Another Peek Beyond the Point · Lesson 5 of 13
Dividing Decimals by 10, 100 and 1000
“Explain decimal division by powers of ten using equal ribbon pieces, fractions, and changing place values.”
• Interpret division as sharing a quantity into equal parts. • Derive division by powers of ten using fractions. • Use zero placeholders when a quotient is smaller than one. • Convert the resulting measurements into suitable units.
Sharing a decimal quantity
A ribbon is 3.9 m long and must be cut into ten equal pieces. Division tells us the length of one piece: the total length is shared equally among the ten pieces. The answer must be smaller than 3.9 m, and multiplying the answer by 10 must restore the original length. These two checks help us interpret the calculation before using a shortcut.
Write 3.9 as 39/10. Dividing this fraction by 10 means multiplying it by 1/10, the reciprocal of 10. A reciprocal reverses the numerator and denominator of a non-zero fraction. Here, (39/10) × (1/10) = 39/100 = 0.39. Each piece therefore measures 0.39 m.
Problem
Find the length of each piece when a 3.9 m ribbon is divided into ten equal parts.
- 1.Calculate 3.9 ÷ 10 = 39/100 = 0.39 m.
- 2.Check the total: 0.39 × 10 = 3.9 m.
- 3.In centimetres, 0.39 × 100 = 39 cm. Ten 39 cm pieces total 390 cm, the same as 3.9 m.
Why the point moves left
Dividing by 10 makes the value of each digit one-tenth as large. Ones become tenths, tenths become hundredths, and so on. Dividing by 100 performs two such changes, while dividing by 1000 performs three. In the written decimal this is represented by moving the point left by one, two, or three places. The digit values, rather than the order of the digits, change.
| Starting number | ÷ 10 | ÷ 100 | ÷ 1000 | ÷ 10000 |
|---|---|---|---|---|
| 18.7 | 1.87 | 0.187 | 0.0187 | 0.00187 |
| 0.13 | 0.013 | 0.0013 | 0.00013 | 0.000013 |
If moving the point would pass the first digit, insert zeroes to hold the empty places. In 0.0058 ÷ 100, the digit 5 changes from thousandths to hundred-thousandths, and the digit 8 from ten-thousandths to millionths. The result is 0.000058. Removing those zeroes would give a different value.
Problem
Find 0.0058 ÷ 100.
- 1.Write 0.0058 = 58/10000.
- 2.Dividing by 100 gives 58/(10000 × 100) = 58/1000000.
- 3.Write 58 millionths as 0.000058. Verify that 0.000058 × 100 = 0.0058.
Keep the amount and unit connected
A change of unit uses the same multiplication or division ideas. Smaller units require more units to name the same amount. Larger units require fewer. For example, going from metres to centimetres multiplies the numerical value by 100. Going from millimetres to centimetres divides it by 10. The physical length remains unchanged.
Problem
The same 3.9 m ribbon is cut into 100 equal pieces. Express one piece in metres, centimetres, and millimetres.
- 1.Each piece is 3.9 ÷ 100 = 0.039 m.
- 2.Multiply by 100 to express metres as centimetres: 0.039 × 100 = 3.9 cm.
- 3.Multiply by 1000 to express metres as millimetres: 0.039 × 1000 = 39 mm.
- 4.The three answers name one length. A numerical value below 1 does not mean a length must be less than 1 in every possible unit.
Division by 100 uses two places, not the number of digits in 100. There are two zeroes in 100. Also, do not reverse a unit conversion: 39 mm is 3.9 cm, not 390 cm.
Quiz
What is 3.9 ÷ 10?
How many places left are used when dividing by 1000?
Which is equal to 0.039 m?
What is 0.13 ÷ 100?
Which multiplication checks 18.7 ÷ 1000 = 0.0187?
Practice Problems
- Find 21.1 ÷ 10, ÷ 100, ÷ 1000, and ÷ 10000 in a table.
- Complete the same table for 2.146 and 0.0058.
- Explain why 0.13 ÷ 1000 needs zeroes between the point and the digit 1.
- A 7.5 m ribbon is cut into 100 equal pieces. Express one piece in metres and centimetres.
- Convert 125.5 mL into litres and 35 cm into metres.
- Describe the place-value changes in 24.67 ÷ 100, then check your result by multiplication.
Key Takeaways
• Division shares the total into equal parts. • Dividing by a power of ten reduces each digit’s place value by that factor. • The point moves left by the number of zeroes in 10, 100, or 1000. • Zero placeholders are needed when the new places would otherwise be empty. • Unit conversions change the numerical value used to name an unchanged quantity.