Multiplying and dividing in your head · applications

Applications: Multiplying and Dividing in Your Head

10 question types · Primary 4 · each worked step by step with a figure that follows the steps

PSLE · GCSE Higher

01

Packs of 10 and Boxes of 100

strategyPlace-Value Chart / Move Each Digit to the Left

A shop sells pencils in packs of 10 and in boxes of 100. It has 36 packs and 36 boxes. (a) How many pencils are in the packs? (b) How many pencils are in the boxes?

ThHTO3636
In 36 the 3 is in the tens column and the 6 is in the ones column.
Write 36 in a place-value chart. The 3 is in the tens column and the 6 is in the ones column.
step 1 of 5

Write 36 in a place-value chart. Multiplying by 10 moves each digit one column to the left, and multiplying by 100 moves each digit two columns to the left.

  1. Write 36 in a place-value chart. The 3 is in the tens column and the 6 is in the ones column.
  2. To multiply by 10, move each digit one column to the left. The 3 moves to the hundreds and the 6 moves to the tens. Write 0 in the empty ones column.
  3. (a) 36 × 10 = 360, so there are 360 pencils in the packs.
  4. To multiply by 100, move each digit two columns to the left. The 3 moves to the thousands and the 6 moves to the hundreds. Write 0 in the tens column and in the ones column.
  5. (b) 36 × 100 = 3600, so there are 3600 pencils in the boxes.

answer(a) 360 pencils; (b) 3600 pencils

techniqueMultiplying and Dividing by 10, 100 and 1,000

examsPSLE

Common pitfalls

  • Moving the digits only one column for the boxes and writing 360 again. A box holds 100 pencils, so each digit moves two columns.
  • Leaving the empty columns blank and writing 36. The zeros hold the 3 and the 6 in their new columns, so they must be written.
02

A Price That Is One Dollar Less Than a Round Number

strategyRound the Price Up, Multiply, Then Take Off the Extra

A T-shirt costs $29. A coach buys 7 T-shirts for her team. (a) How much would 7 T-shirts cost if each one cost $30? (b) How much does the coach really pay?

One T-shirt$29$1 less than $30
$29 is $1 less than $30.
$29 is $1 less than $30. Work with $30 first, because 30 is easy to multiply.
step 1 of 4

$29 is $1 less than $30, and 30 is easy to multiply. Multiply by 30 first, then take off the extra $1 for each T-shirt.

  1. $29 is $1 less than $30. Work with $30 first, because 30 is easy to multiply.
  2. (a) 7 × 3 = 21, so 7 × 30 = 210. At $30 each, the T-shirts would cost $210.
  3. Each T-shirt really costs $1 less. So $210 is 7 × $1 = $7 too much.
  4. (b) Take off the extra: 210 − 7 = $203. Check: 7 × 20 = 140 and 7 × 9 = 63, and 140 + 63 = 203.

answer(a) $210; (b) $203

techniqueMultiplying by Splitting · Multiplying and Dividing by 10, 100 and 1,000

examsPSLE

Common pitfalls

  • Taking off only $1 and writing $209. The price was rounded up by $1 for each of the 7 T-shirts, so $7 must come off.
  • Adding $7 and writing $217. The rounded price of $30 is more than the real price, so the real total is less than $210.
03

Rows of Seats Split into Tens and Ones

strategySplit One Number into Tens and Ones / Multiply Each Part and Add

A hall has 6 rows of seats with 34 seats in each row. In every row the first 30 seats are blue and the last 4 seats are red. (a) How many blue seats are there? (b) How many seats are there altogether?

6 rows30 blue4 red
Each row of 34 seats is 30 blue seats and 4 red seats.
Split 34 into 30 and 4. Each row has 30 blue seats and 4 red seats.
step 1 of 4

Split 34 into 30 and 4. Multiply each part by 6, then add the two products.

  1. Split 34 into 30 and 4. Each row has 30 blue seats and 4 red seats.
  2. (a) 6 × 3 = 18, so 6 × 30 = 180. There are 180 blue seats.
  3. Multiply the other part: there are 6 × 4 = 24 red seats.
  4. (b) Add the two parts: 180 + 24 = 204. There are 204 seats altogether.

answer(a) 180 blue seats; (b) 204 seats

techniqueMultiplying by Splitting · Multiplying and Dividing by 10, 100 and 1,000

examsPSLE

Common pitfalls

  • Multiplying only the tens and giving 180 as the total. The 4 red seats in each row must be multiplied by 6 as well.
  • Working out 6 × 3 = 18 and 6 × 4 = 24, then adding them to get 42. The 3 in 34 stands for 3 tens, so that part is 6 × 30 = 180.
04

Trays of Eggs Doubled for 4 and for 8

strategyDouble Twice to Multiply by 4 / Double Three Times to Multiply by 8

A tray holds 27 eggs. (a) How many eggs are in 4 trays? (b) How many eggs are in 8 trays?

1 tray27
One tray holds 27 eggs. Each doubling gives twice as many trays.
Multiplying by 4 is doubling twice. Multiplying by 8 is doubling three times. Start with the 27 eggs in one tray.
step 1 of 5

Doubling gives 2 trays, doubling again gives 4 trays, and doubling once more gives 8 trays. To double a number, double the tens and the ones and add.

  1. Multiplying by 4 is doubling twice. Multiplying by 8 is doubling three times. Start with the 27 eggs in one tray.
  2. Double 27. Double 20 is 40 and double 7 is 14, so 2 trays hold 40 + 14 = 54 eggs.
  3. (a) Double 54. Double 50 is 100 and double 4 is 8, so 4 trays hold 108 eggs.
  4. Double once more for 8 trays. Double 100 is 200 and double 8 is 16.
  5. (b) 200 + 16 = 216, so 8 trays hold 216 eggs.

answer(a) 108 eggs; (b) 216 eggs

techniqueDoubling to Multiply

examsPSLE · GCSE Higher

Common pitfalls

  • Doubling only twice for 8 trays and writing 108. Two doublings multiply by 4. A third doubling is needed to multiply by 8.
  • Doubling only the tens, for example writing double 27 as 47. Both parts are doubled: double 20 is 40 and double 7 is 14.
05

Five Notebooks as Half of Ten Notebooks

strategyMultiply by 10, Then Halve

A notebook has 68 pages. (a) How many pages are in 10 notebooks? (b) How many pages are in 5 notebooks?

10 notebooks68686868686868686868
5 notebooks are half of 10 notebooks.
5 is half of 10. So find the pages in 10 notebooks first, then halve.
step 1 of 4

5 is half of 10. Multiply by 10 first, because that only moves the digits, and then halve the answer.

  1. 5 is half of 10. So find the pages in 10 notebooks first, then halve.
  2. (a) To multiply by 10, move each digit one column to the left: 68 × 10 = 680. There are 680 pages in 10 notebooks.
  3. 5 notebooks are half of 10 notebooks, so halve 680. Half of 600 is 300 and half of 80 is 40.
  4. (b) 300 + 40 = 340, so there are 340 pages in 5 notebooks. Check: 5 × 60 = 300 and 5 × 8 = 40.

answer(a) 680 pages; (b) 340 pages

techniqueMultiplying and Dividing by 10, 100 and 1,000 · Halving to Divide

examsPSLE

Common pitfalls

  • Stopping at 680. That is the number of pages in 10 notebooks, and it still has to be halved for 5 notebooks.
  • Halving only the hundreds and writing 380. Both parts of 680 are halved: half of 600 is 300 and half of 80 is 40.
06

A Ribbon Halved Twice into 4 Equal Pieces

strategyHalve, Then Halve Again to Divide by 4

Mrs Lim has a ribbon that is 92 cm long. She cuts it in half. Then she cuts each half in half again, so she has 4 equal pieces. (a) How long is each half? (b) How long is each of the 4 pieces?

Ribbon92 cm
Dividing by 4 is halving twice.
Dividing by 4 is halving twice, because 2 × 2 = 4.
step 1 of 4

Dividing by 4 is the same as halving twice. To halve a number, halve the tens and the ones and add.

  1. Dividing by 4 is halving twice, because 2 × 2 = 4.
  2. (a) Halve 92. Half of 90 is 45 and half of 2 is 1, so each half is 45 + 1 = 46 cm long.
  3. Halve 46. Half of 40 is 20 and half of 6 is 3.
  4. (b) 20 + 3 = 23, so each of the 4 pieces is 23 cm long. Check: double 23 is 46 and double 46 is 92.

answer(a) 46 cm; (b) 23 cm

techniqueHalving to Divide

examsPSLE · GCSE Higher

Common pitfalls

  • Halving only once and giving 46 cm for each of the 4 pieces. One halving makes 2 pieces. A second halving is needed to make 4.
  • Writing half of 90 as 40. Half of 80 is 40 and half of 10 is 5, so half of 90 is 45.
07

Eggs Packed in Sixes by Splitting the Total

strategySplit the Total into Ten Groups and the Rest / Divide Each Part and Add

A farmer packs 84 eggs into cartons of 6. He packs the first 60 eggs, and then he packs the rest. (a) How many cartons do the first 60 eggs fill? (b) How many cartons does he fill altogether?

84 eggs60 eggs24 eggs
Ten cartons hold 60 eggs, so 84 is split into 60 and 24.
Ten cartons hold 10 × 6 = 60 eggs. So split 84 into 60 and the rest: 84 = 60 + 24.
step 1 of 4

Ten cartons hold 10 × 6 = 60 eggs, so split 84 into 60 and the rest. Divide each part by 6 and add the two answers.

  1. Ten cartons hold 10 × 6 = 60 eggs. So split 84 into 60 and the rest: 84 = 60 + 24.
  2. (a) 60 ÷ 6 = 10, so the first 60 eggs fill 10 cartons.
  3. Divide the rest: 24 ÷ 6 = 4, so the other 24 eggs fill 4 cartons.
  4. (b) Add the two parts: 10 + 4 = 14 cartons. Check: 14 × 6 = 60 + 24 = 84.

answer(a) 10 cartons; (b) 14 cartons

techniqueDividing by Splitting

examsPSLE · GCSE Higher

Common pitfalls

  • Stopping at 10 cartons. The other 24 eggs still have to be packed, and they fill 4 more cartons.
  • Splitting 84 into 80 and 4. Neither 80 nor 4 can be shared into sixes exactly. Choose a first part that is in the 6 times table, such as 60.
08

Marbles of 25 g Grouped in Fours

strategyFour 25s Make 100 / Count the Groups of Four

Each marble has a mass of 25 g. A bag holds 32 marbles. (a) How many groups of 4 marbles can be made from the bag? (b) What is the mass of all the marbles in the bag?

4 marbles25 g25 g25 g25 g100 g
Four 25s make 100, so 4 marbles have a mass of 100 g.
Four 25s make 100: 4 × 25 = 100. So a group of 4 marbles has a mass of 100 g.
step 1 of 5

Four 25s make 100. Put the marbles in groups of 4, so that each group has a mass of 100 g, and count the groups.

  1. Four 25s make 100: 4 × 25 = 100. So a group of 4 marbles has a mass of 100 g.
  2. Find the number of groups of 4 in 32 by halving twice. Half of 32 is 16, and half of 16 is 8.
  3. (a) 32 ÷ 4 = 8, so 8 groups of 4 marbles can be made.
  4. Each group has a mass of 100 g, so the total mass is 8 × 100 g.
  5. (b) 8 × 100 = 800, so the marbles have a mass of 800 g.

answer(a) 8 groups; (b) 800 g

techniqueHalving to Divide · Multiplying and Dividing by 10, 100 and 1,000

examsPSLE · GCSE Higher

Common pitfalls

  • Multiplying 32 by 100 and writing 3200 g. One marble is 25 g. It takes 4 marbles to make 100 g, so the number of marbles is divided by 4 first.
  • Halving 32 only once and using 16 groups. Halving once divides by 2. The marbles are in groups of 4, so halve twice.
09

Grams Changed to Kilograms Before a Comparison

strategyDivide by 1000 by Moving Each Digit Three Columns / Compare in the Same Unit

A pumpkin has a mass of 7000 g. A watermelon has a mass of 3 kg. (a) What is the mass of the pumpkin in kilograms? (b) How many kilograms heavier is the pumpkin than the watermelon?

ThHTO7000 g7000
There are 1000 g in 1 kg, so divide 7000 by 1000.
There are 1000 g in 1 kg. To change grams to kilograms, divide the number of grams by 1000.
step 1 of 5

There are 1000 g in 1 kg, so divide the number of grams by 1000. Dividing by 1000 moves each digit three columns to the right. Then both masses are in kilograms and can be compared.

  1. There are 1000 g in 1 kg. To change grams to kilograms, divide the number of grams by 1000.
  2. To divide by 1000, move each digit three columns to the right. The 7 moves from the thousands column to the ones column.
  3. (a) 7000 ÷ 1000 = 7, so the pumpkin has a mass of 7 kg.
  4. Now both masses are in kilograms: the pumpkin is 7 kg and the watermelon is 3 kg.
  5. (b) 7 − 3 = 4, so the pumpkin is 4 kg heavier than the watermelon.

answer(a) 7 kg; (b) 4 kg

techniqueMultiplying and Dividing by 10, 100 and 1,000

examsPSLE

Common pitfalls

  • Subtracting 3 from 7000 and writing 6997. One mass is in grams and the other is in kilograms, so change them to the same unit before subtracting.
  • Moving the digits only two columns and writing 70 kg. Dividing by 1000 moves each digit three columns, one for each zero in 1000.
10

A Calculator Answer Tested with an Estimate

strategyRound Both Numbers to Estimate / Compare the Size of the Answer

Ravi works out 49 × 62 on a calculator. The calculator shows 338. (a) Estimate 49 × 62 by rounding both numbers to the nearest ten. (b) The answer on the calculator is wrong. What is 49 × 62?

Estimate50 × 60
49 rounds up to 50 and 62 rounds down to 60.
Round each number to the nearest ten. 49 rounds up to 50 and 62 rounds down to 60.
step 1 of 5

An estimate shows how big the answer should be. If the calculator shows a number of a very different size, a key was pressed wrongly. The exact product can then be found from 50 × 62.

  1. Round each number to the nearest ten. 49 rounds up to 50 and 62 rounds down to 60.
  2. (a) 5 × 6 = 30, so 50 × 60 = 3000. The answer should be about 3000.
  3. 338 is nowhere near 3000. It is about ten times too small, so it cannot be right.
  4. Find 50 × 62 first. 100 × 62 = 6200, and half of 6200 is 3100.
  5. (b) 49 lots of 62 is one 62 less than 50 lots: 3100 − 62 = 3038. This is close to the estimate of 3000.

answer(a) 3000; (b) 3038

techniqueRounding to Estimate · Multiplying by Splitting

examsO-Level · GCSE Higher

Common pitfalls

  • Accepting 338 because 8 is the correct last digit of 49 × 62. The last digit does not show the size of a number. The estimate of 3000 shows that a digit is missing.
  • Taking off 49 instead of 62 and writing 3100 − 49 = 3051. 50 lots of 62 is one lot of 62 too many, so 62 is taken off.
Mr. Chalk Drill the times tables