Basic 9 · Mathematics · Number Operations
Multiplying and Dividing by Powers of 10 and the Commutative Property
In this topic you will multiply and divide numbers by powers of 10, including decimals and benchmark fractions, and determine the commutative properties of addition and multiplication.
By the end of these notes you should be able to:
- Multiply and divide decimals by powers of 10
- Use powers of 10 with benchmark fractions
- Apply the commutative property
What you already know
- Indices and decimals
Copy everything that has a blue line on its left into your notebook. The other parts are for reading.
Lesson 1
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.1.1 – Multiply and divide given numbers by powers of 10 including decimals and benchmark fractions
Introduction
Scientists measure stars and atoms. Powers of 10 let them work with such numbers quickly.
1. Powers of 10
| Power | Value |
|---|---|
| 10⁻² | 0.01 |
| 10⁻¹ | 0.1 |
| 10⁰ | 1 |
| 10¹ | 10 |
| 10² | 100 |
| 10³ | 1,000 |
| 10⁶ | 1,000,000 |
2. The Rule
Multiplying by 10ⁿ moves each digit n places to the left. Dividing by 10ⁿ moves each digit n places to the right. It is common to think of the decimal point moving: it moves right when you multiply and left when you divide. Zeros may have to be added as place holders.
Multiplying
Calculate 0.036 × 1,000.
- 1,000 = 10³, so move the digits 3 places left.
Answer: 36
Dividing
Calculate 4.5 ÷ 10³.
- Move the digits 3 places right.
- Add place holders.
Answer: 0.0045
Negative powers
Calculate 250 × 10⁻².
- Multiplying by 10⁻² is the same as dividing by 100.
Answer: 2.5
3. Using Powers of 10 in Real Life
Metric units are based on powers of 10. To change kilometres to metres, multiply by 1,000. To change centimetres to metres, divide by 100. A bacterium might be 0.000002 m long, which is 2 × 10⁻⁶ m. The Earth is about 150,000,000 km from the Sun, or 1.5 × 10⁸ km. Standard form is a tidy way of writing such numbers.
4. Linking with Indices
Since 10² × 10³ = 10⁵, you can multiply by powers of 10 using the laws of indices. For example, 3 × 10⁴ × 2 × 10³ = 6 × 10⁷. For division, (8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴. Multiply or divide the front numbers, then deal with the powers of 10. If the front number goes over 10, adjust it: 5 × 10² × 4 × 10³ = 20 × 10⁵ = 2 × 10⁶.
New words
- power of 10
- 10 raised to an index.
- place holder
- A zero keeping digits in position.
- standard form
- A number written as A × 10ⁿ.
- metric
- A system of units based on 10.
Summary
- Multiplying moves digits left, and dividing moves them right.
- Negative powers of 10 divide.
- Standard form uses powers of 10.
Class activity: Power Slide
- Make a place value chart.
- Slide digits to multiply and divide.
- Record your results.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. 3.2 × 100 =
- A. 320
- B. 32
- C. 3,200
- D. 0.32
Show answer
Answer: 320Move left.2. 45 ÷ 10³ =
- A. 0.045
- B. 0.45
- C. 4.5
- D. 450
Show answer
Answer: 0.045Move right.3. 10⁰ = 1.
- A. True
- B. False
Show answer
Answer: TrueZero index.4. 0.7 × 10 = ______.
Show answer
Answer: 7Move left.5. Calculate 6.5 × 10⁻².
Show answer
Answer: 0.065Divide by 100.6. Calculate (6 × 10⁵) × (3 × 10²) in standard form.
Show answer
Answer: 1.8 × 10⁸18 × 10⁷.
Lesson 2
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.1.1 – Multiply and divide given numbers by powers of 10 including decimals and benchmark fractions
Introduction
Quick calculations with ½, ¼ and ¾ are a trader's secret weapon.
1. Benchmark Fractions
| Fraction | Decimal | Percentage |
|---|---|---|
| ½ | 0.5 | 50% |
| ¼ | 0.25 | 25% |
| ¾ | 0.75 | 75% |
| ⅕ | 0.2 | 20% |
| ⅒ | 0.1 | 10% |
| ⅛ | 0.125 | 12.5% |
Benchmark fractions are familiar fractions used as reference points. Knowing them lets you estimate quickly. Combined with powers of 10, they make many calculations simple.
2. Multiplying
Benchmark multiplication
Calculate ¾ × 10³ and 0.25 × 10⁴.
- ¾ × 1,000 = 750.
- 0.25 × 10,000 = 2,500.
Answer: 750 and 2,500
3. Dividing by Fractions
Dividing by a fraction means multiplying by its reciprocal. Dividing by ½ is the same as multiplying by 2, and dividing by ¼ is the same as multiplying by 4. So 3.6 ÷ ¼ = 14.4. To divide by 0.2, multiply by 5, since 0.2 is ⅕. These shortcuts are very useful in timed examinations.
Shortcuts
Calculate 240 ÷ 0.5 and 7.5 ÷ 0.25.
- 240 ÷ ½ = 240 × 2 = 480.
- 7.5 ÷ ¼ = 7.5 × 4 = 30.
Answer: 480 and 30
4. Multiplying by 25, 50 and 125
To multiply by 50, multiply by 100 and halve: 48 × 50 = 4,800 ÷ 2 = 2,400. To multiply by 25, multiply by 100 and divide by 4: 36 × 25 = 3,600 ÷ 4 = 900. To multiply by 125, multiply by 1,000 and divide by 8: 16 × 125 = 16,000 ÷ 8 = 2,000. These work because 50 = 100 × ½, 25 = 100 × ¼ and 125 = 1,000 × ⅛.
5. In Real Life
A trader with 2,000 mangoes who sells ¾ of them has sold 1,500. If a shop gives 25% off a GH₵800 phone, you pay ¾ of 800, or GH₵600. Thinking with benchmark fractions lets you check prices quickly, without a calculator.
New words
- benchmark fraction
- A familiar fraction used as a reference.
- reciprocal
- One divided by a number.
- shortcut
- A quicker way of calculating.
- halve
- To divide by 2.
Summary
- Benchmark fractions make estimates easy.
- Dividing by a fraction means multiplying by the reciprocal.
- Use 100 and 1,000 to multiply by 25, 50 and 125.
Class activity: Shortcut Race
- Solve ten benchmark questions.
- Time yourselves.
- Explain your shortcuts.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. ¼ of 1,000 is…
- A. 250
- B. 25
- C. 400
- D. 100
Show answer
Answer: 250Divide by 4.2. 9 ÷ 0.25 =
- A. 36
- B. 2.25
- C. 18
- D. 90
Show answer
Answer: 36× 4.3. Dividing by ½ is the same as multiplying by 2.
- A. True
- B. False
Show answer
Answer: TrueReciprocal.4. 0.2 as a fraction is ______.
Show answer
Answer: ⅕Benchmark.5. Calculate 64 × 25 mentally.
Show answer
Answer: 1,6006,400 ÷ 4.6. A GH₵1,200 phone has a 25% discount. Find the price.
Show answer
Answer: GH₵900¾ of 1,200.
Lesson 3
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B9.1.2.1.2 – Demonstrate the ability to determine commutative properties of addition and multiplication
Introduction
Does 3 + 5 equal 5 + 3? Yes. Does 3 − 5 equal 5 − 3? No. This is the idea behind the commutative property.
1. What Is the Commutative Property?
Commutative property: Changing the order of two numbers does not change the result.
| Operation | Example | Commutative? |
|---|---|---|
| addition | 7 + 4 = 4 + 7 = 11 | yes |
| multiplication | 6 × 3 = 3 × 6 = 18 | yes |
| subtraction | 7 − 4 ≠ 4 − 7 | no |
| division | 8 ÷ 2 ≠ 2 ÷ 8 | no |
In algebra, we write a + b = b + a and a × b = b × a. The property works for all rational numbers, including fractions and decimals. For example, ½ + ¼ = ¼ + ½ and 0.5 × 4 = 4 × 0.5.
2. Using the Property
Mental calculation
Calculate 0.25 × 37 × 4.
- Swap the order: 0.25 × 4 × 37.
- 0.25 × 4 = 1.
- 1 × 37.
Answer: 37
Algebra
Simplify 3x + 5 + 2x + 7.
- Rearrange: 3x + 2x + 5 + 7.
- Collect like terms.
Answer: 5x + 12
3. Why It Works
Think of a rectangle made from rows of counters. Three rows of four and four rows of three contain the same number of counters, because you are looking at the same rectangle from a different side. This is why multiplication is commutative. For addition, joining two groups gives the same total, whichever you count first.
4. Commutative or Not?
Since subtraction is not commutative, take care: "6 less than 10" means 10 − 6, not 6 − 10. Likewise, in division, the order matters. You can, however, rewrite subtraction as addition of a negative: 10 − 6 = 10 + (−6) = (−6) + 10. In this form, the terms can be swapped. This is useful in algebra. Division can be rewritten as multiplication by a reciprocal in a similar way. The property has real-life uses too. The cost of 25 books at GH₵4 is the same as 4 books at GH₵25, and this may be easier to calculate.
New words
- commutative
- Order does not change the result.
- operation
- A mathematical process.
- like terms
- Terms with the same letter part.
- array
- Objects in rows and columns.
Summary
- Addition and multiplication are commutative.
- Subtraction and division are not.
- Use the property to simplify calculations and algebra.
Class activity: Test the Property
- Try each operation with fractions and decimals.
- Decide whether it is commutative.
- Share your evidence.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. Which is commutative?
- A. multiplication
- B. subtraction
- C. division
- D. none
Show answer
Answer: multiplicationProperty.2. Simplify 2y + 3 + 4y.
- A. 6y + 3
- B. 9y
- C. 6y³
- D. 5y + 4
Show answer
Answer: 6y + 3Like terms.3. 8 ÷ 4 = 4 ÷ 8.
- A. True
- B. False
Show answer
Answer: FalseDivision is not commutative.4. a + b = b + ______.
Show answer
Answer: aAlgebra.5. Calculate 4 × 73 × 0.5 mentally.
Show answer
Answer: 1462 × 73.6. Show that subtraction is not commutative.
Show answer
Answer: 5 − 3 = 2, but 3 − 5 = −2.Counterexample.
Homework
- Write five calculations that are easier using the commutative property.