Ten times each part
45 is 4 tens and 5 ones. Multiplying it by 10 makes each part ten times as big. Ten times 4 tens is 4 hundreds, and ten times 5 ones is 5 tens. So 45 × 10 is 4 hundreds and 5 tens, which is 450.
Each digit has moved one place to the left, into a column worth ten times as much: the 4 from the tens to the hundreds, and the 5 from the ones to the tens. The ones column is now empty, and a 0 is written there to keep the 4 and the 5 in their new places.
45 is 4 tens and 5 ones.
45 × 10: the 4 tens become 4 hundreds and the 5 ones become 5 tens, which is 450.
By 100 and by 1,000
Multiplying by 100 is multiplying by 10 and then by 10 again, so every digit moves two places to the left. The 4 goes to the thousands and the 5 to the hundreds, and two zeros fill the empty tens and ones: 45 × 100 = 4500.
Multiplying by 1,000 is three moves, with three zeros to fill the gaps: 45 × 1,000 = 45,000. Count the zeros in 10, 100 or 1,000, and that is how many places each digit moves.
45 × 100: every part has moved up two places, to 4 thousands and 5 hundreds, which is 4500.
× 1 moves no digit, so the value is unchanged
Turn 45 into 4500
Choose × 10, × 100 or × 1,000 and watch the digits of 45 move left, one column for each zero, while zeros fill the empty places up to the ones.
Dividing moves the digits back
Dividing by 10, 100 or 1,000 undoes the multiplying, so the digits move the other way, to the right. To work out 4500 ÷ 100, move every digit two places to the right. The 4 goes from the thousands back to the tens, and the 5 from the hundreds back to the ones. The two zeros are no longer needed: 4500 ÷ 100 = 45.
Multiplying by 300
300 is 3 × 100, so multiplying by 300 is multiplying by 3 and then by 100. For 6 × 300, work out the digit first: 6 × 3 = 18. Then multiply by 100, which moves the digits of 18 two places left: 18 × 100 = 1,800.
The same split works for 20, 40, 500 or 7,000: multiply by the digit, then put the zeros back at the end.
6 × 300 is 18 hundreds, which is 1 thousand and 8 hundreds: 1,800.
Where the zero goes
A 0 fills the empty place at the end, never one in the middle. 45 × 10 is 450, not 405: both digits move up one place, so the 5 goes to the tens, and the 0 goes in the ones.
The number of moves matters too. Writing 4500 for 45 × 10 is a mistake: 4500 is two moves, and that is 45 × 100. Multiplying by 10 moves each digit one place, no more.
Worked example: Packs of 10 and Boxes of 100
Question 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?
1.Write 36 in a place-value chart. The 3 is in the tens column and the 6 is in the ones column.
In 36 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.
Each digit moves one column to the left, and a 0 fills the ones column. 3.(a) 36 × 10 = 360, so there are 360 pencils in the packs.
(a) 36 × 10 = 360 pencils are 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.
For 100, each digit moves two columns to the left, and zeros fill the tens and the ones. 5.(b) 36 × 100 = 3600, so there are 3600 pencils in the boxes.
(b) 36 × 100 = 3600 pencils are in the boxes.
Answer: (a) 360 pencils; (b) 3600 pencils
Common mistakes
- 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.
More multiplying and dividing in your head problems, worked step by step →
Worked example: Grams Changed to Kilograms Before a Comparison
Question 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?
1.There are 1000 g in 1 kg. To change grams to kilograms, divide the number of grams by 1000.
There are 1000 g in 1 kg, so divide 7000 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.
Each digit moves three columns to the right. The 7 moves to the ones column. 3.(a) 7000 ÷ 1000 = 7, so the pumpkin has a mass of 7 kg.
(a) 7000 ÷ 1000 = 7, so the pumpkin is 7 kg. 4.Now both masses are in kilograms: the pumpkin is 7 kg and the watermelon is 3 kg.
Both masses are now in kilograms: 7 kg and 3 kg. 5.(b) 7 − 3 = 4, so the pumpkin is 4 kg heavier than the watermelon.
(b) The pumpkin is 7 − 3 = 4 kg heavier.
Answer: (a) 7 kg; (b) 4 kg
Common mistakes
- 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.
More multiplying and dividing in your head problems, worked step by step →