Lesson 11 ~30 min Unit 1 · Decimals +90 XP

Operations with Decimals

Line up the dots for adding and subtracting. Count decimal places for multiplying. Shift the decimal for dividing. Master all four operations.

Today’s hook: If petrol costs $1.749 per litre and you buy 35.5 litres, how much do you pay? $1.749 × 35.5 = ? Decimals are everywhere in real life. Let’s master them.

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Think First

warm-up

Before you read on — quickly estimate: 2.8 × 4.1 and 15.6 ÷ 3.9. Use rounding to check your answers make sense.

Record your answer in your workbook.

The Big Idea

+5 XP

Adding/subtracting: line up the decimal points, then add or subtract as normal. Multiplying: ignore decimals, multiply as whole numbers, then count total decimal places in the question. Dividing: shift the decimal point in both numbers to make the divisor a whole number, then divide.

Think of decimals as whole numbers with a dot. For addition: line up the dots. For multiplication: treat as whole numbers, then place the dot. For division: shift dots equally to make the divisor whole. The golden rule: estimation checks your answer. 2.8 × 4.1 ≈ 3 × 4 = 12, so the answer should be close to 12 (it’s 11.48).

$2.45 + 1.3 = 3.75$ | $2.5 imes 0.3 = 0.75$ | $6.4 \div 0.8 = 8$

Line up decimal points

For + and −, the dots must be in a column. Add trailing zeros.

Count decimal places

For ×, total d.p. in question = d.p. in answer.

Shift both dots

For ÷, shift both numbers equally to make divisor whole.

What You’ll Master

objectives

Know

Line up decimals for addition and subtraction
Count decimal places for multiplication
Shift decimals for division

Understand

Why the decimal point moves in multiplication/division
How estimation validates answers
Why we shift both numbers in division

Can Do

Add and subtract decimals accurately
Multiply decimals by counting places
Divide decimals by shifting points

Words You Need

vocabulary

Decimal pointThe dot that separates the whole number from the fractional part.

Line upWrite numbers so their decimal points are in the same column.

Trailing zeroA zero added after the last non-zero digit. Doesn’t change value but helps alignment.

Shift the decimalMove the decimal point equally in both numbers when dividing.

EstimationUsing rounded numbers to check if an answer is reasonable.

Decimal placesThe number of digits after the decimal point.

Spot the Trap

heads-up

Wrong: 2.5 + 1.35 = 3.40 by adding from the right. No! Line up the dots first: 2.50 + 1.35.

Right: 2.50 + 1.35 = 3.85. The dots line up, so tenths add to tenths and hundredths to hundredths.

Wrong: 0.2 × 0.3 = 0.6. No! 2 × 3 = 6, and there are 2 decimal places total (1 + 1), so 0.06.

Right: 0.2 × 0.3: 2 × 3 = 6. Two decimal places total, so 0.06. Estimate: 0.2 × 0.3 ≈ 0, so 0.06 makes sense.

Adding and Subtracting Decimals

+5 XP

Line up the decimal points. Add trailing zeros so all numbers have the same number of decimal places. Then add or subtract as normal. The decimal point in the answer goes directly below the others.

Calculate 4.75 + 2.3. Line up: 4.75 + 2.30. Add: 5 + 0 = 5 (hundredths). 7 + 3 = 10, write 0 carry 1 (tenths). 4 + 2 + 1 = 7 (ones). Decimal point below. Answer: 7.05. Check: 4.75 + 2.3 ≈ 5 + 2 = 7. Close enough!

$4.75 + 2.3 = 7.05$

Dots in a column

The decimal point must be directly below the others.

Trailing zeros

Add zeros so all numbers have the same decimal places.

Estimate first

Round to whole numbers, do rough check, then calculate.

Multiplying Decimals

+5 XP

Ignore the decimal points. Multiply as whole numbers. Count the total number of decimal places in the original numbers. Place the decimal point in the answer so it has the same total number of decimal places.

Calculate 1.25 × 0.4. Ignore dots: 125 × 4 = 500. Count d.p.: 1.25 has 2 d.p., 0.4 has 1 d.p., total = 3 d.p. So 500 becomes 0.500 = 0.5. Estimate: 1.25 × 0.4 ≈ 1 × 0.4 = 0.4. 0.5 is close!

$1.25 imes 0.4 = 0.5$

Multiply as whole numbers

Completely ignore the decimal points during multiplication.

Count total d.p.

Add the decimal places from both original numbers.

Estimate to check

1.25 × 0.4 ≈ 0.5. Your answer should be close.

Dividing Decimals

+5 XP

Shift the decimal point in both numbers equally to make the divisor a whole number. Then divide as normal. The decimal point in the answer goes directly above the new position in the dividend.

Calculate 4.32 ÷ 0.6. Shift both dots 1 place right: 43.2 ÷ 6. Now divide: 6 into 43 goes 7 remainder 1. 6 into 12 goes 2. Answer: 7.2. Check: 4.32 ÷ 0.6 ≈ 4 ÷ 0.6 ≈ 6-7. 7.2 makes sense!

$4.32 \div 0.6 = 7.2$

Make divisor whole

Shift the decimal in the divisor until it’s a whole number.

Shift both equally

Move the decimal the same amount in both numbers.

Dot goes above

The decimal in the answer lines up with the new dividend dot.

Watch Me Solve It · 3 examples

step-by-step

Example 1: Subtracting decimals

Calculate 8.3 − 2.47

Line up the dots. Add trailing zero: 8.30 − 2.47.

Subtract: hundredths: 0 − 7 (need to borrow). 10 − 7 = 3. Tenths: 2 − 4 (borrowed, so 2 − 4, need to borrow again). 12 − 4 = 8. Ones: 7 − 2 = 5.

Check: 8.3 − 2.47 ≈ 8 − 2.5 = 5.5. Answer 5.83 is close. Also: 5.83 + 2.47 = 8.30. Correct!

8.3 − 2.47 = 5.83

Example 2: Multiplying decimals

Calculate 0.25 × 1.6

Ignore dots: 25 × 16. 25 × 10 = 250. 25 × 6 = 150. Total = 400.

Count d.p.: 0.25 has 2 d.p., 1.6 has 1 d.p. Total = 3 d.p. So 400 → 0.400 = 0.4.

Check: 0.25 = 1/4, so 1/4 × 1.6 = 1.6/4 = 0.4. Also: 0.25 × 1.6 ≈ 0.25 × 1.5 = 0.375. Close!

0.25 × 1.6 = 0.4

Example 3: Dividing decimals

Calculate 7.56 ÷ 0.09

Shift both dots 2 places right: 756 ÷ 9.

Divide: 9 × 80 = 720. 756 − 720 = 36. 9 × 4 = 36. So 80 + 4 = 84.

Check: 7.56 ÷ 0.09 ≈ 7.5 ÷ 0.1 = 75. Answer 84 is in the right ballpark. Also: 84 × 0.09 = 7.56.

7.56 ÷ 0.09 = 84

Common Pitfalls

avoid these

Mistake: 3.5 + 0.25 = 3.30. No! 3.50 + 0.25 = 3.75. You cannot just add the numbers as they appear without lining up.

Fix: Line up dots: 3.50 + 0.25 = 3.75. Tenths: 5 + 2 = 7. Hundredths: 0 + 5 = 5.

Mistake: 0.6 × 0.5 = 3.0. No! 6 × 5 = 30, and there are 2 d.p. total, so 0.30 = 0.3.

Fix: 0.6 × 0.5: 6 × 5 = 30. Two decimal places, so 0.30 = 0.3. Multiplying two decimals under 1 gives a smaller answer.

Mistake: 4.5 ÷ 0.5 = 4.5 ÷ 5 = 0.9. No! You only shifted the divisor, not the dividend too!

Fix: Shift both 1 place: 45 ÷ 5 = 9. Check: 4.5 ÷ 0.5 = 4.5 × 2 = 9 (dividing by 0.5 = multiplying by 2).

Copy Into Your Books

essential notes

+ / −: Line up decimal points, add trailing zeros, then calculate.

×: Ignore dots, multiply as whole numbers, count total d.p., place dot.

÷: Shift both dots equally to make divisor whole, then divide.

Always estimate first to check your answer is reasonable.

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Decimal Ops

4 problems

Four drill problems to build your decimal operation fluency. Work each, then reveal the answer.

1 5.74 + 2.8

Line up: 5.74 + 2.80 = 8.54.8.54
2 9.6 − 4.35

Line up: 9.60 − 4.35. 0−5 borrow, 10−5=5. 5−3=2. 9−4=5. = 5.25.5.25
3 0.4 × 0.15

Ignore dots: 4 × 15 = 60. Total d.p.: 1 + 2 = 3. So 0.060 = 0.06.0.06
4 3.24 ÷ 0.4

Shift both 1 place: 32.4 ÷ 4. 4 into 32 = 8, 4 into 4 = 1. = 8.1.8.1

Complete in your workbook.

Quick Check · 5 questions

3.45 + 2.8 = ?

+10 XP

0.6 × 0.7 = ?

+10 XP

3.15 ÷ 0.07 = ?

+10 XP

12.5 − 4.75 = ?

+10 XP

2.5 × 0.4 = ?

+10 XP

Show Your Working · 3 questions

Show Your Working

11 marks total

ApplyMedium4 MARKS

Q6. (a) Calculate 6.35 + 2.8 + 0.75. Show your working with dots lined up. (b) Calculate 10.2 − 3.67. Show borrowing.

Answer in your workbook.

ApplyMedium4 MARKS

Q7. (a) Calculate 0.25 × 3.6. Show the multiplication without decimals, then place the dot. (b) Calculate 8.4 ÷ 0.2. Show the shift and division.

Answer in your workbook.

ReasonHard3 MARKS

Q8. A student calculates 4.8 ÷ 0.6 and gets 0.8. They say “dividing makes numbers smaller.” Explain why their answer and reasoning are both wrong, and show the correct calculation.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — 3.45 + 2.80 = 6.25.

2. A — 6 × 7 = 42, 2 d.p. = 0.42.

3. D — Shift 2 places: 315 ÷ 7 = 45.

4. B — 12.50 − 4.75 = 7.75.

5. C — 25 × 4 = 100, 2 d.p. = 1.00 = 1.

Show Your Working Model Answers

Q6 (4 marks): (a) 6.35 + 2.80 + 0.75 [1]. 5+0+5=10, 3+8+7+1=19, 6+2+0+1=9 [1]. = 9.90. (b) 10.20 − 3.67 [0.5]. Borrow twice: 10−7=3, 1−6 borrow, 11−6=5, 9−3=6 [1.5]. = 6.53.

Q7 (4 marks): (a) 25 × 36 = 900 [1]. D.p.: 2 + 1 = 3, so 0.900 = 0.9 [1]. (b) Shift 1 place: 84 ÷ 2 = 42 [1]. Check: 42 × 0.2 = 8.4 [1].

Q8 (3 marks): 4.8 ÷ 0.6 should be larger than 4.8 (dividing by less than 1) [1]. Correct: shift 1 place, 48 ÷ 6 = 8 [1]. Dividing by a decimal < 1 makes the answer bigger [1].

Stretch Challenge · +25 XP, +10 coins

The Shopping Spree

You have $50 to spend at a stationery shop. Prices: pens $2.45 each, notebooks $4.80 each, rulers $1.25 each, erasers $0.95 each. (a) You buy 3 pens, 2 notebooks, and 4 erasers. What is the total cost? (b) If you also want to buy as many rulers as possible with the remaining money, how many can you buy? (c) How much change do you get? Show all working with estimation checks.

Reveal solution

(a) Pens: 3 × $2.45 = $7.35. Notebooks: 2 × $4.80 = $9.60. Erasers: 4 × $0.95 = $3.80. Total: $7.35 + $9.60 + $3.80 = $20.75. (b) Remaining: $50 − $20.75 = $29.25. Rulers: $29.25 ÷ $1.25 = 23.4, so 23 rulers. Cost: 23 × $1.25 = $28.75. (c) Change: $29.25 − $28.75 = $0.50. Total spent: $20.75 + $28.75 = $49.50. Change from $50: $0.50.

Quick Review

+ / −

Line up decimal points

×

Count d.p., ignore dots, place dot

÷

Shift both dots equally

Trailing zeros

Helps alignment

Estimate

Check answer is reasonable

Divisor whole

Goal for division

Interactive: Order of Operations — Spot the Error

Someone made a mistake in their working. Can you find which line breaks BOMDAS? Watch out for trick questions!

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