Sub­trac­tion is every­where: work­ing out change at the shop, how many pages are left in a book, or how many runs a team still needs. Many chil­dren learn one way to sub­tract, with bor­row­ing, and use it for every sum. There is a quicker way, and you can often do it in your head. It works for small num­bers and for very large ones.

The one big idea

We are going to take the trick from the ear­lier sub­trac­tion les­son and use it on num­bers of any size. Let me say the idea once more, because every­thing below is really that one sen­tence used again and again.

Remem­ber. A sub­trac­tion asks how far apart two num­bers are. Move both of them up by the same amount, or both down by the same amount, and the gap does not change.

So you move them until one of them becomes a round num­ber, and then the sum is easy.

Think of two chil­dren stand­ing on a stair­case, one on step 1212 and one on step 3030. They are 1818 steps apart. If both climb 88 steps, they stand on steps 2020 and 3838, still 1818 steps apart. That is why 30−1230 - 12 and 38−2038 - 20 have the same answer, and the sec­ond one is much eas­ier.

Two number lines from 10 to 40: a bar from 12 to 30 and a bar from 20 to 38, each marked gap 18, showing 30 - 12 and 38 - 20 give the same answer.
Mov­ing both num­bers up by the same amount keeps the gap between them the same.

Remem­ber. Add the same to both, or take the same from both. Never add to one and take from the other.

Which way should I move?

Look at the num­ber being taken away. If it is just below a round num­ber, like 2828, 9898 or 19981998, move up to make it round. If it is just above a round num­ber, like 1212 or 103103, move down.

Warm­ing up

Sin­gle dig­its

1) 9−09 - 07) 8−48 - 413) 8−78 - 7
2) 7−17 - 18) 7−47 - 414) 9−79 - 7
3) 8−28 - 29) 5−55 - 515) 8−88 - 8
4) 5−25 - 210) 9−59 - 516) 9−89 - 8
5) 9−39 - 311) 7−67 - 617) 9−59 - 5
6) 7−37 - 312) 9−69 - 618) 0−00 - 0

From a two-digit num­ber

1) 10−010 - 07) 13−213 - 213) 26−826 - 8
2) 15−815 - 88) 15−415 - 414) 56−956 - 9
3) 12−112 - 19) 27−727 - 715) 78−978 - 9
4) 14−614 - 610) 46−546 - 516) 26−826 - 8
5) 50−550 - 511) 11−711 - 717) 35−735 - 7
6) 80−780 - 712) 16−816 - 818) 56−956 - 9

Tak­ing away whole tens

Only the tens digit changes, so these can be done at a glance.

50−10=4050 - 10 = 4038−10=2838 - 10 = 28
60−10=5060 - 10 = 5046−40=646 - 40 = 6
80−10=7080 - 10 = 7057−30=2757 - 30 = 27
40−10=3040 - 10 = 3098−30=6898 - 30 = 68

Two-digit take away two-digit

Take the tens off both num­bers first. That is the move-both-down step, done ten at a time, and it shrinks the sum to some­thing small.

35 - 12

35−12=25−2=23\begin{aligned}35 - 12 &= 25 - 2 \\ &= 23\end{aligned}

Ten came off each. Then two off each again would give 23−023 - 0, which is the same answer.

25 - 13

25−13=15−3=12\begin{aligned}25 - 13 &= 15 - 3 \\ &= 12\end{aligned}

56 - 34

56−34=26−4=22\begin{aligned}56 - 34 &= 26 - 4 \\ &= 22\end{aligned}

Some­times mov­ing up is eas­ier, because it makes the num­ber being taken away into a round ten:

30 - 12, mov­ing down

30−12=20−2=18\begin{aligned}30 - 12 &= 20 - 2 \\ &= 18\end{aligned}

30 - 12, mov­ing up

30−12=38−20=18\begin{aligned}30 - 12 &= 38 - 20 \\ &= 18\end{aligned}

Ques­tionMov­ing downMov­ing upAnswer
80−3780 - 3750−750 - 783−4083 - 404343
52−3652 - 3622−622 - 656−4056 - 401616
92−4792 - 4752−7=50−552 - 7 = 50 - 595−5095 - 504545

Tak­ing away from a hun­dred

A hun­dred is the friend­liest num­ber to take from, because you can use the pairs that make ten.

100−1=99100 - 1 = 99100−10=90100 - 10 = 90
100−4=96100 - 4 = 96100−15=85100 - 15 = 85
100−7=93100 - 7 = 93100−73=27100 - 73 = 27
100−9=91100 - 9 = 91100−56=44100 - 56 = 44

100 - 12

100−12=90−2=88\begin{aligned}100 - 12 &= 90 - 2 \\ &= 88\end{aligned}

100 - 36

100−36=70−6=64\begin{aligned}100 - 36 &= 70 - 6 \\ &= 64\end{aligned}

Thirty came off both num­bers, leav­ing 70−670 - 6.

Three dig­its and more

123 - 8, mov­ing down

123−8=120−5=115\begin{aligned}123 - 8 &= 120 - 5 \\ &= 115\end{aligned}

123 - 8, mov­ing up

123−8=125−10=115\begin{aligned}123 - 8 &= 125 - 10 \\ &= 115\end{aligned}

125 - 82

125−82=105−62=45−2=43\begin{aligned}125 - 82 &= 105 - 62 \\ &= 45 - 2 \\ &= 43\end{aligned}

Or in one move: 125−82=143−100=43125 - 82 = 143 - 100 = 43.

342 - 57

342−57=300−15=290−5=285\begin{aligned}342 - 57 &= 300 - 15 \\ &= 290 - 5 \\ &= 285\end{aligned}

Or in one move: 342−57=385−100=285342 - 57 = 385 - 100 = 285.

600 - 123

600−123=500−23=480−3=477\begin{aligned}600 - 123 &= 500 - 23 \\ &= 480 - 3 \\ &= 477\end{aligned}

535 - 258

535−258=335−58=300−23=280−3=277\begin{aligned}535 - 258 &= 335 - 58 \\ &= 300 - 23 \\ &= 280 - 3 \\ &= 277\end{aligned}

Or in one move: 535−258=577−300=277535 - 258 = 577 - 300 = 277.

927 - 649

927−649=327−49=328−50=308−30=278\begin{aligned}927 - 649 &= 327 - 49 \\ &= 328 - 50 \\ &= 308 - 30 \\ &= 278\end{aligned}

Or in one move: 927−649=978−700=278927 - 649 = 978 - 700 = 278.

More worked exam­ples

71 - 28

71−28=73−30=43\begin{aligned}71 - 28 &= 73 - 30 \\ &= 43\end{aligned}

Two was added to both, so 2828 became the round num­ber 3030.

63 - 27

63−27=66−30=36\begin{aligned}63 - 27 &= 66 - 30 \\ &= 36\end{aligned}

400 - 168

400−168=399−167=232\begin{aligned}400 - 168 &= 399 - 167 \\ &= 232\end{aligned}

Tak­ing one off both turns 400400 into 399399. Now no digit needs bor­row­ing: 9−7=29 - 7 = 2, 9−6=39 - 6 = 3, 3−1=23 - 1 = 2.

5003 - 1998

5003−1998=5005−2000=3005\begin{aligned}5003 - 1998 &= 5005 - 2000 \\ &= 3005\end{aligned}

Adding two to both makes 19981998 into 20002000, and the rest is easy.

Tak­ing away from round num­bers

A thou­sand, ten thou­sand, a lakh: the work­ing is the same each time, and the answer is always a string of nines with the last dig­its made up to ten.

1000−1=9991000 - 1 = 9991000−70=9301000 - 70 = 930
1000−7=9931000 - 7 = 9931000−100=9001000 - 100 = 900
1000−10=9901000 - 10 = 9901000−300=7001000 - 300 = 700
1000−20=9801000 - 20 = 9801000−900=1001000 - 900 = 100
10000−1=999910000 - 1 = 999910000−100=990010000 - 100 = 9900
10000−5=999510000 - 5 = 999510000−800=920010000 - 800 = 9200
10000−10=999010000 - 10 = 999010000−1000=900010000 - 1000 = 9000
10000−50=995010000 - 50 = 995010000−6000=400010000 - 6000 = 4000
100000−1=99999100000 - 1 = 99999100000−500=99500100000 - 500 = 99500
100000−10=99990100000 - 10 = 99990100000−1232=98768100000 - 1232 = 98768
100000−50=99950100000 - 50 = 99950100000−567=99433100000 - 567 = 99433
100000−100=99900100000 - 100 = 99900100000−37012=62988100000 - 37012 = 62988

And the very same work­ing holds how­ever long the num­bers are: 325201−23578=301623325201 - 23578 = 301623, and 7125687920−2374392467=47512954537125687920 - 2374392467 = 4751295453.

The last trick with round num­bers is the same take-one-off idea: 1000−468=999−467=5321000 - 468 = 999 - 467 = 532. With a row of nines on top, each digit is taken from 99 and noth­ing is ever bor­rowed.

Your turn

Go back to the two warm­ing-up tables at the top of the les­son and write down each answer. Then try the mixed ques­tions below, choos­ing whether to move up or down each time.

Check your answers

Work these out, then look at the answers beside them.

Ques­tionAnswerQues­tionAnswer
500−123500 - 123377377224−76224 - 76148148
801−347801 - 34745445453−2953 - 292424
127−46127 - 46818193−3793 - 375656
723−439723 - 439284284934−277934 - 277657657
81−4981 - 493232123−8123 - 8115115
56−2856 - 282828357−123357 - 123234234
56−2356 - 233333100−46100 - 465454
94−2794 - 276767100−26100 - 267474
204−79204 - 7912512537−1837 - 181919
614−523614 - 5239191436−298436 - 298138138
500−327500 - 327173173705−214705 - 214491491

Check­ing an answer

Every sub­trac­tion can be checked with an addi­tion. If 342−57=285342 - 57 = 285, then 285+57285 + 57 must give back 342342. Add the ones: 5+7=125 + 7 = 12, write 22 and carry 11. Add the tens: 8+5+1=148 + 5 + 1 = 14, write 44 and carry 11. Add the hun­dreds: 2+1=32 + 1 = 3. The total is 342342, so the answer was right.

A sec­ond quick check is size. The answer to 927−649927 - 649 must be a lit­tle less than 300300, because 649649 is a lit­tle more than 600600 and 927927 is a lit­tle more than 900900. An answer like 378378 or 178178 would be easy to spot as wrong. The real answer, 278278, fits.

Use both checks on the ques­tions below. They take only a few sec­onds, and they catch most slips before a teacher does.

Com­mon mis­takes

  • Mov­ing the two num­bers in oppo­site direc­tions. 30−1230 - 12 is not 32−1032 - 10.
  • Mov­ing only one of the num­bers, which changes the gap.
  • For­get­ting that 9−59 - 5 and 26−826 - 8 appear twice in the tables; they have the same answer each time.
  • Mis­count­ing the zeros in a round num­ber such as 100000100000. Say it aloud as one lakh.
  • Not check­ing. Add your answer to the num­ber you took away; you should get back the first num­ber.

Key terms

Sub­trac­tion
Find­ing how far apart two num­bers are, or how many are left.
Dif­fer­ence
The answer to a sub­trac­tion.
Round num­ber
A num­ber end­ing in zeros, such as 3030, 100100 or 10001000.
Mov­ing up or down
Adding the same amount to both num­bers, or tak­ing the same amount from both.
Bor­row­ing
The step in col­umn sub­trac­tion where a ten is moved to the next col­umn; the tricks here avoid it.

Answers

Sin­gle dig­its

  1. 99
  2. 66
  3. 66
  4. 33
  5. 66
  6. 44
  7. 44
  8. 33
  9. 00
  10. 44
  11. 11
  12. 33
  13. 11
  14. 22
  15. 00
  16. 11
  17. 44
  18. 00

From a two-digit num­ber

  1. 1010
  2. 77
  3. 1111
  4. 88
  5. 4545
  6. 7373
  7. 1111
  8. 1111
  9. 2020
  10. 4141
  11. 44
  12. 88
  13. 1818
  14. 4747
  15. 6969
  16. 1818
  17. 2828
  18. 4747

Check your answers

The answers to the mixed ques­tions are printed beside each ques­tion in the table above: 377377, 454454, 8181, 284284, 3232, 2828, 3333, 6767, 125125, 9191, 173173 in the first col­umn, and 148148, 2424, 5656, 657657, 115115, 234234, 5454, 7474, 1919, 138138, 491491 in the sec­ond.