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

Decimals are not whole numbers with a dot inserted. They are place-value numbers in which every position has a value ten times the position to its right.

Wait, What? 0.5 Is Greater Than 0.45 Even Though 45 Is Greater Than 5

Comparing decimals by ignoring the decimal point creates predictable errors. 0.5 means five tenths. 0.45 means forty-five hundredths, which is four tenths and five hundredths. Five tenths is therefore larger.

The reliable method is to align place values, not count digits or compare the strings as if they were whole numbers.

The Decimal Control Frame

1. Identify the value of each digit by position.
2. Align decimal points when comparing, adding or subtracting.
3. Use zeros as placeholders where useful.
4. Estimate the size of the result before or after exact work.
5. When multiplying or dividing by powers of ten, reason from place value rather than “moving the decimal point”.
6. Check whether the operation should make the answer larger or smaller.

Decimal Place Value

Number	Ones	Tenths	Hundredths	Thousandths
4.307	4	3	0	7
0.582	0	5	8	2

In 4.307, the 3 represents 3 tenths, the 0 represents 0 hundredths and the 7 represents 7 thousandths.

Trailing Zeros Do Not Change Value

0.5 = 0.50 = 0.500. The extra zeros show finer place-value positions but do not change the quantity.

Worked Example 1: Compare Decimals

Compare 2.7 and 2.65.

Write 2.7 as 2.70. Then compare 2.70 and 2.65.

2.70 > 2.65.

Worked Example 2: Order Decimals

Order 0.8, 0.75 and 0.805 from smallest to largest.

Write them as 0.800, 0.750 and 0.805.

0.75, 0.8, 0.805.

Rounding Decimals

To round to a given place, identify that digit, then inspect the digit immediately to its right. If the next digit is 5 or more, increase the rounding digit by 1; otherwise leave it unchanged.

Worked Example 3: Round to the Nearest Tenth

Round 6.47 to the nearest tenth.

The tenths digit is 4. The hundredths digit is 7, so round the tenths up.

6.47 ≈ 6.5.

Worked Example 4: Round to the Nearest Hundredth

Round 3.842 to the nearest hundredth.

The hundredths digit is 4. The thousandths digit is 2, so keep 4.

3.84.

Rounding Is Not Truncating

Truncating 5.98 to one decimal place gives 5.9 by simply removing digits. Rounding to the nearest tenth gives 6.0 because the hundredths digit is 8. These are different operations.

Adding Decimals

Align decimal points so ones are added to ones, tenths to tenths and hundredths to hundredths.

Worked Example 5: Decimal Addition

3.75 + 2.8.

Write 2.8 as 2.80.

3.75 + 2.80 = 6.55.

Worked Example 6: Add Three Decimals

1.25 + 0.8 + 3.075.

1.250 + 0.800 + 3.075 = 5.125.

Subtracting Decimals

Again, align decimal points and use placeholder zeros where needed.

Worked Example 7: Decimal Subtraction

8.2 – 3.47.

Write 8.2 as 8.20.

8.20 – 3.47 = 4.73.

Estimate Before Subtracting

8.2 – 3.47 should be about 8 – 3.5 = 4.5. The exact 4.73 is reasonable. An answer such as 47.3 would fail the magnitude check.

Multiplying Decimals by Whole Numbers

Multiplication can be understood as repeated groups while place value determines the final scale.

Worked Example 8

2.35 × 4.

2 × 4 = 8, 0.35 × 4 = 1.40, so total = 9.40.

Answer: 9.4.

Worked Example 9: Money Context

Five items cost $3.75 each.

3.75 × 5 = 18.75.

Total = $18.75.

Dividing Decimals by Whole Numbers

Division can represent equal sharing. Preserve place value and check by multiplication.

Worked Example 10

7.2 ÷ 3.

72 tenths ÷ 3 = 24 tenths = 2.4.

Check: 2.4 × 3 = 7.2.

Worked Example 11

12.6 ÷ 6 = 2.1 because 2.1 × 6 = 12.6.

Multiplying by 10, 100 and 1000

When a number is multiplied by 10, every digit becomes ten times its previous value. The digits shift one place to the left relative to the place-value chart.

• 3.47 × 10 = 34.7
• 3.47 × 100 = 347
• 0.058 × 1000 = 58

A safer explanation is “the digits change place value” rather than “the decimal point moves”, because the decimal point itself marks the ones boundary.

Dividing by 10, 100 and 1000

• 47.2 ÷ 10 = 4.72
• 47.2 ÷ 100 = 0.472
• 5.8 ÷ 1000 = 0.0058

Worked Example 12: Place-Value Scaling

0.36 × 100 = 36 because 3 tenths becomes 3 tens and 6 hundredths becomes 6 ones.

Decimal × Decimal

At upper-primary levels, decimal multiplication can be checked by estimating size first.

Worked Example 13

1.2 × 0.5.

0.5 means one half, so half of 1.2 is 0.6.

This meaning is stronger than counting decimal places mechanically.

Worked Example 14: Check Magnitude

2.4 × 0.3 should be less than 2.4 because multiplying by 0.3 takes three tenths of 2.4.

2.4 × 0.3 = 0.72.

An answer such as 7.2 is ten times too large and fails the magnitude check.

Decimal Division and Magnitude

Division does not always make a number smaller. Dividing by a positive number less than 1 asks how many small groups fit, so the quotient can be larger.

Worked Example 15

2 ÷ 0.5 = 4 because four halves fit into two wholes.

Decimals and Fractions

Decimals and fractions are two representations of rational quantities.

• 0.5 = 1/2
• 0.25 = 1/4
• 0.75 = 3/4
• 0.2 = 1/5

Moving between representations can make a calculation easier. For example, 0.25 of 80 is one quarter of 80 = 20.

Worked Example 16: Representation Choice

Find 0.75 of 60.

0.75 = 3/4. One quarter of 60 = 15; three quarters = 45.

Rounding for Estimation

Rounding can create a quick estimate before exact computation.

Worked Example 17

Estimate 19.8 × 4.1.

Use 20 × 4 = 80. The exact answer should be near 80.

Exact: 19.8 × 4.1 = 81.18, which is consistent with the estimate.

Common Decimal Errors

• Comparing decimal strings as whole numbers.
• Misaligning decimal points in addition or subtraction.
• Thinking more decimal digits means a larger number.
• Rounding the wrong place value.
• Truncating when rounding is required.
• Using “move the decimal point” without place-value understanding.
• Assuming multiplication always makes larger and division always makes smaller.
• Forgetting placeholder zeros when aligning values.

Error Repair 1: Compare 0.6 and 0.58

Wrong: 0.58 is larger because 58 > 6.

Repair: 0.6 = 0.60. Compare 60 hundredths with 58 hundredths. Therefore 0.6 > 0.58.

Error Repair 2: Subtract Misaligned Decimals

Wrong setup: 8.2 – 3.47 with digits aligned at the right edge rather than the decimal point.

Repair: write 8.20 above 3.47 and subtract matching place values.

Practice Set A: Place Value and Rounding

1. Which is greater: 0.47 or 0.5?
2. Order 1.08, 1.8, 1.18 from smallest to largest.
3. Round 7.46 to the nearest tenth.
4. Round 3.875 to the nearest hundredth.
5. Write the value of the 6 in 4.062.

Practice Set B: Four Operations

1. 4.75 + 2.6
2. 9.2 – 4.85
3. 3.45 × 6
4. 8.4 ÷ 4
5. 0.36 × 100
6. 57.2 ÷ 100
7. 1.5 × 0.4
8. 3 ÷ 0.5

Practice Set C: Context

1. Four tickets cost $6.75 each. Find the total.
2. A 12.5 m rope is cut into 5 equal pieces. Find the length of each piece.
3. Find 0.25 of 72.
4. A container has 8.4 L. 2.75 L is used. How much remains?

Answers

Set A: 0.5; 1.08, 1.18, 1.8; 7.5; 3.88; 0.06.

Set B: 7.35; 4.35; 20.7; 2.1; 36; 0.572; 0.6; 6.

Set C: $27.00; 2.5 m; 18; 5.65 L.

Transfer Test

A tank contains 12.8 L of water. 0.25 of the water is poured out, then 1.6 L is added. How much water is in the tank now?

0.25 = 1/4. One quarter of 12.8 L = 3.2 L. Remaining = 12.8 – 3.2 = 9.6 L. Add 1.6 L → 11.2 L.

Delayed Return

Several days later, solve one decimal comparison, one rounding problem, one addition/subtraction, one multiplication/division and one decimal-fraction conversion without notes. Estimate each answer before exact work.

Parent and Tutor Guide

Use a place-value chart when the child repeatedly misaligns decimals or treats digits as a whole-number string. Ask “What does this digit represent?” before teaching another procedure. For multiplication and division, ask whether the answer should be larger or smaller and why.

Money and measurement provide useful contexts, but the aim is not context dependence. The learner should eventually transfer the same decimal relationships to abstract numbers, lengths, masses, capacities and rates.

Mastery Receipt

• I compare decimals by place value.
• I round to a named place correctly.
• I align decimal points in addition and subtraction.
• I multiply and divide decimals with magnitude checks.
• I explain ×10, ×100 and ÷10, ÷100 using place value.
• I move between common decimal and fraction representations.
• I estimate before trusting an exact answer.
• I know multiplication/division do not always change size in the same direction.

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

Decimal fluency is place-value fluency extended to smaller units. Keep the positions meaningful, estimate the scale of the result and the procedures become much easier to control.