Primary Mathematics: Decimals, Rounding and Four Operations | Worked Learning Guide

A decimal point does not create a new kind of arithmetic. It changes the size of the units being counted, so place value has to stay visible while the operations continue.

BTT Mathematics / Primary Mathematics Learning Hub / Guide 16

Wait, What? 3.5 and 3.50 Are the Same Value

The zero in 3.50 does not make the number larger than 3.5. Three and five tenths is the same quantity as three and fifty hundredths. Decimal notation works because each place is ten times smaller than the place to its left: ones, tenths, hundredths and thousandths.

Many decimal errors come from treating the digits as a whole-number string. A learner may say 0.45 is greater than 0.7 because 45 is greater than 7. But 0.45 is forty-five hundredths, while 0.7 is seven tenths or seventy hundredths. The place-value units, not the number of digits, determine the comparison.

The Decimal Control System

1. Identify the value of each digit.
2. Rename decimals using equivalent forms when useful.
3. Align place values before adding or subtracting.
4. Track how multiplication or division changes the magnitude.
5. Estimate before accepting an exact answer.
6. Use inverse operations or context to check.

1. Decimal Place Value

In 4.382, the 4 represents four ones, the 3 represents three tenths, the 8 represents eight hundredths and the 2 represents two thousandths.

Worked Example 1: Rename a Decimal

Write 5.47 as a sum of place values.

5.47 = 5 + 0.4 + 0.07.

It can also be written as 5 ones 4 tenths 7 hundredths or 547 hundredths.

Worked Example 2: Equivalent Decimals

0.6 = 0.60 = 0.600. Adding zeros to the right of the final decimal digit does not change the value because six tenths equals sixty hundredths equals six hundred thousandths.

2. Comparing Decimals

Compare place value from left to right. If the whole-number parts are equal, compare tenths, then hundredths, then thousandths.

Worked Example 3: Compare 2.45 and 2.5

Rename 2.5 as 2.50. Now compare hundredths: 2.45 < 2.50.

Therefore 2.45 < 2.5.

Worked Example 4: Order Three Decimals

Order 0.9, 0.89 and 0.905 from smallest to largest.

Rename: 0.900, 0.890, 0.905.

0.89 < 0.9 < 0.905.

3. Rounding Decimals

Rounding asks which stated place-value landmark is nearest. To round to the nearest tenth, inspect the hundredths digit. To round to the nearest hundredth, inspect the thousandths digit.

Worked Example 5: Round to the Nearest Tenth

Round 6.47 to the nearest tenth.

The tenths digit is 4. The hundredths digit is 7, so 6.47 rounds up to 6.5.

Worked Example 6: Boundary Case

Round 3.95 to the nearest tenth.

The hundredths digit is 5, so 3.9 rounds up to 4.0. The answer crosses the whole-number boundary.

4. Estimation Before Decimal Calculation

Estimation creates a size check. 12.48 + 7.61 should be near 12.5 + 7.6 = 20.1. An answer such as 200.9 should be rejected immediately.

Worked Example 7: Estimate and Calculate

Estimate 19.8 × 4 as about 20 × 4 = 80. The exact product is 79.2. The estimate confirms the magnitude.

5. Decimal Addition

Align place values, not merely the right-hand edge of the digits. The decimal point is a useful visual marker because tenths must be added to tenths and hundredths to hundredths.

Worked Example 8: Add Unlike Decimal Lengths

3.7 + 2.46.

Rename 3.7 as 3.70.

3.70 + 2.46 = 6.16.

Worked Example 9: Money Addition

$12.75 + $8.60 = $21.35. Writing $8.60 rather than $8.6 makes the cents structure visible, though both represent the same value.

6. Decimal Subtraction

Subtraction may require regrouping across the decimal point. One whole can be renamed as ten tenths; one tenth can be renamed as ten hundredths.

Worked Example 10: Regroup Across Tenths

5.2 – 1.86.

Rename 5.2 as 5.20. Regroup as needed.

5.20 – 1.86 = 3.34.

Worked Example 11: Change From a Payment

An item costs $17.85 and payment is $20.00.

$20.00 – $17.85 = $2.15.

7. Multiplying a Decimal by a Whole Number

Multiplication can be understood through repeated equal groups. 2.4 × 3 means three groups of 2.4, giving 7.2.

Worked Example 12: Decimal × Whole Number

4.35 × 6.

Think 435 hundredths × 6 = 2610 hundredths = 26.10 = 26.1.

Estimate first: 4.35 is near 4.5, and 4.5 × 6 = 27, so 26.1 is plausible.

Worked Example 13: Money Rate

Five identical items cost $3.80 each.

$3.80 × 5 = $19.00.

8. Dividing a Decimal by a Whole Number

Division can represent equal sharing. $18.60 shared among three people gives $6.20 each because 1860 cents ÷ 3 = 620 cents.

Worked Example 14: Decimal ÷ Whole Number

7.2 ÷ 4 = 1.8. Check: 1.8 × 4 = 7.2.

Worked Example 15: Place-Value Sharing

9.45 ÷ 5.

945 hundredths ÷ 5 = 189 hundredths = 1.89.

9. Multiplying and Dividing by 10, 100 and 1000

The decimal point does not physically “move”. The digits take on new place values because the number is scaled.

Worked Example 16: Multiply by 100

3.47 × 100 = 347. Each unit becomes one hundred times as large, so the digits shift two place-value positions relative to the decimal point.

Worked Example 17: Divide by 1000

58 ÷ 1000 = 0.058.

Estimate direction first: dividing by one thousand must make a positive number much smaller.

10. Decimal Fractions and Percentage Connections

Decimals connect directly to fractions with denominators that are powers of ten. 0.25 = 25/100 = 1/4. 0.6 = 6/10 = 3/5. These equivalences help learners move between fraction, decimal and percentage representations when the same quantity appears in different forms.

Worked Example 18: Decimal to Percentage

0.35 = 35 hundredths = 35%.

Worked Example 19: Fraction to Decimal

3/4 = 75/100 = 0.75.

11. Decimals in Measurement

Measurement often uses decimals because one unit is partitioned into smaller equal units. 2.4 L = 2 L 400 ml; 1.35 m = 1 m 35 cm. The decimal must remain attached to its unit.

Worked Example 20: Litres and Millilitres

2.75 L = 2750 ml.

Check direction: millilitres are smaller units, so more of them are needed to name the same capacity.

12. Decimal Word Problems

A decimal calculation is only useful after the relationship is correct. Decide whether the problem requires a total, difference, equal sharing, repeated groups, rate or comparison.

Worked Example 21: Multi-Step Cost

Four tickets cost $6.75 each. A family pays with $30.

Total cost = 4 × $6.75 = $27.00.

Change = $30.00 – $27.00 = $3.00.

Worked Example 22: Measurement Difference

A rope is 8.5 m long. 2.75 m is cut away.

8.50 – 2.75 = 5.75 m.

13. Common Decimal Errors

• Comparing decimal digits as though they form whole numbers.
• Thinking more decimal places means a larger number.
• Aligning digits rather than place values in addition or subtraction.
• Dropping zeros before regrouping.
• Multiplying or dividing by 10 without checking whether the answer should grow or shrink.
• Rounding before the requested final stage and creating avoidable error.
• Using money notation without two decimal places when cents need to be explicit.
• Accepting a decimal answer without an estimate or inverse check.

14. Error Repair

Error A: 0.8 < 0.75 because 8 < 75

Rename 0.8 as 0.80. Eighty hundredths is greater than seventy-five hundredths, so 0.8 > 0.75.

Error B: 3.6 + 0.48 = 3.108

The learner combined digits without aligning place-value units. Write 3.60 + 0.48 = 4.08.

Error C: 5.2 – 1.86 = 4.66

Estimate first: 5.2 – about 1.9 should be about 3.3, so 4.66 is implausible. Correct result: 3.34.

Error D: 4.35 × 6 = 261

The whole-number product 435 × 6 = 2610 represents hundredths, not whole units. 2610 hundredths = 26.10.

15. Practice: 24 Questions

Questions 1–8: Place Value, Comparison and Rounding

1. Write 6.482 as a sum of place values.
2. Write 0.7 as an equivalent decimal with two decimal places.
3. Which is greater: 0.56 or 0.6?
4. Order 1.08, 1.8 and 1.18 from smallest to largest.
5. Round 4.67 to the nearest tenth.
6. Round 9.945 to the nearest hundredth.
7. Estimate 12.48 + 7.61 by rounding suitably.
8. Estimate 19.8 × 4 before calculating exactly.

Questions 9–16: Four Operations

9. 3.7 + 2.46
10. 8.05 + 4.9
11. 5.2 – 1.86
12. 10 – 3.475
13. 4.35 × 6
14. 2.75 × 8
15. 7.2 ÷ 4
16. 9.45 ÷ 5

Questions 17–24: Application and Checking

17. Convert 2.75 L to millilitres.
18. Convert 0.35 to a percentage.
19. Write 3/4 as a decimal.
20. Four tickets cost $6.75 each. Find the total.
21. A $30 payment is used for the tickets in Question 20. Find the change.
22. A rope is 8.5 m long. 2.75 m is cut away. Find the remaining length.
23. A learner says 0.9 < 0.85 because 9 < 85. Explain and correct the comparison.
24. A learner calculates 4.8 × 7 = 336. Give an estimate and use it to identify the place-value error.

16. Worked Answers

1. 6 + 0.4 + 0.08 + 0.002. 2. 0.70. 3. 0.6, because 0.60 > 0.56. 4. 1.08 < 1.18 < 1.8. 5. 4.7. 6. 9.95. 7. About 20.1. 8. About 80; exact 79.2.

9. 6.16. 10. 12.95. 11. 3.34. 12. 6.525. 13. 26.1. 14. 22.0. 15. 1.8. 16. 1.89.

17. 2750 ml. 18. 35%. 19. 0.75. 20. $27.00. 21. $3.00. 22. 5.75 m. 23. 0.9 = 0.90, so 0.9 > 0.85. 24. 4.8 is about 5, and 5 × 7 is about 35. The exact result should be near 35, not 336. Correct: 4.8 × 7 = 33.6.

17. Checking Ladder

• Magnitude: should the answer be around 3, 30 or 300?
• Place value: are tenths aligned with tenths?
• Inverse: does subtraction undo addition, or multiplication undo division?
• Units: is the answer dollars, metres, litres or a pure number?
• Boundary: can the answer exceed the original whole or available amount?

18. Transfer Test

A school buys 18 packs at $4.85 each and receives a $100 budget. Estimate the total before calculating exactly, find the exact total, find the remaining budget, then explain one independent check. Next, a 7.5 L container is filled equally into 6 bottles. Find the amount per bottle and state the answer in litres and millilitres.

19. Delayed Return

Three days later, compare three decimals with different numbers of decimal places, round one value, add and subtract two decimals, multiply and divide by whole numbers, and solve one money problem without looking at the examples. Explain which place-value relationship controlled each step.

20. Parent and Tutor Guide

When a decimal error appears, ask the child to name each digit’s unit before correcting the arithmetic. If 0.45 and 0.7 are confused, rewrite both in hundredths. If addition fails, use a place-value grid or align decimal points. If multiplication produces an answer ten times or one hundred times too large, estimate before reteaching the algorithm.

Fluency grows when the learner can switch between fraction, decimal, money and measurement representations without losing the underlying quantity.

21. Mastery Receipt

• I identify tenths, hundredths and thousandths correctly.
• I recognise equivalent decimals.
• I compare decimals by place value rather than digit count.
• I round to a stated decimal place.
• I add and subtract with aligned place values.
• I multiply and divide decimals by whole numbers appropriately.
• I connect decimals to money, measurement, fractions and percentages.
• I estimate and use inverse operations to check magnitude and accuracy.

Official Reference Route

Singapore Ministry of Education — Primary Mathematics Syllabus P1–P6, updated October 2025

Continue the Primary Mathematics Worked Series

• BTT Primary Mathematics Learning Hub
• Mental Calculation, Number Bonds and Compensation
• Time, Money and Schedules
• Fractions of a Quantity, Equivalence and Operations

The Quiet Return

Decimals become stable when the learner stops treating the point as punctuation and starts seeing the units on both sides of it. Keep place value visible, estimate the scale and the four operations remain connected to the same number system.