, an answer near 440 makes sense.

Two-Number Adjustment

For addition only, you can use equal transfer: move an amount from one addend to the other so the total stays unchanged. For 46 + 37, transfer 4 from 37 to 46, creating 50 + 33 = 83. This needs no final correction because one addend rose by 4 while the other fell by 4.

Do not apply equal transfer unchanged to subtraction. Adding the same amount to both numbers preserves a difference, while moving an amount from one number to the other changes it. Keep this extension separate until the basic correction logic feels secure.

Common Correction Traps

  • Stopping at the friendly calculation: 699 + 245 is not 945; using 700 introduced an extra 1.
  • Correcting subtraction as though it were addition: After subtracting too much, the result must rise.
  • Choosing a distant landmark: A large correction defeats the method’s purpose.
  • Adjusting by the wrong distance: The gap from 596 to 600 is 4, not 6.
  • Applying the correction twice: Equal transfer already balances an addition sum, so no later repair is needed.

A Calm Practice Routine

Before solving, circle mentally the value nearest a ten or hundred. State the adjustment aloud: “up 3,” “down 2,” or “subtracting 4 extra.” Calculate, correct, then estimate. Practise addition and subtraction in separate small groups at first. When mixing them, write only a plus or minus correction sign if memory is overloaded. Speed can develop later from repeated accurate decisions.

Compensation Challenge

  1. 199 + 347
  2. 602 - 298
  3. 78 + 49
  4. 451 - 99
  5. 596 + 238
  6. 825 - 397
  7. 304 + 198
  8. 710 - 203
  9. 69 + 57
  10. 943 - 499
  11. 388 + 206
  12. 650 - 98
  13. 997 + 125
  14. 534 - 301
  15. 89 + 46
  16. 800 - 394
  17. 487 + 299
  18. 721 - 198

Four Routes Unpacked

  • 199 + 347: Use 200, obtaining 547, then remove the extra 1 for 546.
  • 602 - 298: Subtract 300 to get 302; restore 2 because the temporary subtraction was too large, producing 304.
  • 596 + 238: Raise 596 by 4: 600 + 238 = 838; reverse the added 4 for 834.
  • 943 - 499: Remove 500 for 443, then return 1; the exact difference is 444.
  • 650 - 98: Taking away 100 leaves 550, and restoring the excess 2 gives 552.

Solutions

  1. 546; 2. 304; 3. 127; 4. 352; 5. 834; 6. 428; 7. 502; 8. 507; 9. 126; 10. 444; 11. 594; 12. 552; 13. 1,122; 14. 233; 15. 135; 16. 406; 17. 786; 18. 523.

FAQ

How close should a number be before I compensate?

There is no fixed limit. A correction of 1, 2, or 3 is often convenient, but use any adjustment that makes the entire route easier for you.

Can I round to 50 instead of 100?

Certainly. Friendly values depend on the calculation. For 48 + 27, using 50 creates a tiny correction and keeps the arithmetic simple.

Is compensation the same as estimation?

No. Estimation may intentionally leave a rounded answer. Compensation starts with a rounded value but reverses the adjustment, so its final answer is exact.

Practice Examples

Example 1: 298 + 156 = ?

  1. Change 298 to the friendly number 300, an increase of 2
  2. Add with the friendly value: 300 + 156 = 456
  3. The temporary sum includes 2 extra, so calculate 456 - 2
  4. The corrected result is 454

Answer: 454

Example 2: 503 - 197 = ?

  1. Change the amount being subtracted from 197 to 200
  2. Work out the easier difference: 503 - 200 = 303
  3. This removed 3 too much, so restore 3
  4. Calculate 303 + 3 = 306

Answer: 306

Example 3: 49 + 38 = ?

  1. Increase 49 by 1 to make 50
  2. Find the temporary sum: 50 + 38 = 88
  3. Reverse the increase by taking 1 from 88
  4. The original sum equals 87

Answer: 87

Example 4: 84 - 31 = ?

  1. Replace 31 with 30, which is 1 less
  2. Calculate the simpler subtraction: 84 - 30 = 54
  3. Only 30 was removed, so remove the remaining 1
  4. The exact difference is 54 - 1 = 53

Answer: 53