Multiplication and division · applications

Applications: Multiplication and Division

10 question types · Primary 2 · each worked step by step with a figure that follows the steps

PSLE

01

Cakes on Plates in Equal Groups

strategyDraw the Equal Groups / Count in Equal Steps

There are 5 plates on a table. Each plate has 3 cakes on it. (a) How many cakes are there altogether? (b) Mum brings 1 more plate with 3 cakes on it. How many cakes are there now?

Each plate is one group of 3 cakes. There are 5 equal groups.
Draw 5 plates. Draw 3 cakes on each plate. These are 5 equal groups of 3.
step 1 of 4

Every plate has the same number of cakes, so the plates are equal groups. Draw the groups and count them in threes.

  1. Draw 5 plates. Draw 3 cakes on each plate. These are 5 equal groups of 3.
  2. Count in threes, one plate at a time: 3, 6, 9, 12, 15.
  3. (a) 5 groups of 3 is 5 × 3 = 15. There are 15 cakes altogether.
  4. (b) One more plate brings 3 more cakes: 15 + 3 = 18. Now there are 6 groups of 3, and 6 × 3 = 18 cakes.

answer(a) 15 cakes; (b) 18 cakes

techniqueEqual Groups

examsPSLE

Common pitfalls

  • Adding the two numbers: 5 + 3 = 8. The 5 is the number of plates and the 3 is the number of cakes on each plate, so the cakes are 3 + 3 + 3 + 3 + 3, not 5 + 3.
  • Adding 1 in part (b) to get 16. One more plate is one more group of 3, so 3 cakes are added, not 1.
02

Rows of Chairs Seen from the Front and from the Side

strategyDraw Rows and Columns / Count by Rows, Then by Columns

The chairs in a hall stand in 3 rows. There are 4 chairs in each row. (a) How many chairs are there? (b) Mei stands at the side of the hall. From there she sees the same chairs in 4 rows. How many chairs are in each of her rows?

4 in a row3 rows
One dot is one chair. There are 3 rows with 4 chairs in each row.
Draw the chairs as dots. Make 3 rows and put 4 dots in each row.
step 1 of 5

Draw one dot for each chair, in rows and columns. The same dots can be counted by rows or by columns, and the total does not change.

  1. Draw the chairs as dots. Make 3 rows and put 4 dots in each row.
  2. Count by rows. Each row has 4 chairs: 4, 8, 12.
  3. (a) 3 rows of 4 is 3 × 4 = 12. There are 12 chairs.
  4. Now look from the side. Each column is a row for Mei. There are 4 columns, so she sees 4 rows.
  5. (b) Each column has 3 dots, so each of her rows has 3 chairs. Check: 3, 6, 9, 12, so 4 × 3 = 12, the same 12 chairs.

answer(a) 12 chairs; (b) 3 chairs

techniqueRows and Columns · Equal Groups

examsPSLE

Common pitfalls

  • Saying that Mei's rows also have 4 chairs. From the side, the 4 is the number of rows. The number in each row is the number of dots in one column, which is 3.
  • Thinking that 4 rows of 3 is a different number of chairs from 3 rows of 4. No chair has moved, so both are 12: 4 × 3 = 3 × 4.
03

Sweets Shared Equally Among Children

strategyGive Out One at a Time / Count What Each One Gets

12 sweets are shared equally among 3 children. (a) How many sweets does each child get? (b) One more child comes, and the 12 sweets are shared equally among the 4 children instead. How many sweets does each child get now?

9 sweets still to give
One plate for each child. The first round gives each child 1 sweet.
Draw 3 plates, one for each child. Give 1 sweet to each child in turn. One round uses 3 sweets.
step 1 of 4

Sharing equally means every child gets the same number. Draw a plate for each child and give out the sweets one at a time until none are left.

  1. Draw 3 plates, one for each child. Give 1 sweet to each child in turn. One round uses 3 sweets.
  2. Keep going until all 12 sweets are given out. That takes 4 rounds, so each plate has 4 sweets.
  3. (a) 12 ÷ 3 = 4. Each child gets 4 sweets. Check: 3 × 4 = 12.
  4. (b) Now draw 4 plates and share the 12 sweets again. Each plate gets 3 sweets, so 12 ÷ 4 = 3. Each child gets 3 sweets.

answer(a) 4 sweets; (b) 3 sweets

techniqueSharing Equally

examsPSLE

Common pitfalls

  • Subtracting: 12 − 3 = 9. That takes away 3 sweets once. Sharing among 3 children means making 3 equal groups, which is 12 ÷ 3.
  • Thinking that more children means more sweets each. The 12 sweets do not change, so when 4 children share them each child gets fewer: 3 instead of 4.
04

Cards Put into Packs of 5

strategyMake Groups of the Same Size / Count the Groups

Ben has 20 cards. He puts them into packs of 5. (a) How many packs does he make? (b) Ben gets 10 more cards and packs them in the same way. How many packs does he have now?

15 cards still to pack
Five cards in a ring make 1 pack. Do the same with the cards that are left.
Take 5 cards and put a ring round them. That is 1 pack. Do the same with the cards that are left.
step 1 of 4

The size of each group is known, and the number of groups is asked for. Take away 5 cards at a time and count how many times this can be done.

  1. Take 5 cards and put a ring round them. That is 1 pack. Do the same with the cards that are left.
  2. Count in fives until 20 is reached: 5, 10, 15, 20. That is 4 fives.
  3. (a) 20 ÷ 5 = 4. He makes 4 packs. Check: 4 × 5 = 20.
  4. (b) Now there are 20 + 10 = 30 cards. Count on in fives: 25, 30. That is 6 fives, so 30 ÷ 5 = 6. He has 6 packs.

answer(a) 4 packs; (b) 6 packs

techniqueSharing Equally · Equal Groups

examsPSLE

Common pitfalls

  • Drawing 5 packs and sharing the cards among them. The 5 is the number of cards in each pack, not the number of packs. Here the number of packs is what must be found.
  • Answering 10 ÷ 5 = 2 packs in part (b). Those are only the new packs. He still has the first 4 packs, so he has 4 + 2 = 6 packs.
05

Bags of Apples with the Number of Bags Missing

strategyWrite the Missing-Number Sentence / Divide to Undo the Multiplying

Some bags have 4 apples in each. There are 28 apples in all the bags together. (a) How many bags are there? (b) Dad buys 2 more bags of 4 apples. How many apples are there now?

? × 4 = 28
One bag has 4 apples. The number of bags is missing: ? × 4 = 28.
Write what is known. A number of bags times 4 apples makes 28 apples: ? × 4 = 28.
step 1 of 4

The story is a multiplication with one number missing. Dividing undoes multiplying, so the missing number is found by dividing.

  1. Write what is known. A number of bags times 4 apples makes 28 apples: ? × 4 = 28.
  2. Dividing undoes multiplying, so find 28 ÷ 4. Count in fours up to 28: 4, 8, 12, 16, 20, 24, 28. That is 7 fours.
  3. (a) 28 ÷ 4 = 7. There are 7 bags. Check: 7 × 4 = 28.
  4. (b) Now there are 7 + 2 = 9 bags. Count on in fours from 28: 32, 36. So 9 × 4 = 36 apples.

answer(a) 7 bags; (b) 36 apples

techniqueMultiplying and Dividing Undo Each Other · Equal Groups

examsPSLE

Common pitfalls

  • Multiplying the two numbers in the question: 28 × 4. The 28 is already the total. The missing number is the number of bags, and it is found by dividing the total by 4.
  • Adding 2 apples in part (b) to get 30. Dad buys 2 bags, and each bag has 4 apples, so 2 × 4 = 8 apples are added.
06

Stickers Shared with Some Left Over

strategyShare in Rounds / Stop When There Are Too Few for Another Round

A teacher has 14 stickers. She shares them equally among 4 children and gives each child as many as she can. (a) How many stickers does each child get? (b) How many stickers are left over?

10 stickers still to give
One plate for each child. The first round gives each child 1 sticker.
Draw 4 plates, one for each child. Give 1 sticker to each child in turn. One round uses 4 stickers.
step 1 of 4

Give out the stickers in rounds, one to each child. When there are not enough stickers for every child to get one more, the sharing stops and the rest are left over.

  1. Draw 4 plates, one for each child. Give 1 sticker to each child in turn. One round uses 4 stickers.
  2. After 3 rounds, 3 × 4 = 12 stickers are given out and 14 − 12 = 2 are left. Another round needs 4 stickers, and 2 is not enough.
  3. (a) Each child gets 3 stickers.
  4. (b) 2 stickers are left over. Check: 4 × 3 + 2 = 14.

answer(a) 3 stickers; (b) 2 stickers

techniqueSharing Equally

examsPSLE

Common pitfalls

  • Giving the 2 stickers that are left to two of the children. Then two children have 4 stickers and two have 3, and the sharing is no longer equal.
  • Stopping after 2 rounds and saying that 6 stickers are left over. 6 is enough for another round of 4. What is left over must be smaller than the number of children.
07

Equal Jumps Along a Number Line

strategyRepeated Addition on a Number Line / Write It as a Multiplication

A number line is painted on the playground. Ravi starts at 0. Each jump takes him 3 numbers along the line. (a) Which number does he land on after 6 jumps? (b) How many jumps from 0 does he need to land on 30?

036912151821242730+3
Start at 0. Every jump is 3 numbers long.
Draw the number line. Start at 0 and mark jumps of 3.
step 1 of 5

Every jump adds the same number, so the jumps are a repeated addition. Adding the same number again and again is a multiplication.

  1. Draw the number line. Start at 0 and mark jumps of 3.
  2. Six jumps is 3 + 3 + 3 + 3 + 3 + 3. The jumps land on 3, 6, 9, 12, 15, 18.
  3. (a) 6 jumps of 3 is 6 × 3 = 18. He lands on 18.
  4. Keep jumping in threes and count the jumps: 21, 24, 27, 30. These are jumps 7, 8, 9 and 10.
  5. (b) He needs 10 jumps to land on 30, because 10 × 3 = 30.

answer(a) 18; (b) 10 jumps

techniqueEqual Groups · Skip Counting

examsPSLE

Common pitfalls

  • Adding the two numbers: 6 + 3 = 9. That is one jump of 3 starting from 6. Six jumps of 3 starting from 0 is 3 added six times.
  • Counting the marks on the line and not the jumps. From 0 to 30 there are 11 marks but only 10 jumps, because the mark at 0 is where he starts.
08

Boxes of Pencils with Some Broken

strategyTwo Steps: Multiply to Find the Total, Then Subtract

A teacher has 3 boxes of pencils. There are 5 pencils in each box. (a) How many pencils does she have altogether? (b) 2 of the pencils break. How many pencils are still good?

Each box is one group of 5 pencils. There are 3 equal groups.
Draw 3 boxes. Draw 5 pencils in each box.
step 1 of 4

There are two steps. First find how many pencils there are altogether. Then take away the broken ones.

  1. Draw 3 boxes. Draw 5 pencils in each box.
  2. (a) Count in fives: 5, 10, 15. So 3 × 5 = 15. She has 15 pencils altogether.
  3. Cross out the 2 broken pencils.
  4. (b) 15 − 2 = 13. There are 13 good pencils.

answer(a) 15 pencils; (b) 13 pencils

techniqueEqual Groups · Two-Step Word Problems

examsPSLE

Common pitfalls

  • Taking the 2 away from the 5 first and then finding 3 × 3 = 9. That breaks 2 pencils in every box. Only 2 pencils break altogether, so find the total first and subtract once.
  • Stopping at 15. That is the number of pencils before any broke. Part (b) asks for the pencils that are still good, so the 2 broken pencils must be taken away.
09

Pairs of Socks

strategyDouble to Multiply by 2 / Halve to Divide by 2

A pair of socks is 2 socks. (a) Sam has 8 pairs of socks. How many socks does he have? (b) There are 18 socks hanging on a line. How many pairs is that?

Each pair is a group of 2 socks. Here are 8 pairs.
Draw 8 pairs. Each pair is a group of 2 socks.
step 1 of 4

Pairs are groups of 2. Going from pairs to socks is multiplying by 2, which is doubling. Going from socks to pairs is dividing by 2, which is halving.

  1. Draw 8 pairs. Each pair is a group of 2 socks.
  2. (a) 8 groups of 2 is double 8: 8 × 2 = 16. Sam has 16 socks.
  3. For the line, put the 18 socks into twos. Making pairs is dividing by 2, which is finding half.
  4. (b) Half of 18 is 9, so 18 ÷ 2 = 9. There are 9 pairs. Check: 9 × 2 = 18.

answer(a) 16 socks; (b) 9 pairs

techniqueMultiplying and Dividing Undo Each Other · Equal Groups · Sharing Equally

examsPSLE

Common pitfalls

  • Answering 8 socks in part (a). 8 is the number of pairs. Each pair is 2 socks, so the number of socks is double the number of pairs.
  • Doubling in part (b) to get 36. The 18 are socks already. Putting socks into pairs makes fewer groups than socks, so halve: 18 ÷ 2 = 9.
10

Buns Packed in Rows of 3 or Rows of 4

strategyDraw Both Ways of Packing / Compare the Number of Rows

A baker has 24 buns to put on a tray. (a) If he puts them in rows of 3, how many rows are there? (b) If he puts them in rows of 4 instead, how many rows are there?

36912151821243 in a row
The 24 buns in rows of 3. Count in threes: 3, 6, 9, 12, 15, 18, 21, 24.
Put the buns in rows of 3 and count in threes until 24 is reached: 3, 6, 9, 12, 15, 18, 21, 24.
step 1 of 4

The number of buns stays the same. Draw them in rows of 3 and then in rows of 4, and count the rows each time.

  1. Put the buns in rows of 3 and count in threes until 24 is reached: 3, 6, 9, 12, 15, 18, 21, 24.
  2. (a) That is 8 threes, so 24 ÷ 3 = 8. There are 8 rows of 3.
  3. Now put the same 24 buns in rows of 4 and count in fours: 4, 8, 12, 16, 20, 24.
  4. (b) That is 6 fours, so 24 ÷ 4 = 6. There are 6 rows of 4. Check: 8 × 3 = 24 and 6 × 4 = 24.

answer(a) 8 rows; (b) 6 rows

techniqueRows and Columns · Multiplying and Dividing Undo Each Other

examsPSLE

Common pitfalls

  • Thinking that rows of 4 must give more rows because 4 is more than 3. Each row of 4 uses up more buns, so there are fewer rows: 6 instead of 8.
  • Finding 8 rows in part (a) and then answering 8 + 1 = 9 rows in part (b) because each row has 1 more bun. The buns must be packed again from the start: 24 ÷ 4 = 6.
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