The Compensation Strategy: Round, Operate, Then Undo the Change

The check that catches a reversed correction

Estimate before you start, not after. For 532 − 198, the estimate 530 − 200 = 330 tells you the answer sits a little above 330. If your correction takes you to 330 exactly, or below it, the correction went the wrong way.

A second check costs nothing: compensation and counting up are independent routes to the same answer, so if one of them matters, run both. Counting up from 198 to 532 gives 2 to reach 200, then 332 to reach 532, and 2 + 332 = 334 — the same result, obtained without any correction step at all. That method is worth having for its own sake, and counting up as a subtraction strategy covers when to prefer it.

The magnitude discipline behind both checks is the same one described in estimation techniques that make mental math useful: fix the size of the answer first, and the exact calculation then has something to fail against.

When compensation is the wrong tool

Compensation pays when a number sits close to a benchmark — a multiple of ten, a hundred, a familiar anchor like 250 or 1,000. It costs more than it saves when neither number is near one. For 463 − 287 there is no cheap rounding: you would move 287 up by 13, which is not obviously easier to track than the original subtraction.

For subtraction specifically, there is usually a better move available. Instead of changing one number and correcting, change both by the same amount. The difference is unaffected, so there is no correction at all, and the borrow disappears — the method covered in subtraction without borrowing. For 463 − 287, add 13 to both: 476 − 300 = 176.

Multiplication has its own compensation family — multiply by the round number, then take away one lot of the original — and it behaves like the addition case, with a single consistent correction direction. That family is treated separately in multiplying by 9, 99 and 999.

Building the habit

The method only becomes useful when the decision is automatic, and that takes repetition on numbers you did not choose. IQ Test & Brain Training Games, our own app at Reign Creative, drills exactly the shapes this method is for: Addition Sprint runs single-digit, two-digit, three-digit and mixed chains against a 60-second clock, Subtraction Sprint includes a borrowing-practice difficulty where compensation and its alternatives both apply, and the Accuracy Trainer runs ten questions with no timer at all, with a hint and an explanation available, which is the right setting for a method you are still getting backwards.

The app's technique library carries compensation as two separate entries — one for addition, one for subtraction — for the same reason this article splits them: the correction direction is not shared, and teaching them as one rule is what produces the reversal. The app itself is described on the IQ Test & Brain Training Games app page.

Practise it in one direction at a time. Do twenty additions with a rounded addend until the correction stops needing thought, then twenty subtractions where only the subtrahend moves, then twenty where only the minuend moves. Mixing all three before any of them is automatic is how the rule gets blurred in the first place.

Where this sits among the other methods

Compensation is one of a small number of moves that make mental arithmetic tractable; the others are partitioning, counting up, doubling and halving, and outright recall. The mental math practice guide puts them in order and explains which to reach for when. More of our writing on this sits under education and brain training articles, and the full set of arithmetic methods is collected in the mental math topic hub.

England's national curriculum expects mental strategies of this kind to be developed alongside written methods rather than after them, and specifically expects pupils to draw on known facts to calculate mentally. That ordering matters for adults too: compensation is fast only because 487 + 200 is instant, and that instant step rests on facts that are already secure.