Multiplying and dividing by powers of ten

Before this page: know the place-value columns on either side of the decimal point, and that each is worth ten times the column to its right.

Ten times the value

In 52 × 10 = 520, the 5 moves from tens to hundreds and the 2 moves from ones to tens. In 5.2 × 10 = 52, the same movement carries the 5 from ones to tens and the 2 from tenths to ones.

The decimal point does not move at all — the digits move past it.

Multiplication moves digits left each place is worth ten times as much
Division moves digits right each place is worth a tenth as much
10

one place

100

two places

1,000

three places

52.8 × 100 5,280

Interactive place-value machine Each digit in 52.8 moves two places left, producing 5,280. two places left

The calculation step by step

  1. The multiplier 100 contains two factors of 10, so every digit moves two places to the left.
  2. The 5 moves from the tens place to the thousands place; the 2 moves from the ones place to the hundreds place; and the 8 moves from the tenths place to the tens place.
  3. The ones place is empty, so a zero holds it. The number is 5,280.

Multiplying by 10, 100 and 1,000

Multiplying by 100 makes every digit worth a hundred times as much: 6.47 × 100 = 647. Two multiplications by 10 move the digits two places to the left.

Example 1: calculate 6.47 × 100

  1. 6.47 Start with 6 ones, 4 tenths and 7 hundredths.
  2. Move every digit two places left. Ones become hundreds, tenths become tens and hundredths become ones.
  3. 6.47 × 100 = 647 Six hundreds, four tens and seven ones.

Example 2: calculate 0.39 × 1,000

  1. 3 tenths → 3 hundreds Three places to the left, past the ones.
  2. 9 hundredths → 9 tens The same three places.
  3. 0.39 × 1,000 = 390 A zero holds the empty ones place.

Dividing by 10, 100 and 1,000

Dividing a whole number can give a decimal, because the places carry on past the point: the divisor 10 moves every digit of 725 one place to the right, and 725 ÷ 10 = 72.5.

Example 3: calculate 725 ÷ 10

  1. 7 hundreds → 7 tens The new place is worth one tenth of the original place.
  2. 2 tens → 2 ones One tenth of the original place again.
  3. 5 ones → 5 tenths The 5 crosses the point, which stays between ones and tenths.
  4. 725 ÷ 10 = 72.5 Seven tens, two ones and five tenths.

Example 4: calculate 4 ÷ 100

  1. 4 ones → 4 hundredths Two factors of 10 make the place value one hundred times smaller.
  2. Write zeros in the empty places. Both the ones and tenths places must be held by zeros.
  3. 4 ÷ 100 = 0.04 The placeholder zero in the tenths place prevents 4 being read as 4 tenths.

A negative number keeps its sign, and its digits move as before: −3.6 × 100 = −360 and −48 ÷ 1,000 = −0.048.

Multiplying and dividing by tenths

The numbers 0.1 and 0.01 are powers of ten smaller than 1.

Multiplying finds a fraction of the starting value

0.1 is one tenth, so multiplying by 0.1 is equivalent to dividing by 10; multiplying by 0.01 is dividing by 100:

Calculate 6 × 0.1

  1. One tenth of 6 is 6 ÷ 10.
  2. So 6 × 0.1 = 0.6.

Calculate 6 × 0.01

  1. One hundredth of 6 is 6 ÷ 100.
  2. So 6 × 0.01 = 0.06.

Division counts how many fractional pieces fit

The calculation 3 ÷ 0.1 asks how many tenths are in 3:

3÷ 0.1= ?
1 whole 10 tenths
1 whole 10 tenths
1 whole 10 tenths
  1. Each whole contains 10 tenths.
  2. Three wholes contain 3 × 10 = 30 tenths.
  3. So 3 ÷ 0.1 = 30.

Dividing by 0.01 counts hundredths: one whole contains 100 hundredths, so 3 ÷ 0.01 = 300.

Key points

  • Multiplying by 10, 100 or 1,000 increases every digit’s place value by one, two or three places.
  • Dividing by those numbers decreases every digit’s place value by the same number of places.
  • Multiplying by 0.1 or 0.01 divides by 10 or 100. Dividing by 0.1 or 0.01 multiplies by 10 or 100.

Continue

Next lesson

Column addition

Adding integers and decimals in columns, lining up the point and carrying.
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