Subtraction Without Borrowing: Slide Both Numbers

Choosing between the two directions

Ask which number is closer to a benchmark and move the pair so that number lands on it.

• If the number being taken away is just below a round figure — 298, 467, 1,876 — add.
• If the number you are starting from is just above one — 712, 5,000, 1,003 — subtract.
• If both are awkward, this method has nothing to offer, and counting the gap in hops is usually the better route. Counting up as a subtraction strategy covers that case, and gives an independent way to check any answer produced here.

The mistake that turns this into a different method

Sliding the numbers in opposite directions destroys the difference rather than preserving it. Adding 33 to 803 while subtracting 33 from 467 gives 836 − 434 = 402, which is 66 away from the correct 336 — the two changes did not cancel, they doubled.

The check is the same one that protects every mental subtraction: estimate first. 803 − 467 is roughly 800 − 470, so the answer sits somewhere near 330. An answer of 402 fails that test immediately, and an answer of 303 fails it by a smaller but still visible margin.

A second habit worth building is simply noticing when a written method is being used out of reflex. NRICH's Subtraction Surprise task makes the point neatly in passing: in the worked video the presenter uses column subtraction for all three calculations even though one of them had a quicker mental route, which the task flags as a discussion point about choosing a method rather than defaulting to one.

What the research does and does not settle

The equal-addition idea has a long history as a taught written algorithm, in competition with decomposition — the familiar crossing-out-and-borrowing method. A comparison published in Arithmetic Teacher in 1979 tested the two approaches on subtraction problems requiring regrouping and reported results favouring decomposition.

That is a real finding and it is worth stating, but it is narrow. It concerns which algorithm to teach children for written computation, measured by accuracy on written problems. It says nothing about whether an adult who already has a written method should add a mental shortcut for the specific shape where the smaller number sits just below a round figure. Those are different questions, and the honest position is that the classroom evidence points one way about instruction while leaving the mental-shortcut case untouched.

England's national curriculum takes the same practical line for adults' purposes: mental methods are expected alongside written ones, with pupils drawing on known facts to calculate mentally, rather than treating the column algorithm as the only legitimate route.

Practising it

Three things need to be automatic before this method is fast: complements to 100 and 1,000, the decision about which direction to slide, and the restraint not to correct an answer that needs no correction.

IQ Test & Brain Training Games, our own app at Reign Creative, drills the first of those in Subtraction Sprint, which includes a borrowing-practice difficulty built from precisely the shapes this method dissolves, and the Accuracy Trainer runs the same material untimed, ten questions at a time, with a hint and an explanation available when one goes wrong. The app's technique library carries the slide-both-numbers move as a named method with its own worked example and its own named common mistake, which is the opposite-directions error above. The app itself is described on the IQ Test & Brain Training Games app page.

Work one shape at a time. Twenty subtractions where the second number ends in 98 or 99, then twenty from a number ending in zeros, then a mixed set where you have to choose the direction. The mixed set is the real skill; the first two are just installing the moves. The broader sequence of methods, and the facts each assumes, is laid out in the mental math practice guide, with the rest of this material in the mental math topic hub and under education and brain training articles.