Decimals and place value Revision Notes for AQA GCSE Maths

Decimals and place value

Understanding place value in decimals

Place value tells you what each digit in a number is worth based on its position. This concept becomes especially important when working with decimal numbers.

In decimal numbers, each position has a specific value:

Tens - worth 10 times more than units
Units - the whole number part
Tenths - the first decimal place (0.1)
Hundredths - the second decimal place (0.01)
Thousandths - the third decimal place (0.001)

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A key point to remember is that having more digits doesn't automatically make a decimal number larger. For example, $0.8$ is actually greater than $0.758$ because the tenths place is more significant.

Comparing decimal numbers

When comparing decimals, you need to look at each place value column systematically, starting from the left.

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To compare decimal numbers effectively:

Start with the tenths column
If the tenths are the same, move to the hundredths
Continue this process until you find a difference
The number with the larger digit in the first different place is greater

For instance, when comparing $0.76$ and $0.758$ :

Both have 7 in the tenths place
Both have 6 in the hundredths place
$0.76$ has no digit in the thousandths (equivalent to 0), while $0.758$ has 8
Since $0 < 8$ , we have $0.76 < 0.758$

Wait, that's incorrect based on the image. Let me reconsider: $0.76 = 0.760$ , and $0.760 > 0.758$ because in the thousandths place, $0 > 8$ is wrong - actually $0.760$ means 6 hundredths and 0 thousandths, while $0.758$ means 5 hundredths and 8 thousandths. Since $6 > 5$ in the hundredths place, $0.760 > 0.758$ .

Ordering decimal numbers

When putting decimal numbers in order, follow a systematic approach:

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Method for ordering decimals:

Write out all the numbers clearly
Compare the tenths place first
If tenths are equal, compare hundredths
Continue comparing place values until you can order them
Cross out each number as you place it to avoid confusion

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Worked Example: Ordering Decimal Numbers

Order $0.43$ , $0.425$ , $0.4$ , $0.48$ , $0.459$ from smallest to largest

Solution:

$0.4 = 0.400$ (smallest - only 4 tenths)
$0.425$ (next - 4 tenths, 2 hundredths, 5 thousandths)
$0.43 = 0.430$ (4 tenths, 3 hundredths)
$0.459$ (4 tenths, 5 hundredths, 9 thousandths)
$0.48 = 0.480$ (largest - 4 tenths, 8 hundredths)

Working with powers of 10

Understanding how decimal numbers change when multiplied or divided by 10, 100, or 1000 is crucial for many calculations.

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Key rules:

Multiplying by 10 moves the decimal point one place right
Multiplying by 100 moves the decimal point two places right
Dividing by 10 moves the decimal point one place left
Dividing by 100 moves the decimal point two places left

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Worked Example: Using Powers of 10

If $58 \times 71 = 4118$ , find: (a) $58 \times 0.71$ (b) $5800 \times 7.1$

Solution: (a) 71 has been divided by 100, so the answer must be divided by 100 $4118 \div 100 = 41.18$

(b) 58 has been multiplied by 100, so the answer must be multiplied by 100 $4118 \times 100 = 411800$

Wait, let me check this again. Looking at the image: (b) shows $5800 \times 7.1 = 41180$ , which means 58 was multiplied by 100 and 71 was divided by 10, so the overall effect is multiplication by 10: $4118 \times 10 = 41180$ .

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Exam Tips:

Always write trailing zeros to help compare decimals (e.g., write $0.4$ as $0.400$ )
Cross out numbers as you order them to keep track
Check your work by ensuring your ordered list makes sense
Remember that fewer decimal places doesn't mean a smaller number

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Key Points to Remember:

Place value determines the worth of each digit in a decimal number
When comparing decimals, start from the leftmost place value and work right
More digits doesn't necessarily mean a bigger number
Adding zeros after the last decimal place doesn't change the value ( $0.4 = 0.400$ )
Moving decimal points follows predictable patterns when multiplying or dividing by powers of 10

Decimals and place value (AQA GCSE Maths): Revision Notes