Decimals: Place Value and Rounding
Every digit has a place. Tenths, hundredths, thousandths. Know the value, compare with confidence, round with precision.
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Before you read on — quickly: Write these decimals in order from smallest to largest: 0.7, 0.07, 0.707, 0.77. What strategy did you use?
Decimal place value extends the number system to the right of the decimal point. Each position is 10 times smaller than the one to its left. Tenths ($ rac{1}{10}$), hundredths ($ rac{1}{100}$), thousandths ($ rac{1}{1000}$). Understanding place value lets you compare, order, and round decimals with confidence.
In 34.527: the 3 is in the tens place (30), the 4 is in the ones place (4), the 5 is in the tenths place ($ rac{5}{10}$), the 2 is in the hundredths place ($ rac{2}{100}$), and the 7 is in the thousandths place ($ rac{7}{1000}$). Each digit’s value depends entirely on its position.
Know
- Place values to the right of the decimal point
- Decimals are fractions with powers of 10 denominators
- Rounding rules: look at the next digit
Understand
- Why 0.4 = 0.40 = 0.400 (trailing zeros)
- Why more digits doesn’t always mean a bigger number
- How rounding approximates while keeping accuracy
Can Do
- Compare and order decimals
- Convert between decimals and fractions
- Round to any decimal place
Wrong: “0.37 is bigger than 0.4 because 37 > 4.” No! Compare digit by digit from the left. 0.4 = 0.40, and 40 > 37.
Right: Write both with the same number of decimal places. 0.4 = 0.40. Now compare: 40 > 37, so 0.4 > 0.37.
Wrong: Rounding 2.345 to 2 decimal places = 2.35. No! Look at the third decimal digit (5). Since 5 ≥ 5, round the 4 up to 5. So 2.35.
Right: Rounding 2.345 to 2 decimal places: look at the third digit (5). 5 ≥ 5, so round 4 up to 5. Answer = 2.35. Wait, that’s the same! Let me try again: 2.344 to 2 d.p. = 2.34 (4 < 5, so 4 stays).
To compare decimals, write them with the same number of decimal places, then compare digit by digit from left to right. Add trailing zeros if needed — they don’t change the value.
Which is bigger: 0.608 or 0.62? Write 0.62 as 0.620. Now compare: tenths are both 6 (equal). Hundredths: 0 vs 2. Since 2 > 0, 0.62 > 0.608. You don’t need to look at the thousandths place once you find a difference.
To round to a given decimal place: (1) Find the digit in the target place. (2) Look at the digit immediately to its right. (3) If that digit is 5 or more, round up. If less than 5, leave it. (4) Drop all digits after the target place.
Round 3.8472 to 2 decimal places. Target digit: 4 (hundredths). Next digit: 7. Since 7 ≥ 5, round 4 up to 5. Answer: 3.85. To 1 decimal place: target digit is 8, next digit is 4, 4 < 5, so answer = 3.8.
Decimals are just fractions with denominators that are powers of 10. Converting between them is straightforward once you know the place values.
Convert 0.375 to a fraction. 0.375 = $ rac{3}{10} + rac{7}{100} + rac{5}{1000} = rac{375}{1000}$. Simplify: HCF(375, 1000) = 125. $ rac{375 \div 125}{1000 \div 125} = rac{3}{8}$. Convert $ rac{7}{20}$ to a decimal: $ rac{7}{20} = rac{7 imes 5}{20 imes 5} = rac{35}{100} = 0.35$.
Write in order from smallest to largest: 0.45, 0.405, 0.5, 0.045
Write all with 3 decimal places: 0.450, 0.405, 0.500, 0.045.
Compare digit by digit. Tenths: 0, 4, 4, 5. So 0.045 is smallest, 0.5 is largest. For 0.450 and 0.405: hundredths 5 > 0, so 0.405 < 0.450.
Final order: 0.045 < 0.405 < 0.45 < 0.5.
0.045, 0.405, 0.45, 0.5
Round 6.8495 to (a) 1 decimal place, (b) 2 decimal places, (c) 3 significant figures.
(a) 1 d.p.: target digit is 8, next digit is 4. 4 < 5, so 8 stays. 6.8.
(b) 2 d.p.: target digit is 4, next digit is 9. 9 ≥ 5, so round 4 up to 5. 6.85.
(c) 3 sig figs: first 3 non-zero digits are 6, 8, 4. Next digit is 9. 9 ≥ 5, round 4 up to 5. 6.85 (3 sig figs).
Convert $\frac{5}{8}$ to a decimal.
Method: convert to equivalent fraction with denominator 1000. 8 × 125 = 1000, so multiply top and bottom by 125.
$\frac{5}{8} = \frac{5 \times 125}{8 \times 125} = \frac{625}{1000} = 0.625$.
Check: 0.625 = $\frac{6}{10} + \frac{2}{100} + \frac{5}{1000} = \frac{625}{1000}$. Simplify: HCF(625, 1000) = 125. $\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}$. Correct!
$\frac{5}{8} = 0.625$
Mistake: “0.5 < 0.45 because 5 < 45.” No! 0.5 = 0.50, and 50 > 45. The number of digits does not determine size.
Fix: Write both with the same decimal places: 0.50 vs 0.45. Tenths: 5 > 4, so 0.5 > 0.45.
Mistake: Rounding 2.498 to 2 d.p. = 2.50 but writing 2.5. Keep the trailing zero to show precision!
Fix: 2.498 ≈ 2.50 (2 d.p.). The zero is important — it shows we rounded to hundredths.
Mistake: Converting $\frac{1}{3}$ to 0.3 exactly. $\frac{1}{3}$ = 0.3333... forever. It never terminates!
Fix: $\frac{1}{3} \approx 0.333$ (3 d.p.) or exactly $\frac{1}{3}$. Some fractions have recurring decimals.
How are you completing this lesson?
Brain Trainer · 4 problems
Four drill problems to build your decimal fluency. Work each, then reveal the answer.
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1 Order from smallest to largest: 0.6, 0.06, 0.66, 0.606
Write as 0.600, 0.060, 0.660, 0.606. Compare: 0.060 < 0.600 < 0.606 < 0.660.0.06, 0.6, 0.606, 0.66 -
2 Round 4.7382 to (a) 1 d.p. (b) 2 d.p.
(a) Target 7, next digit 3 (less than 5), so 4.7. (b) Target 3, next digit 8 (5+), round up: 4.74.(a) 4.7 (b) 4.74 -
3 Convert $\frac{3}{8}$ to a decimal.
3/8 = (3 × 125)/(8 × 125) = 375/1000 = 0.375.0.375 -
4 Convert 0.65 to a fraction in simplest form.
0.65 = 65/100 = 13/20 (HCF = 5).13/20
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. In the number 807.354: (a) What is the value of the 8? (b) What is the value of the 5? (c) What is the value of the 4?
Q7. Round 9.996 to (a) 1 decimal place, (b) 2 decimal places, (c) the nearest whole number, (d) 1 significant figure. Show your working for each.
Q8. A student says 0.6 and 0.60 are different numbers because one has more digits. Explain why this is incorrect, and describe a real-world situation where writing 0.60 matters.
Quick Check
1. B — 0.4 = 0.40, 0.07 = 0.07. 40 > 7, so 0.4 > 0.07.
2. C — Target 7, next digit 6 (5+), so 3.48.
3. A — 3/5 = 6/10 = 0.6.
4. D — 0.050 < 0.500 < 0.505 < 0.550.
5. B — 0.125 = 125/1000 = 1/8 (HCF = 125).
Show Your Working Model Answers
Q6 (3 marks): (a) 8 is in hundreds place, value = 800 [1]. (b) 5 is in hundredths place, value = 0.05 or 5/100 [1]. (c) 4 is in thousandths place, value = 0.004 or 4/1000 [1].
Q7 (4 marks): (a) 10.0 (9 rounds up, carrying over) [1]. (b) 10.00 [1]. (c) 10 [1]. (d) 10 [1].
Q8 (3 marks): 0.6 = 0.60 = 6/10 = 60/100, same value [1]. The extra zero shows precision [1]. Example: money ($0.60), measurements (0.60 m shows precision to cm) [1].
The Recurring Detective
Some fractions have decimals that go on forever. $\frac{1}{3} = 0.333...$ and $\frac{1}{7} = 0.142857142857...$ These are called recurring decimals. Using a calculator, find the decimal for $\frac{2}{7}$, $\frac{3}{7}$, $\frac{4}{7}$, $\frac{5}{7}$, and $\frac{6}{7}$. What pattern do you notice? Can you predict $\frac{7}{7}$ without calculating?
Reveal solution
$\frac{2}{7} = 0.\overline{285714}$, $\frac{3}{7} = 0.\overline{428571}$, $\frac{4}{7} = 0.\overline{571428}$, $\frac{5}{7} = 0.\overline{714285}$, $\frac{6}{7} = 0.\overline{857142}$. All use the same 6 digits (142857) in the same cyclic order! $\frac{7}{7} = 1$ exactly.
Tenths
1st digit after point
Hundredths
2nd digit after point
Thousandths
3rd digit after point
Compare
Same decimal places, left to right
Round
Target digit, look right, 5+ rounds up
Trailing zeros
Don’t change value but show precision
Interactive: FDP Triple Match
Match the fraction, decimal, and percentage that represent the same value. Three cards, one value — can you find all the triples?
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