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Topic 1.3

Decimal Operations

1. Place Value

  • Each place is worth 10 times the place to its right: hundreds, tens and ones to the left of the decimal point, and tenths (0.1), hundredths (0.01), thousandths (0.001) to its right.
  • Like decimals have the same number of decimal places, e.g. 3.45 and 2.10. Adding zeros at the end does not change the value: 2.1 = 2.10 = 2.100.
  • Multiplying by 10, 100 or 1000 moves each digit 1, 2 or 3 places to the left; dividing moves each digit to the right.
100
2
10
5
1
7
 
.
0.1
3
0.01
4
0.001
9
0.0001
6
257.3496 = 200 + 50 + 7 + 0.3 + 0.04 + 0.009 + 0.0006

2. Addition and Subtraction

  • Always line up the decimal points vertically, then add or subtract as with whole numbers.
  • Unlike decimals have different numbers of decimal places: add zeros to match, e.g. write 40 as 40.00 before subtracting 17.83.
  • Borrow from the left when needed, just as with whole numbers.
  • Check: addition and subtraction are inverse operations, e.g. 22.17 + 17.83 = 40.
3.45 + 2.1: write 2.1 as 2.10
3.45+2.105.55
1.Calculate 25.71 + 138.9
Line up the decimal points and write 138.9 as 138.90
25.71+138.90164.61
2.Calculate 40 − 17.83
Write 40 as 40.00, then subtract, borrowing where needed
40.00−17.8322.17
3.A rope is 12.45 m long. A 3.7 m piece is cut off. How much remains?
Original
12.45 m
−
Cut off
3.70 m
=
Remaining
8.75 m
12.45−3.708.75
8.75 m of rope remains
4.Calculate 7.006 − 2.384
Borrow across the zeros: 7.006 = 6 + 0.9 + 0.09 + 0.016
7.006−2.3844.622

3. Multiplication

  • Ignore the decimal points and multiply as whole numbers.
  • Count the total number of decimal places in the two numbers being multiplied.
  • Place the decimal point so the answer has that many decimal places, e.g. 0.18 × 1.4: 18 × 14 = 252, and 2 + 1 = 3 decimal places give 0.252.
  • Remove trailing zeros at the end: 0.360 = 0.36.
  • Money answers are usually rounded to 2 decimal places (the nearest cent).
0.3 × 0.2
3 × 2 = 6
1 + 1 = 2 decimal places
0.3 × 0.2 = 0.06
1.Calculate 15.2 × 3.7
Multiply 152 × 37 as whole numbers
152×371064+45605624
1 + 1 = 2 decimal places, so 15.2 × 3.7 = 56.24
2.Calculate 0.18 × 1.4
Multiply 18 × 14 as whole numbers
18×1472+180252
2 + 1 = 3 decimal places, so 0.18 × 1.4 = 0.252
3.Milk costs $1.25 per litre. What is the cost of 5.5 litres?
Per litre
$1.25
×
Litres
5.5
=
Cost
$6.88
Multiply 125 × 55 as whole numbers
125×55625+62506875
2 + 1 = 3 decimal places: 6.875, so the cost is $6.88 to the nearest cent
4.Calculate 0.04 × 0.25
Multiply as whole numbers, then place the decimal point
4 × 25 = 100
2 + 2 = 4 decimal places
100 with 4 decimal places is 0.0100
0.04 × 0.25 = 0.01

4. Division

  • Make the divisor a whole number first by multiplying both numbers by 10, 100 or 1000.
  • Move the decimal point in both numbers by the same number of places, e.g. 7.5 ÷ 0.05 = 750 ÷ 5 = 150.
  • Then divide, placing the decimal point in the answer directly above the one in the dividend.
  • Dividing by a number less than 1 gives a larger answer.
2.4 ÷ 0.08 = 240 ÷ 8
3082240
2.4 ÷ 0.08 = 30
1.Calculate 9.12 ÷ 0.4
Multiply both numbers by 10 so the divisor is whole: 9.12 ÷ 0.4 = 91.2 ÷ 4
22.84911.32
9.12 ÷ 0.4 = 22.8
2.Calculate 7.5 ÷ 0.05
Multiply both numbers by 100 so the divisor is whole: 7.5 ÷ 0.05 = 750 ÷ 5
15057250
7.5 ÷ 0.05 = 150
3.Calculate 123.48 ÷ 6
The divisor is already whole: put the decimal point in the answer above the one in 123.48
20.5861123.3448
123.48 ÷ 6 = 20.58
4.A box of 100 pens costs $18.50. Find the cost of one pen.
Total
$18.50
÷
Pens
100
=
Per pen
$0.185
Dividing by 100 moves every digit two places to the right of the decimal point
18.50 ÷ 100 = 0.185
$0.19 per pen to the nearest cent

5. Combined Operations and Word Problems

  • Follow BIDMAS: Brackets, Indices, Division and Multiplication, then Addition and Subtraction.
  • Rates: distance ÷ fuel = km per litre, cost ÷ quantity = unit price.
  • Metric conversions: × 1000 from kg to g, ÷ 1000 from g to kg.
  • Proportion: find the value for one first, then scale up or down.
  • Always check that the answer is reasonable in context.
(1.5 + 2.5) × 0.2: brackets first
1.5 + 2.5 = 4
4 × 0.2 = 0.8
1.Calculate (12.6 − 3.78) ÷ 0.3
Brackets first: 12.6 − 3.78, writing 12.6 as 12.60
12.60−3.788.82
Then 8.82 ÷ 0.3 = 88.2 ÷ 3
29.43828.12
(12.6 − 3.78) ÷ 0.3 = 29.4
2.A car travels 385.5 km using 32.5 litres of fuel. Find the fuel consumption in km per litre.
Distance
385.5 km
÷
Fuel
32.5 L
=
km per L
11.9
Fuel consumption = distance ÷ fuel used
385.5 ÷ 32.5 = 3855 ÷ 325
= 11.861…
11.9 km per litre (3 s.f.)
3.Convert 3.75 kg to grams.
1 kg = 1000 g, so multiply by 1000: each digit moves three places to the left
3.75 × 1000
3.75 kg = 3750 g
4.A recipe needs 0.75 kg of flour for 6 people. How much flour is needed for 10 people?
Find the amount for one person, then multiply by 10
0.75 ÷ 6 = 0.125 kg per person
0.125 × 10 = 1.25
1.25 kg for 10 people