Unit 2: Equivalent Fractions
Topic A: Equivalent Fractions
Start from the left side of each drawing and shade in the fraction shown. Below is the whole shape.
Shade [latex]\tfrac{1}{2}[/latex]
Shade [latex]\tfrac{2}{4}[/latex]
Shade [latex]\tfrac{3}{6}[/latex]
Shade [latex]\tfrac{4}{8}[/latex]
Shade [latex]\tfrac{5}{10}[/latex]
Did you notice that the amount you shaded was the same in each drawing?
The fractions that you were asked to shade are equivalent fractions. Equivalent fractions are fractions that are equal.
Now shade the fractions asked for in these drawings, the same way.
Below is the whole shape.
Shade [latex]\tfrac{1}{3}[/latex]
Shade [latex]\tfrac{2}{6}[/latex]
Shade [latex]\tfrac{3}{9}[/latex]
Shade [latex]\tfrac{4}{12}[/latex]
Shade [latex]\tfrac{5}{15}[/latex]
These above examples are all equivalent fractions.
[latex]\dfrac{1}{3}[/latex] = [latex]\dfrac{2}{6}[/latex] = [latex]\dfrac{3}{9}[/latex] = [latex]\dfrac{4}{12}[/latex] = [latex]\dfrac{5}{15}[/latex]
To work with common fractions, it is often necessary to use an equivalent fraction in place of the fraction that is given. There are several processes to learn which will help you to find equivalent fractions.
Factors
Factors are the numbers which are multiplied together to make a product. An understanding of factors is needed to express fractions in lowest terms.
Example A
We say, “The factors of 12 are 3 and 4.”
Does 12 have any other factors?
What other numbers can be multiplied together to equal 12?
- [latex]1 \times 12 = 12[/latex] or [latex]12 \times 1 = 12[/latex]
- [latex]2 \times 6 = 12[/latex] or [latex]6 \times 2 = 12[/latex]
- [latex]3 \times 4 = 12[/latex] or [latex]4 \times 3 = 12[/latex]
The factors of 12 are 1, 2, 3, 4, 6, 12.
Example B
Find the factors of 10.
- [latex]1 \times 10 = 10[/latex]
- [latex]2 \times 5 = 10[/latex]
The factors of 10 are 1, 2, 5, 10.
Example C
Find the factors of 9.
- [latex]1 \times 9 = 9[/latex]
- [latex]3 \times 3 = 9[/latex]
The factors of 9 are 1, 3, 9.
Exercise 1
Find all the factors
- The factors of 16: [latex]1 \times 16 = 16; 2 \times 8 = 16; 4 \times 4 = 16[/latex] The factors of 16 are 1, 2, 4, 8, 16.
- The factors of 4: [latex]1 \times 4 = 4; 2 \times 2 = 4[/latex] The factors of 4 are 1, 2, 4.
- The factors of 8:
- The factors of 20:
- The factors of 5:
- The factors of 15:
- The factors of 21:
- The factors of 6:
- The factors of 25:
Answers to Exercise 1
- 1, 2, 4, 8
- 1, 2, 4, 5, 10, 20
- 1, 5
- 1, 3, 5, 15
- 1, 3, 7, 21
- 1, 2, 3, 6
- 1, 5, 25
Some numbers only have two factors, 1 and the number itself. These numbers are called prime numbers. Look at the chart for some prime numbers.
| Prime Numbers | Factors |
|---|---|
| 1 | 1, 1 |
| 2 | 1, 2 |
| 3 | 1, 3 |
| 5 | 1, 5 |
| 7 | 1, 7 |
| 11 | 1, 11 |
| 13 | 1, 13 |
| 17 | 1, 17 |
| 19 | 1, 19 |
| 23 | 1, 23 |
| 29 | 1, 29 |
Add other prime numbers to the chart as you find them.
Reminder: Prime numbers only have two, prime factors.
Finding Common Factors
A common factor