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Multiplying by 5, 25 and 50 by Dividing Instead

Five is ten halved and twenty-five is a hundred quartered, so the multiplication becomes a shift and a division. Pick which number to divide first.

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Key takeaways

  • Five is ten divided by two and twenty-five is a hundred divided by four, so each multiplication becomes a shift followed by a division most people find easier.
  • The family extends: 50 is a hundred halved, 125 is a thousand divided by eight, and 625 is ten thousand divided by sixteen.
  • You may divide either factor, so pick whichever divides cleanly — quartering 48 before the shift is easier than quartering 4,800 after it.
  • Run the identity backwards for division: dividing by five is doubling then shifting, and dividing by twenty-five is multiplying by four then shifting two places.
  • The mistake to guard against is doing the division and forgetting the shift, which puts the answer out by a factor of ten or a hundred.

Five is ten halved. Twenty-five is a hundred quartered. So 86 × 5 = 860 ÷ 2 = 430 and 48 × 25 = 4,800 ÷ 4 = 1,200, and in both cases the multiplication has been replaced by a free shift and a division nobody finds difficult. This is the mirror image of the round-then-subtract family covered in multiplying by 9, 99 and 999: instead of rounding up to a power of ten and taking something off, you go up to the power of ten and divide.

The family

Each of these numbers is a power of ten divided by a small factor, and the table below gives the shift and the division that each one turns into.

Multiplier Written as Procedure
5 10 ÷ 2 shift one place, halve
25 100 ÷ 4 shift two places, quarter
50 100 ÷ 2 shift two places, halve
125 1,000 ÷ 8 shift three places, divide by 8
625 10,000 ÷ 16 shift four places, divide by 16

The table reads the same in practice as it does on the page:

  • 23 × 5 = 230 ÷ 2 = 115
  • 248 × 5 = 2,480 ÷ 2 = 1,240
  • 36 × 25 = 3,600 ÷ 4 = 900
  • 86 × 50 = 8,600 ÷ 2 = 4,300
  • 48 × 125 = 48,000 ÷ 8 = 6,000
  • 32 × 625 = 320,000 ÷ 16 = 20,000

Odd numbers are no obstacle, since halving an odd multiple of ten is still clean: 37 × 5 = 370 ÷ 2 = 185, and 17 × 25 = 1,700 ÷ 4 = 425.

Choose which number to divide

Multiplication is associative, so the shift and the division can happen in either order, and either factor can absorb the division. Pick the arrangement where the division is easiest.

For 48 × 25, quartering 48 first gives 12, and 12 × 100 = 1,200. Quartering 4,800 gives the same answer with an extra digit to carry through the division. The small number was the better one to divide.

For 25 × 14, halving 14 gives 7, and 7 × 50 = 350. Alternatively 1,400 ÷ 4 = 350. Both work; the first avoids dividing a four-digit number by four.

For 86 × 5, halving 86 gives 43 and 43 × 10 = 430 — quicker than halving 860, though only barely.

The rule of thumb: if the other factor divides cleanly by 2, 4 or 8, divide it first and then shift. If it does not, shift first and divide the round number.

This is the same flexibility that makes arrays such a useful picture of multiplication. As NRICH's article on illustrating number properties with arrays sets out, rearranging or partitioning the rectangle changes nothing about its area, which is why the order of operations here is free.

The same identity runs backwards

Dividing by five is multiplying by two and shifting right, because 1 ÷ 5 = 2 ÷ 10.

  • 340 ÷ 5 = 680 ÷ 10 = 68
  • 1,600 ÷ 25 = 6,400 ÷ 100 = 64
  • 4,500 ÷ 50 = 9,000 ÷ 100 = 90
  • 96 ÷ 125 = 768 ÷ 1,000 = 0.768

That last one shows the general shape: multiply by the small factor, then shift by as many places as the power of ten. Division by awkward-looking divisors in general is covered in how to divide in your head, which treats factor splitting and partial quotients for the cases this shortcut does not reach.

Combinations worth knowing

Because 75 = 3 × 25, the quartering move handles it too: 75 × 16 = 3 × (16 ÷ 4) × 100 = 3 × 4 × 100 = 1,200. Check it as 16 × 75 = 16 × 70 + 16 × 5 = 1,120 + 80 = 1,200.

Similarly 250 = 1,000 ÷ 4, so 250 × 36 = 36,000 ÷ 4 = 9,000, and 500 = 1,000 ÷ 2, so 500 × 47 = 47,000 ÷ 2 = 23,500.

Money makes the same move constantly. Twenty-four items at £2.50 is 24 × (10 ÷ 4) = 240 ÷ 4 = £60. Anything priced at a quarter, a half or an eighth of a round figure is a candidate.

Where these numbers actually come up

Quarters and halves of a hundred are everywhere in practical arithmetic: 25% is a quarter, 50% is a half, 12.5% is an eighth. The percentage anchors in percentages in your head are the same identities used in the other direction, and someone who can quarter a number quickly gets 25% discounts for free.

The other common home is unit conversion and pricing — quarter-pound weights, 25-count packs, prices ending in .25, .50 and .75. Whenever a factor is a familiar fraction of a round number, converting the multiplication into a division is the shortest route.

The mistake to watch for

Doing the division and forgetting the shift. Halving 86 gives 43, and if the ten never arrives the answer is out by a factor of ten. The same slip at 25 costs a factor of a hundred.

The guard is the estimate you make before starting. 86 × 5 must be somewhere near 86 × 5 ≈ 450, so an answer of 43 fails immediately. Fixing the magnitude first is the habit that catches nearly all shift errors, and it is the central argument of estimation techniques that make mental math useful.

A second, quieter error is dividing the wrong factor when neither divides cleanly. For 27 × 25 there is no clean quarter of 27, so shift first: 2,700 ÷ 4 = 675. Trying to quarter 27 produces 6.75, which is correct but harder to carry.

Practising it

Two skills carry the method: halving and quartering fluently, and shifting without losing the place. Both respond to short, frequent repetition rather than long sessions.

IQ Test & Brain Training Games, our own app at Reign Creative, drills them in Multiplication Rush and Division Dash, and the Accuracy Trainer runs ten untimed questions with a hint and an explanation available, which is the right setting while a new method is still being installed. The applied modes put the same arithmetic in context — a tip calculator round, a discount round and a unit-price round all lean on quarters and halves of round numbers. The app's technique library carries the divide-instead-of-multiply move as a named method with its own worked example and named common mistake. The app itself is described on the IQ Test & Brain Training Games app page.

The What Works Clearinghouse practice guide on mathematics intervention in the elementary grades lists timed activities as one of six recommendations for building fluency, alongside systematic instruction and the use of number lines. The useful reading for an adult is that the clock belongs at the end of the process rather than the start: install the method untimed, then use timing to make it automatic.

England's national curriculum expects pupils to multiply and divide mentally by drawing on known facts, which is exactly what this method does — it converts an unfamiliar multiplication into two operations you already own. The wider ordering of methods sits 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.

Good to know

Frequently asked questions

How do you multiply by 25 quickly?
Multiply by 100 and take a quarter, or take a quarter first and then multiply by 100. For 48 times 25, a quarter of 48 is 12, so the answer is 1,200. Both orders give the same result because multiplication and division can be applied in any order.
What is the trick for multiplying by 5?
Multiply by 10 and halve. For 86 times 5, 860 halved is 430. If the number is even it is usually quicker to halve first: half of 86 is 43, and 43 times 10 is 430.
How do you divide by 5 in your head?
Double the number and shift one place right, because dividing by five is the same as multiplying by two and dividing by ten. For 340 divided by 5, double to 680 and shift to get 68. Dividing by 25 works the same way with four and two places.
Why does this work at all?
Because 5 equals 10 divided by 2 exactly, and multiplication is associative. Multiplying by 5 is multiplying by a ten and a half in sequence, and the order of those two steps does not affect the result, so you can take whichever is easier first.

Sources

  1. Multiplication Series: Illustrating Number Properties With Arrays (opens in a new tab)Way, J., NRICH, Millennium Mathematics Project, University of Cambridge — accessed
  2. National curriculum in England: mathematics programmes of study (opens in a new tab)Department for Education (GOV.UK) — accessed
  3. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades (opens in a new tab)What Works Clearinghouse, Institute of Education Sciences (2021) — accessed

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