Written Addition and Subtraction: 10 Applications, Worked Step by Step

Two Parts Joined to Make a Whole

strategyPart-Whole Bar / Add the Parts in Columns

A library has 2467 story books and 2385 fact books. (a) How many books does the library have altogether? (b) The library then buys 579 new books. How many books does it have now?

The two kinds of books are two parts. The whole bar is the total, so add.

Draw one bar with two parts, 2467 story books and 2385 fact books. The whole bar is the total, so add the two parts.

step 1 of 5

The two kinds of books are two parts of one bar. The whole bar is the total, so the parts are added. The numbers are large, so the adding is done in columns.

Draw one bar with two parts, 2467 story books and 2385 fact books. The whole bar is the total, so add the two parts.
Write 2385 under 2467 with the ones under the ones. Add the ones: 7 + 5 = 12. Write 2 and carry 1 to the tens. Add the tens: 6 + 8 + 1 = 15. Write 5 and carry 1 to the hundreds.
(a) Add the hundreds: 4 + 3 + 1 = 8. Add the thousands: 2 + 2 = 4. The library has 4852 books altogether.
The 579 new books are one more part. Write 579 under 4852. Ones: 2 + 9 = 11, so write 1 and carry 1. Tens: 5 + 7 + 1 = 13, so write 3 and carry 1.
(b) Hundreds: 8 + 5 + 1 = 14, so write 4 and carry 1. Thousands: 4 + 1 = 5. The library has 5431 books now. Check: 5431 − 579 = 4852.

answer(a) 4852 books; (b) 5431 books

techniqueCarrying in More Than One Column · Carrying · Lining Up the Columns

examsPSLE

Common pitfalls

Forgetting the 1 that was carried. In the tens, 6 + 8 is 14, but the carried 1 makes it 15. Write the small 1 above the next column so that it is not forgotten.
Writing 579 with its 5 under the 4 of 4852. The 5 means 5 hundreds, so it goes under the 8 in the hundreds column.

A Whole and One Part, the Other Part Missing

strategyPart-Whole Bar / Subtract the Part You Know in Columns

There are 5342 people at a concert. 2678 of them are adults and the rest are children. (a) How many children are at the concert? (b) 1395 of the children are girls. How many boys are at the concert?

The people are the whole. The adults are one part, so subtract to find the children.

Draw one bar for the 5342 people. One part is the 2678 adults. The other part is the children, so subtract: 5342 − 2678.

step 1 of 5

All the people are the whole bar. The adults are one part and the children are the other part. When the whole and one part are known, the other part is found by subtracting.

Draw one bar for the 5342 people. One part is the 2678 adults. The other part is the children, so subtract: 5342 − 2678.
Write 2678 under 5342. Ones: 2 is less than 8, so borrow a ten. The tens become 3 and the ones become 12. 12 − 8 = 4.
(a) Tens: 3 is less than 7, so borrow a hundred: 13 − 7 = 6. Hundreds: 2 is less than 6, so borrow a thousand: 12 − 6 = 6. Thousands: 4 − 2 = 2. There are 2664 children.
The children are a whole too. The girls are one part, so the boys are 2664 − 1395. Ones: borrow a ten, 14 − 5 = 9. Tens: 5 is less than 9, so borrow a hundred: 15 − 9 = 6.
(b) Hundreds: 5 − 3 = 2. Thousands: 2 − 1 = 1. There are 1269 boys. Check: 1269 + 1395 = 2664.

answer(a) 2664 children; (b) 1269 boys

techniqueBorrowing

examsPSLE · GCSE Higher

Common pitfalls

Taking the smaller digit from the larger digit in every column, such as 8 − 2 in the ones. The top number is 5342, so the ones are 12 − 8 after a ten is borrowed.
Borrowing a ten and leaving the tens digit as 4. The ten that was borrowed has left the tens column, so the 4 must be crossed out and 3 written above it.

Two Amounts Compared: How Many More

strategyComparison Bars / The Difference Is a Subtraction

A zoo had 4052 visitors on Saturday and 2786 visitors on Sunday. (a) How many more visitors came on Saturday than on Sunday? (b) On Monday there were 1895 visitors. How many fewer visitors came on Monday than on Sunday?

The dotted part is how much longer the Saturday bar is. It is the difference, so subtract.

Draw a bar for Saturday and a shorter bar for Sunday under it. The part of the Saturday bar that goes past the Sunday bar is the difference, so subtract: 4052 − 2786.

step 1 of 5

Two amounts are compared with two bars, one under the other. The part of the longer bar that goes past the shorter bar is the difference. The difference is found by subtracting the smaller amount from the larger amount.

Draw a bar for Saturday and a shorter bar for Sunday under it. The part of the Saturday bar that goes past the Sunday bar is the difference, so subtract: 4052 − 2786.
Ones: 2 is less than 6, so borrow a ten. The tens become 4 and the ones become 12. 12 − 6 = 6.
(a) Tens: 4 is less than 8, and the hundreds digit is 0, so borrow from the thousands. The thousands become 3, the hundreds become 9 and the tens become 14. 14 − 8 = 6. Hundreds: 9 − 7 = 2. Thousands: 3 − 2 = 1. 1266 more visitors came on Saturday.
Now compare Monday with Sunday. Draw the Monday bar under the Sunday bar and subtract: 2786 − 1895. Ones: 6 − 5 = 1. Tens: 8 is less than 9, so borrow a hundred: 18 − 9 = 9.
(b) Hundreds: 6 is less than 8, so borrow a thousand: 16 − 8 = 8. Thousands: 1 − 1 = 0. 891 fewer visitors came on Monday. Check: 1895 + 891 = 2786.

answer(a) 1266 visitors; (b) 891 visitors

techniqueBorrowing Past a Zero · Borrowing

examsPSLE

Common pitfalls

Adding the two numbers because the question says "more". "How many more" asks for the difference between the two bars, and a difference is found by subtracting.
Trying to borrow from the 0 in the hundreds of 4052 and stopping there. A zero has nothing to lend, so borrow from the thousands first. The 0 becomes 10, and then it lends one hundred to the tens and becomes 9.

One Amount Is More Than Another: Find the Larger Amount

strategyComparison Bars / The Longer Bar Is the Shorter Bar and the Extra Part

A red ship carries 2586 boxes. A blue ship carries 1775 more boxes than the red ship. (a) How many boxes does the blue ship carry? (b) The blue ship has room for 6450 boxes. How many more boxes can it take?

The blue bar is the red bar with 1775 more on the end, so add.

Draw the red bar, 2586. The blue bar is as long as the red bar with 1775 more on the end. So add: 2586 + 1775.

step 1 of 5

The blue ship has more boxes, so its bar is the longer one. The longer bar is as long as the red bar with the extra 1775 on the end, so the larger amount is found by adding.

Draw the red bar, 2586. The blue bar is as long as the red bar with 1775 more on the end. So add: 2586 + 1775.
Ones: 6 + 5 = 11. Write 1 and carry 1. Tens: 8 + 7 + 1 = 16. Write 6 and carry 1.
(a) Hundreds: 5 + 7 + 1 = 13. Write 3 and carry 1. Thousands: 2 + 1 + 1 = 4. The blue ship carries 4361 boxes.
The room on the blue ship is a whole, 6450. The 4361 boxes are one part of it, so subtract: 6450 − 4361. Ones: 0 is less than 1, so borrow a ten: 10 − 1 = 9. Tens: 4 is less than 6, so borrow a hundred: 14 − 6 = 8.
(b) Hundreds: 3 − 3 = 0. Thousands: 6 − 4 = 2. The blue ship can take 2089 more boxes. Check: 4361 + 2089 = 6450.

answer(a) 4361 boxes; (b) 2089 boxes

techniqueCarrying in More Than One Column · Borrowing

examsPSLE

Common pitfalls

Subtracting 1775 from 2586. The blue ship has more boxes than the red ship, so its bar is longer than the red bar. A longer bar needs an addition.
Forgetting the carried 1 in the thousands. 2 + 1 is 3, but a 1 was carried from the hundreds, so the thousands digit is 4.

One Amount Is More Than Another: Find the Smaller Amount

strategyComparison Bars / Take the Extra Part Off the Longer Bar

A shop sold 5430 pens in June. That is 2865 more pens than it sold in May. (a) How many pens did the shop sell in May? (b) The shop hoped to sell 3470 pens in May. How many fewer pens than that did it sell?

June is 2865 more than May, so the May bar is the shorter one. Subtract.

June is the longer bar, 5430. The May bar is shorter by 2865. The word "more" is in the story, but May is the smaller amount, so subtract: 5430 − 2865.

step 1 of 5

June has the extra pens, so the June bar is the longer one and the May bar is the shorter one. The shorter bar is the longer bar with the extra part taken off, so the smaller amount is found by subtracting.

June is the longer bar, 5430. The May bar is shorter by 2865. The word "more" is in the story, but May is the smaller amount, so subtract: 5430 − 2865.
Ones: 0 is less than 5, so borrow a ten. The tens become 2 and the ones become 10. 10 − 5 = 5. Tens: 2 is less than 6, so borrow a hundred: 12 − 6 = 6.
(a) Hundreds: 3 is less than 8, so borrow a thousand: 13 − 8 = 5. Thousands: 4 − 2 = 2. The shop sold 2565 pens in May. Check: 2565 + 2865 = 5430.
Now compare May with the number the shop hoped for. Subtract: 3470 − 2565. Ones: borrow a ten, 10 − 5 = 5. Tens: 6 − 6 = 0.
(b) Hundreds: 4 is less than 5, so borrow a thousand: 14 − 5 = 9. Thousands: 2 − 2 = 0. The shop sold 905 fewer pens than it hoped for.

answer(a) 2565 pens; (b) 905 pens

techniqueBorrowing

examsPSLE · GCSE Higher

Common pitfalls

Adding 5430 and 2865 because the story says "more". June has the extra pens, so May is smaller than June. Draw the two bars first to see which one is longer.
Leaving the tens column empty in part (b). 6 − 6 = 0, and the 0 must be written, or 905 turns into 95.

Some Are Added, Then Some Are Taken Away

strategyTwo Steps / The First Answer Starts the Second Step

A fruit seller has 1465 apples. A lorry brings him 2378 more apples. Then he sells 1856 apples. (a) How many apples does he have after the lorry comes? (b) How many apples does he have left after selling?

First the apples are joined, so add.

First the apples are joined. Draw the 1465 apples and the 2378 apples as two parts of one bar. Add: 1465 + 2378.

step 1 of 5

The story has two changes. First the apples are joined, which is an addition. Then some are sold, which is a subtraction. The answer to the first step is the number that the second step starts from.

First the apples are joined. Draw the 1465 apples and the 2378 apples as two parts of one bar. Add: 1465 + 2378.
(a) Ones: 5 + 8 = 13, so write 3 and carry 1. Tens: 6 + 7 + 1 = 14, so write 4 and carry 1. Hundreds: 4 + 3 + 1 = 8. Thousands: 1 + 2 = 3. He has 3843 apples.
Now 3843 is the whole. The 1856 apples sold are one part, and the apples left are the other part. Subtract: 3843 − 1856.
Ones: 3 is less than 6, so borrow a ten: 13 − 6 = 7. Tens: 3 is less than 5, so borrow a hundred: 13 − 5 = 8.
(b) Hundreds: 7 is less than 8, so borrow a thousand: 17 − 8 = 9. Thousands: 2 − 1 = 1. He has 1987 apples left. Check: 1987 + 1856 = 3843.

answer(a) 3843 apples; (b) 1987 apples

techniqueTwo-Step Word Problems · Carrying in More Than One Column · Borrowing

examsPSLE

Common pitfalls

Subtracting 1856 from 1465, the number at the start. The apples are sold after the lorry comes, so subtract from 3843.
Doing only one step. The story has two changes, first more apples and then fewer apples, so it needs an addition and then a subtraction.

One Amount Is More, Then the Total of Both

strategyTwo Steps / Find the Second Amount Before Adding

Ben has 368 stamps. Ali has 257 more stamps than Ben. (a) How many stamps does Ali have? (b) How many stamps do the two boys have altogether?

Ali has as many stamps as Ben and 257 more. Find Ali's number before the total.

Draw Ben's bar, 368. Ali's bar is as long as Ben's bar with 257 more on the end. The total needs Ali's number, so find it first.

step 1 of 4

The total needs both boys' numbers, and the story gives only Ben's. So the first step finds Ali's number from the comparison, and the second step adds the two numbers.

Draw Ben's bar, 368. Ali's bar is as long as Ben's bar with 257 more on the end. The total needs Ali's number, so find it first.
(a) 368 + 257. Ones: 8 + 7 = 15, so write 5 and carry 1. Tens: 6 + 5 + 1 = 12, so write 2 and carry 1. Hundreds: 3 + 2 + 1 = 6. Ali has 625 stamps.
Now add the two boys' stamps: 368 + 625. Ones: 8 + 5 = 13, so write 3 and carry 1.
(b) Tens: 6 + 2 + 1 = 9. Hundreds: 3 + 6 = 9. The two boys have 993 stamps altogether. Check: 368 + 368 + 257 = 993.

answer(a) 625 stamps; (b) 993 stamps

techniqueCarrying · Two-Step Word Problems

examsPSLE · GCSE Higher

Common pitfalls

Adding 368 and 257 and giving 625 as the total. 257 is not Ali's number of stamps. It is how many more stamps he has than Ben.
Carrying to the hundreds when nothing was carried. In 368 + 625 the tens are 6 + 2 + 1 = 9, which is less than 10, so nothing goes to the hundreds.

Three Amounts in a Chain

strategyThree Bars / Work Along the Chain One Link at a Time

A baker bakes 2657 buns on Friday. On Saturday he bakes 1485 more buns than on Friday. On Sunday he bakes 2586 fewer buns than on Saturday. (a) How many buns does he bake on Saturday? (b) How many buns does he bake on Sunday?

Saturday is Friday and 1485 more. Sunday is Saturday with 2586 taken off. Find Saturday first.

Draw three bars. The Saturday bar is the Friday bar with 1485 more. The Sunday bar is the Saturday bar with 2586 taken off. Find Saturday first, because Sunday is compared with Saturday.

step 1 of 5

Each sentence compares two days. Friday gives Saturday, and Saturday gives Sunday, so the three bars are found in that order.

Draw three bars. The Saturday bar is the Friday bar with 1485 more. The Sunday bar is the Saturday bar with 2586 taken off. Find Saturday first, because Sunday is compared with Saturday.
2657 + 1485. Ones: 7 + 5 = 12, so write 2 and carry 1. Tens: 5 + 8 + 1 = 14, so write 4 and carry 1.
(a) Hundreds: 6 + 4 + 1 = 11, so write 1 and carry 1. Thousands: 2 + 1 + 1 = 4. The baker bakes 4142 buns on Saturday.
4142 − 2586. Ones: 2 is less than 6, so borrow a ten: 12 − 6 = 6. Tens: 3 is less than 8, so borrow a hundred. The hundreds become 0 and the tens become 13. 13 − 8 = 5.
(b) Hundreds: 0 is less than 5, so borrow a thousand: 10 − 5 = 5. Thousands: 3 − 2 = 1. The baker bakes 1556 buns on Sunday. Check: 1556 + 2586 = 4142.

answer(a) 4142 buns; (b) 1556 buns

techniqueTwo-Step Word Problems · Borrowing

examsPSLE · GCSE Higher

Common pitfalls

Taking 2586 away from Friday's 2657. The story says fewer than on Saturday, so subtract from 4142.
Thinking that the 1 in the hundreds of 4142 cannot lend. It lends its one hundred and becomes 0. Then the 0 borrows from the thousands and becomes 10.

Two Things Bought, the Money Left

strategyPart-Whole Bar / Add What Is Spent, Then Borrow Past the Zeros

Mrs Lim has $1000. She buys a bed for $468 and a desk for $275. (a) How much does she spend altogether? (b) How much money does she have left?

The $1000 is the whole. Two parts are spent and the last part is left.

Draw one bar for the $1000. Two parts are spent, $468 and $275. The last part is the money left. Find the money spent first.

step 1 of 4

The $1000 is the whole bar. The two prices are two parts, and the money left is the last part. Add the two prices, then subtract the money spent from $1000.

Draw one bar for the $1000. Two parts are spent, $468 and $275. The last part is the money left. Find the money spent first.
(a) 468 + 275. Ones: 8 + 5 = 13, so write 3 and carry 1. Tens: 6 + 7 + 1 = 14, so write 4 and carry 1. Hundreds: 4 + 2 + 1 = 7. She spends $743 altogether.
The money left is 1000 − 743. Ones: 0 is less than 3. The tens and the hundreds are 0 too, so borrow from the thousands. The thousands become 0, the hundreds become 9, the tens become 9 and the ones become 10.
(b) Ones: 10 − 3 = 7. Tens: 9 − 4 = 5. Hundreds: 9 − 7 = 2. She has $257 left. Check: 1000 − 468 = 532 and 532 − 275 = 257.

answer(a) $743; (b) $257

techniqueBorrowing Past a Zero · Two-Step Word Problems

examsPSLE

Common pitfalls

Writing 9 above every zero. The ones column becomes 10, not 9, because it keeps the ten that it borrowed and lends nothing.
Taking away only the bed and stopping at $532. She buys two things, so both prices come off the $1000.

A Slip in a Pupil's Column Working

strategyCheck the Columns / Ones Under Ones, Tens Under Tens

Jun adds 2465 and 378. He writes the 3 of 378 under the 2 of 2465, so his digits are not lined up, and he gets 6245. (a) What is the correct sum? (b) How far is Jun's answer from the correct sum?

Jun's 8 is under the tens, but it is 8 ones. Every digit of 378 is one column too far left.

Look at Jun's working. His 8 is under the 6 tens, but it is 8 ones. Every digit of 378 is one column too far to the left, so he has added 3780.

step 1 of 5

In a column sum every digit sits in the column of its place value. Find the column that is wrong in Jun's working, do the sum again with the digits lined up, and subtract to see how far off he was.

Look at Jun's working. His 8 is under the 6 tens, but it is 8 ones. Every digit of 378 is one column too far to the left, so he has added 3780.
Write 378 again with the ones under the ones. Ones: 5 + 8 = 13, so write 3 and carry 1. Tens: 6 + 7 + 1 = 14, so write 4 and carry 1.
(a) Hundreds: 4 + 3 + 1 = 8. Thousands: 2. The correct sum is 2843.
To find how far off Jun is, subtract the correct sum from his answer: 6245 − 2843. Ones: 5 − 3 = 2. Tens: 4 − 4 = 0.
(b) Hundreds: 2 is less than 8, so borrow a thousand: 12 − 8 = 4. Thousands: 5 − 2 = 3. Jun's answer is 3402 too large. Check: 3780 − 378 = 3402.

answer(a) 2843; (b) 3402

techniqueLining Up the Columns · Carrying

examsPSLE

Common pitfalls

Lining the numbers up on the left, as words are. Numbers are lined up on the right, so that the ones are under the ones.
Trusting 6245 without an estimate. 2465 + 378 is about 2500 + 400 = 2900, so an answer above 6000 must be wrong.