Place Value, Ordering and Rounding
Section: Number | Syllabus: Cambridge Primary Mathematics (0845)
Decimal Place Value
In a decimal number, each digit's value depends on where it sits relative to the decimal point. Moving one place to the right of the point makes each column worth ten times less than the one before it.
- Each digit in a decimal number has a value that depends on its position after the decimal point
- The first digit after the point is tenths (worth 1/10 of one)
- The second digit after the point is hundredths (worth 1/100 of one)
- The third digit after the point is thousandths (worth 1/1000 of one)
Each digit's position after the decimal point gives it a different value
Worked Example: Reading the Value of a Digit
- Question: What is the value of the digit 7 in 5.734?
- Step 1: Find the position of the 7 - it is the first digit after the decimal point
- Step 2: The first position after the point is tenths
- Answer: The 7 is worth 7 tenths, or 0.7
Common Mistakes
MistakeAssuming that, like whole numbers, the digit further from the decimal point is worth more
Fixdigits to the right of the decimal point get smaller the further they are from the point - tenths, then hundredths, then thousandths
Multiplying and Dividing by 10, 100 and 1000
Multiplying or dividing by 10, 100 or 1000 always has the same effect: it shifts every digit to a new place value column. The decimal point itself never moves - it stays fixed, separating ones from tenths.
- Multiplying or dividing by 10, 100 or 1000 moves every digit to a new place value column
- Multiplying moves digits to the left - the number gets bigger
- Dividing moves digits to the right - the number gets smaller
- It is the digits that move on the place value grid, not the decimal point itself
Worked Example: Multiplying and Dividing by Powers of 10
- 6.5 × 100 = 650 (each digit moves 2 places left)
- 348 ÷ 1000 = 0.348 (each digit moves 3 places right)
- 0.24 × 10 = 2.4 (each digit moves 1 place left)
- 45 × 100 = 4500 (each digit moves 2 places left; a whole number needs placeholder zeros to fill the empty columns)
| Hundreds | Tens | Ones | Tenths | |
|---|---|---|---|---|
| 6.5 | 6 | 5 | ||
| 6.5 × 100 | 6 | 5 | 0 |
Every digit slides 2 columns to the left - the digits move, the place value columns do not
Common Mistakes
MistakeMoving the decimal point instead of the digits when multiplying or dividing by 10, 100 or 1000
Fixthink of the digits sliding across the place value grid - the decimal point stays fixed, separating ones from tenths
Composing, Decomposing and Regrouping Numbers
The same decimal number can be built up, broken apart, or rewritten in a different but equally valid form - these are called composing, decomposing and regrouping.
- Composing a number means building it from its place value parts: 4 + 0.5 + 0.02 = 4.52
- Decomposing a number means breaking it into its place value parts: 7.36 = 7 + 0.3 + 0.06
- Regrouping a number means expressing it in a different, equally valid way: 3.4 = 2 + 1.4 = 34 tenths
3.4 regrouped as 34 tenths - the value stays the same, only the grouping changes
Worked Example: Composing a Decimal
- Question: Compose a decimal from 3 + 0.6 + 0.09
- Step 1: Place each part in its own column: 3 ones, 6 tenths, 9 hundredths
- Answer: 3.69
Worked Example: Decomposing a Decimal
- Question: Decompose 6.208 into its place value parts
- Step 1: Identify the value of each digit: 6 ones, 2 tenths, 0 hundredths, 8 thousandths
- Answer: 6.208 = 6 + 0.2 + 0.008
Worked Example: Regrouping a Decimal
- Question: Show 2.4 as a number of tenths only
- Step 1: 2 ones is the same as 20 tenths
- Step 2: 20 tenths + 4 tenths = 24 tenths
- Answer: 2.4 = 24 tenths
Common Mistakes
MistakeLosing track of place value when decomposing a number, e.g. writing 7.36 = 7 + 3 + 6
Fixeach part must keep its correct place value: 7.36 = 7 + 0.3 + 0.06
MistakeThinking that regrouping changes the value of a number
Fixregrouping only rewrites the same value in a different way - 3.4 and 34 tenths are exactly equal
Rounding Numbers
Rounding replaces a number with a simpler one that is close to its actual value. To round, look at the digit just after the place you are rounding to - that digit alone decides whether to round up or down.
- Numbers with two decimal places can be rounded to the nearest tenth or to the nearest whole number
- Look at the digit after the place you are rounding to
- If it is 5 or more, round up; if it is less than 5, round down
4.36 lies closer to 4.4 than to 4.3, so it rounds up to 4.4
Worked Example: Rounding a Decimal
- Round 4.36 to the nearest tenth: look at the hundredths digit (6), so round up: 4.4
- Round 7.82 to the nearest whole number: look at the tenths digit (8), so round up: 8
- Round 8.25 to the nearest tenth: the hundredths digit is exactly 5, so round up: 8.3
Common Mistakes
MistakeRounding down when the deciding digit is exactly 5
Fixa deciding digit of 5 or more always rounds up
Interactive revision notes, videos and practice questions load below.