Adding and subtracting in your head · applications

Applications: Adding and Subtracting in Your Head

14 problems · Grade 3

PSLE

01

Three Prices Added in the Easiest Order

strategyLook for a Pair That Makes 100 / Add That Pair First

Mei buys a book for $37, a bag for $48 and a lamp for $63. (a) How much do the book and the lamp cost together? (b) How much do the three things cost altogether?

As written$37 book$48 bag$63 lamp
7 + 3 = 10, so 37 and 63 are a good pair.
Look at the ones digits of the three prices. 7 + 3 = 10, so 37 and 63 are a good pair to add first.
step 1 of 4

Numbers can be added in any order. Look for two prices that make 100, add them first, and add the third price last.

All steps
  1. Look at the ones digits of the three prices. 7 + 3 = 10, so 37 and 63 are a good pair to add first.
  2. (a) 30 + 60 = 90 and 7 + 3 = 10, so 37 + 63 = 90 + 10 = 100. The book and the lamp cost $100 together.
  3. Numbers can be added in any order and the total stays the same. So add the price of the bag last.
  4. (b) 100 + 48 = 148, so the three things cost $148 altogether. Check: 37 + 48 = 85 and 85 + 63 = 148.

answer(a) $100; (b) $148

techniqueAdding in Any Order

examsPSLE

Common pitfalls

  • Adding 37 and 63 to make 100 and then stopping. The total needs all three prices, so the $48 for the bag must still be added.
  • Thinking that 48 and 63 make 100 because 4 + 6 = 10. The ones digits 8 and 3 make another ten, so 48 + 63 is more than 100.
02

Stickers in Two Albums

strategySplit Both Numbers into Tens and Ones / Add Each Part, Then Add the Parts

Ali has 46 stickers in his red album and 38 stickers in his blue album. His sister has 253 stickers in one album and 145 stickers in another album. (a) How many stickers does Ali have? (b) How many stickers does his sister have?

TensOnes4640638308
46 is 40 and 6. 38 is 30 and 8.
Split each of Ali's numbers into tens and ones. 46 is 40 and 6. 38 is 30 and 8.
step 1 of 5

Split each number by place value. Add the tens to the tens and the ones to the ones, then add the parts together.

All steps
  1. Split each of Ali's numbers into tens and ones. 46 is 40 and 6. 38 is 30 and 8.
  2. Add the tens: 40 + 30 = 70. Add the ones: 6 + 8 = 14.
  3. (a) 70 + 14 = 84, so Ali has 84 stickers.
  4. Split his sister's numbers into hundreds, tens and ones, and add each part: 200 + 100 = 300, 50 + 40 = 90 and 3 + 5 = 8.
  5. (b) 300 + 90 + 8 = 398, so his sister has 398 stickers.

answer(a) 84 stickers; (b) 398 stickers

techniqueAdding by Splitting

examsPSLE

Common pitfalls

  • Writing the two sums side by side as 714. The 14 ones are 1 ten and 4 ones, so they are added to the 70: 70 + 14 = 84.
  • Adding a tens part to a ones part, such as 40 + 8. Tens are added to tens and ones are added to ones, so that no part is used twice or left out.
03

More People Get On a Train

strategyNumber Line / Jump to the Next Ten or Hundred, Then Jump the Rest

A train has 58 people on it. At the first station 27 more people get on. At the second station 38 more people get on. No one gets off. (a) How many people are on the train after the first station? (b) How many people are on the train after the second station?

5860+2
58 needs 2 more to reach 60, so 27 is split into 2 and 25.
58 needs 2 more to reach the next ten, which is 60. So split 27 into 2 and 25.
step 1 of 4

On a number line, jump first to the next ten or the next hundred. Then jump the rest of the number being added.

All steps
  1. 58 needs 2 more to reach the next ten, which is 60. So split 27 into 2 and 25.
  2. (a) 58 + 2 = 60 and 60 + 25 = 85, so there are 85 people on the train after the first station.
  3. 85 needs 15 more to reach 100. So split 38 into 15 and 23.
  4. (b) 85 + 15 = 100 and 100 + 23 = 123, so there are 123 people on the train after the second station.

answer(a) 85 people; (b) 123 people

techniqueBridging Through Ten

examsPSLE

Common pitfalls

  • Jumping 2 to reach 60 and then adding the whole 27 to get 87. The 2 is part of the 27, so only 25 is left to add.
  • Splitting 38 into 15 and 13. The two parts must add up to 38, and 15 + 13 is only 28. The right split is 15 and 23.
04

Two Classes of Almost the Same Size

strategyNear Doubles / Double the Smaller Number and Add 1

Class 3A has 38 pupils and Class 3B has 39 pupils. (a) What is double 38? (b) How many pupils are in the two classes altogether?

Class 3A38Class 3B381 more
39 is 1 more than 38, so the total is double 38 and 1 more.
39 is 1 more than 38. So 38 + 39 is double 38 and 1 more.
step 1 of 4

39 is 1 more than 38, so the total is double 38 and 1 more. Double 38 by doubling the tens and the ones.

All steps
  1. 39 is 1 more than 38. So 38 + 39 is double 38 and 1 more.
  2. Double 38 by splitting it into 30 and 8. Double 30 is 60 and double 8 is 16.
  3. (a) 60 + 16 = 76, so double 38 is 76.
  4. (b) 76 + 1 = 77, so there are 77 pupils altogether. Check: double 40 is 80, and 80 − 2 − 1 = 77.

answer(a) 76; (b) 77 pupils

techniqueDoubles and Near Doubles · Adding by Splitting

examsPSLE

Common pitfalls

  • Stopping at 76. That is two classes of 38, but Class 3B has 1 more pupil, so 1 must still be added.
  • Doubling 39 to get 78 and then adding 1. Double 39 already counts 1 pupil too many, so 1 must be taken away: 78 − 1 = 77.
05

Prices Just Below a Hundred

strategyRound the Price Up to the Hundred / Add, Then Take Away the Extra

Mrs Tan buys a fan for $56 and a kettle for $99. (a) How much do the fan and the kettle cost together? (b) She then buys a chair for $198. How much does she spend altogether?

56156+100
99 is 1 less than 100, so add 100 first: 56 + 100 = 156.
99 is 1 less than 100, so add 100 first: 56 + 100 = 156.
step 1 of 4

A price just below a hundred is rounded up to the hundred and added. Then the extra that was added is taken away.

All steps
  1. 99 is 1 less than 100, so add 100 first: 56 + 100 = 156.
  2. (a) That is 1 too many, so take 1 away: 156 − 1 = 155. The fan and the kettle cost $155 together.
  3. 198 is 2 less than 200, so add 200 first: 155 + 200 = 355.
  4. (b) That is 2 too many, so take 2 away: 355 − 2 = 353. She spends $353 altogether.

answer(a) $155; (b) $353

techniqueRounding and Adjusting

examsPSLE

Common pitfalls

  • Adding 100 and then adding 1 more to get 157. Adding 100 was already 1 more than the price, so the 1 is taken away.
  • Taking away 1 for the chair as well. 198 is 2 less than 200, so the extra to take away is 2.
06

Change Found by Counting Up

strategyNumber Line / Count Up from the Price to the Money Paid and Add the Jumps

A bicycle costs $367. Mr Lim pays $500 for it. (a) How much must be added to $367 to reach $400? (b) How much change does Mr Lim get?

367370+3
Jump to the next ten: 367 + 3 = 370.
Start at 367 and jump to the next ten: 367 + 3 = 370.
step 1 of 5

The change is the difference between 500 and 367. Count up from 367 to 500 in easy jumps and add the jumps.

All steps
  1. Start at 367 and jump to the next ten: 367 + 3 = 370.
  2. Jump to the next hundred: 370 + 30 = 400.
  3. (a) The two jumps are 3 + 30 = 33, so $33 must be added to $367 to reach $400.
  4. Jump from 400 to 500. This jump is 100.
  5. (b) Add all the jumps: 33 + 100 = 133, so Mr Lim gets $133 change. Check: 367 + 133 = 500.

answer(a) $33; (b) $133

techniqueCounting Up to Subtract · Bridging Through Ten

examsPSLE · GCSE Higher

Common pitfalls

  • Giving 500, the number where the jumps end, as the answer. The change is the sum of the jumps, 3 + 30 + 100.
  • Jumping from 370 to 400 and calling the jump 130. From 370 to 400 is only 30, because 37 tens and 3 tens make 40 tens.
07

Two Savings That Grow by the Same Amount

strategyThe Same Difference / Move Both Numbers by the Same Amount

Sam has saved $503 and Lily has saved $298. Grandma gives each child $2 more. (a) How much does each child have now? (b) How much more does Sam have than Lily?

Sam+2$503$505Lily+2$298$300
(a) 503 + 2 = 505 and 298 + 2 = 300.
(a) 503 + 2 = 505 and 298 + 2 = 300. Sam has $505 and Lily has $300 now.
step 1 of 4

When both numbers grow by the same amount, the difference between them stays the same. So subtract the new numbers, which are easier.

All steps
  1. (a) 503 + 2 = 505 and 298 + 2 = 300. Sam has $505 and Lily has $300 now.
  2. Both children were given the same amount, so the difference between their savings has not changed.
  3. Subtract the new numbers, because 300 is easy to take away: 505 − 300 = 205.
  4. (b) Sam has $205 more than Lily. This means 503 − 298 = 205 as well. Check: 298 + 205 = 503.

answer(a) Sam has $505 and Lily has $300; (b) $205

techniqueThe Same Difference

examsPSLE

Common pitfalls

  • Changing only 298 to 300 and working out 503 − 300 = 203. The difference stays the same only when both numbers move by the same amount.
  • Adding 2 to one number and taking 2 from the other. That keeps the total the same, not the difference. For a difference, both numbers move the same way.
08

Points Won and Lost on a Score Board

strategyCount in Hundreds and in Tens / Only Those Digits Change

Kai has 430 points on the score board. He wins 200 points in the next round. Then he loses 50 points. (a) How many points does Kai have after he wins 200? (b) How many points does he have after he loses 50?

430?+2004 hundreds + 2 hundreds
4 hundreds and 2 hundreds make 6 hundreds. The tens and the ones stay the same.
200 is 2 hundreds and 430 has 4 hundreds. 4 hundreds and 2 hundreds make 6 hundreds. The tens and the ones stay the same.
step 1 of 4

Adding hundreds is counting in hundreds, and subtracting tens is counting back in tens. Think of the numbers as hundreds or as tens.

All steps
  1. 200 is 2 hundreds and 430 has 4 hundreds. 4 hundreds and 2 hundreds make 6 hundreds. The tens and the ones stay the same.
  2. (a) 430 + 200 = 630, so Kai has 630 points after he wins.
  3. 50 is 5 tens and 630 is 63 tens. 63 tens take away 5 tens is 58 tens.
  4. (b) 630 − 50 = 580, so Kai has 580 points after he loses. Check: 580 + 50 = 630.

answer(a) 630 points; (b) 580 points

techniqueAdding and Subtracting Tens and Hundreds

Common pitfalls

  • Adding the 2 of 200 to the tens digit and writing 450. 200 is 2 hundreds, so the hundreds digit changes from 4 to 6.
  • Taking 3 tens from 5 tens and writing 620. The 5 tens are being taken away from 63 tens, and 63 − 5 = 58, so the answer is 580.
09

Buns Made and Sold, a Strategy Chosen for Each Step

strategyLook at the Numbers First / Choose the Strategy That Fits Them

A baker makes 148 buns in the morning and 152 buns in the afternoon. He sells 198 buns. (a) How many buns does he make altogether? (b) How many buns are left?

Morning10048Afternoon1005248 + 52 = 100
148 is 100 and 48. 152 is 100 and 52. The parts 48 and 52 make 100.
For the adding, look at the numbers. 148 is 100 and 48. 152 is 100 and 52. The parts 48 and 52 make 100, because 8 + 2 = 10 and 40 + 50 + 10 = 100.
step 1 of 5

Look at the numbers before choosing a strategy. 48 and 52 make 100, so the adding is done by pairing. 198 is close to 200, so the subtracting is done by rounding and adjusting.

All steps
  1. For the adding, look at the numbers. 148 is 100 and 48. 152 is 100 and 52. The parts 48 and 52 make 100, because 8 + 2 = 10 and 40 + 50 + 10 = 100.
  2. (a) 100 + 100 + 100 = 300, so he makes 300 buns altogether.
  3. For the subtracting, look at 198. It is 2 less than 200, so rounding and adjusting is the easy strategy.
  4. Take away 200 first: 300 − 200 = 100. That takes away 2 buns too many, so add 2 back.
  5. (b) 100 + 2 = 102, so 102 buns are left. Check: 198 + 102 = 300.

answer(a) 300 buns; (b) 102 buns

techniqueAdding in Any Order · Rounding and Adjusting

examsPSLE

Common pitfalls

  • Taking away 2 more after 300 − 200 = 100 and writing 98. Taking away 200 was already 2 too many, so the 2 is added back.
  • Subtracting 198 from 148 or from 152. The buns sold come out of all the buns made, so the two amounts are added first.
10

A Slip in a Pupil's Mental Subtraction

strategyCheck the Adjusting Step / Taking Away Too Much Means Adding Back

Ben has 245 stickers and gives away 98. He works out how many are left in his head. He says: 245 − 100 = 145, then 145 − 2 = 143. (a) How many stickers does Ben really have left? (b) How far is Ben's answer from the right answer?

245145143−100−2Ben: 245 − 100 = 145, then 145 − 2 = 143
The first line is right: 245 − 100 = 145.
Ben's first line is right. 98 is close to 100, and 245 − 100 = 145.
step 1 of 4

Ben rounded 98 up to 100, which is a good start. Check which way he should adjust after taking away 100.

All steps
  1. Ben's first line is right. 98 is close to 100, and 245 − 100 = 145.
  2. Ben took away 100, but he gives away only 98. He took away 2 stickers too many, so 145 is 2 too small.
  3. (a) Add the 2 back: 145 + 2 = 147. Ben really has 147 stickers left. Check: 147 + 100 − 2 = 245.
  4. (b) Ben took away 2 more instead and got 143. 147 − 143 = 4, so his answer is 4 too small.

answer(a) 147 stickers; (b) 4

techniqueRounding and Adjusting

examsPSLE

Common pitfalls

  • Saying that Ben's answer is only 2 away. He moved 2 the wrong way from 145, and the right answer is 2 the other way, so the two answers are 4 apart.
  • Adjusting the same way as in an addition. After adding 100 in place of 98, the 2 is taken away. After taking away 100 in place of 98, the 2 is added back.
11

Nine Dartboard Numbers Added in Tens, and the Two a Player Missed

strategyPair Numbers That Make 10 / Take What Was Hit from the Total

At a school fair, a dartboard has nine rings numbered 3, 7, 4, 8, 6, 2, 9, 1 and 5. Wei Ming throws 7 darts and hits 7 different rings. (a) Find the total of all nine numbers on the board. Pair numbers that make 10 to add them in your head. (b) The 7 numbers he hit add up to 34. What do the two numbers he missed add up to? (c) List every pair of numbers he could have missed.

3748629153 + 7104 + 6108 + 2109 + 11055
3 + 7, 4 + 6, 8 + 2 and 9 + 1 each make 10. Only 5 has no partner.
Look for pairs that make 10: 3 + 7, 4 + 6, 8 + 2 and 9 + 1. That uses eight of the nine numbers, and only 5 is left without a partner.
step 1 of 6

Numbers can be added in any order. Pair the numbers that make 10, add the tens, and then add the number left without a partner. The numbers hit and the numbers missed together make the whole board, so the missed pair is what is left of the total.

All steps
  1. Look for pairs that make 10: 3 + 7, 4 + 6, 8 + 2 and 9 + 1. That uses eight of the nine numbers, and only 5 is left without a partner.
  2. (a) Four tens and the 5: 10 + 10 + 10 + 10 + 5 = 45. The nine numbers add up to 45.
  3. The 7 numbers he hit and the 2 numbers he missed are all nine numbers together. So the numbers he missed make up what is left when 34 is taken from 45.
  4. (b) Count up from 34 to 45: 34 + 6 = 40 and 40 + 5 = 45, so the gap is 6 + 5 = 11. The two numbers he missed add up to 11. Check: 34 + 11 = 45.
  5. Find two different numbers on the board that make 11, starting from the smallest. 1 would need 10, which is not on the board. Then come 2 + 9, 3 + 8, 4 + 7 and 5 + 6. After that the pairs repeat the other way round, as 6 + 5.
  6. (c) He could have missed 2 and 9, 3 and 8, 4 and 7, or 5 and 6.

answer(a) 45; (b) 11; (c) 2 and 9, 3 and 8, 4 and 7, or 5 and 6

techniqueAdding in Any Order · Counting Up to Subtract

examsPSLE

Common pitfalls

  • Adding only the four pairs, 10 + 10 + 10 + 10 = 40, and leaving out the 5. Every number on the board is counted once, whether it has a partner or not, so the total is 45.
  • Listing 1 and 10, or 6 and 5, as more pairs. 10 is not on the board, and 6 and 5 is the same pair as 5 and 6, so there are only 4 pairs.
12

Apples Picked in the Morning and the Afternoon, Added as Near Doubles

strategyNear Doubles / Start from a Double You Know, Then Add 1 or Take 1 Away

Ben picks 6 apples in the morning and 7 apples in the afternoon. Mia picks 8 apples in the morning and 9 apples in the afternoon. (a) How many apples does Ben pick in all? (b) How many apples does Mia pick in all?

Benmorning6afternoon71 more6 + 7 is double 6 and 1 more
7 is 1 more than 6, so 6 + 7 is double 6 and 1 more.
7 is 1 more than 6. So 6 + 7 is double 6 and 1 more.
step 1 of 4

When two numbers are next to each other in counting, their sum is a near double. Start from the double of one of them, then add 1 or take 1 away.

All steps
  1. 7 is 1 more than 6. So 6 + 7 is double 6 and 1 more.
  2. (a) Double 6 is 6 + 6 = 12. One more is 12 + 1 = 13. Ben picks 13 apples.
  3. For Mia, start from double 9. 8 is 1 less than 9, so 8 + 9 is double 9 and 1 less.
  4. (b) Double 9 is 9 + 9 = 18. One less is 18 − 1 = 17. Mia picks 17 apples. Check: double 8 is 16, and 16 + 1 = 17.

answer(a) 13 apples; (b) 17 apples

techniqueDoubles and Near Doubles

examsPSLE

Common pitfalls

  • Doubling 9 and then adding 1 to get 19. Mia picked 8 in the morning, not 9, so double 9 already counts 1 apple too many. Take 1 away: 18 − 1 = 17.
  • Stopping at double 6 and saying 12 in part (a). Ben picked 7 in the afternoon, which is 1 more than 6, so add 1 more: 13.
13

Two Children's Ages, Some Years from Now

strategyThe Same Difference / Both Ages Grow by the Same Amount, So the Gap Stays

Sam is 9 years old. His sister Ana is 6 years old. (a) In 4 years, how many years older than Ana will Sam be? (b) One day Ana will be 15. How old will Sam be then?

now6Ana9Sam3 yearsnow: 9 − 6 = 3
Now Sam is 9 and Ana is 6: 9 − 6 = 3. Sam is 3 years older.
Now Sam is 9 and Ana is 6. The difference is 9 − 6 = 3, so Sam is 3 years older.
step 1 of 4

Every year, Sam and Ana both get 1 year older. Both ages go up by the same amount, so the difference between them stays the same.

All steps
  1. Now Sam is 9 and Ana is 6. The difference is 9 − 6 = 3, so Sam is 3 years older.
  2. Each year, both children get 1 year older. Both ages grow by the same amount, so the difference does not change.
  3. (a) In 4 years, Sam will still be 3 years older than Ana. Check: Sam will be 9 + 4 = 13, Ana will be 6 + 4 = 10, and 13 − 10 = 3.
  4. (b) When Ana is 15, Sam is still 3 years older: 15 + 3 = 18. Sam will be 18. Check: Ana gets 15 − 6 = 9 years older, and so does Sam: 9 + 9 = 18.

answer(a) 3 years older; (b) 18 years old

techniqueThe Same Difference

examsPSLE

Common pitfalls

  • Adding the 4 years to the difference and saying Sam will be 7 years older. Both children get 4 years older, so the gap between their ages does not change. It stays 3.
  • Adding Sam's age now to Ana's age then: 9 + 15 = 24. Sam's age is not the two ages put together. Sam is always 3 years older than Ana, so add the difference: 15 + 3 = 18.
14

Money in a Fair's Cash Box, Given and Paid Out in Tens and Hundreds

strategyAdd or Take Away in One Place / Only That Digit Changes

The school fair has $426 in its cash box. A parent gives $30 more. Then the fair pays $300 for a magic show. (a) How much is in the cash box after the parent's gift, and which digit changes? (b) How much is left after the magic show is paid for, and which digit changes?

HundredsTensOnesStart426
426 is 4 hundreds, 2 tens and 6 ones.
Split 426 by place value: 4 hundreds, 2 tens and 6 ones.
step 1 of 5

30 is 3 tens, so adding it changes only the tens digit. 300 is 3 hundreds, so taking it away changes only the hundreds digit. The other digits stay the same.

All steps
  1. Split 426 by place value: 4 hundreds, 2 tens and 6 ones.
  2. 30 is 3 tens. 2 tens and 3 tens make 5 tens. The hundreds and the ones stay the same.
  3. (a) 426 + 30 = 456. There is $456 in the cash box. Only the tens digit changed, from 2 to 5.
  4. 300 is 3 hundreds. 4 hundreds take away 3 hundreds leaves 1 hundred. The tens and the ones stay the same.
  5. (b) 456 − 300 = 156. There is $156 left. Only the hundreds digit changed, from 4 to 1.

answer(a) $456, and the tens digit changes from 2 to 5; (b) $156, and the hundreds digit changes from 4 to 1

techniqueAdding and Subtracting Tens and Hundreds

Common pitfalls

  • Adding the 3 to the ones digit and writing $429. 30 is 3 tens, not 3 ones, so the tens digit changes.
  • Taking the $300 from $426 in part (b) and writing $126. The parent's $30 is already in the cash box, so start from $456.