Understanding Fractions
2. Equivalent Fractions
Learning outcomes
- I can identify fractions that represent the same value.
- I can generate equivalent fractions by multiplying or dividing the numerator and denominator by the same number.
- I can simplify fractions to their lowest terms.
- I can use visual models to demonstrate equivalent fractions.
- I can explain why equivalent fractions have the same value.
Introduction
At first glance, the fractions 1/2, 2/4, and 4/8 may look different, but they all represent exactly the same amount.
Fractions that represent the same value are called equivalent fractions.
Understanding equivalent fractions is an important skill because it helps us compare fractions, simplify answers, perform calculations, and solve real-world problems involving parts of a whole.
What Are Equivalent Fractions?
Equivalent fractions are fractions that have different numerators and denominators but represent the same value.
For example:
\( \frac{1}{2} = \frac{2}{4} = \frac{4}{8} \)
Although the numbers are different, each fraction represents one-half.
Why Are They Equivalent?
Imagine a chocolate bar.
If it is divided into:
- 2 equal pieces and you eat 1 piece, you have eaten 1/2.
- 4 equal pieces and you eat 2 pieces, you have still eaten 1/2.
- 8 equal pieces and you eat 4 pieces, you have still eaten 1/2.
The size of each piece changes, but the total amount eaten remains the same.
Generating Equivalent Fractions
To create an equivalent fraction:
Multiply both the numerator and denominator by the same number.
Example 1
Multiply both numbers by 2.
\( \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{8} \)
Example 2
Multiply by 3.
\( \frac{4}{7} = \frac{12}{21} \)
The value has not changed because both parts of the fraction were multiplied by the same number.
Simplifying Fractions
We can also work backwards.
To simplify a fraction:
Divide the numerator and denominator by the same number.
Example:
\( \frac{12}{16} \)
Both numbers are divisible by 4.
\( \frac{12 \div4 }{16 \div4 } = \frac{3}{4} \)
The fraction 3/4 is in its lowest terms.
Lowest Terms
A fraction is in lowest terms when:
- the numerator and denominator have no common factors except 1.
Examples:
| Fraction | Lowest Terms |
|---|---|
| 6/8 | 3/4 |
| 15/20 | 3/4 |
| 12/18 | 2/3 |
| 18/24 | 3/4 |
Always simplify your final answers whenever possible.
Using Visual Models
Pictures help us understand why fractions are equivalent.
A fraction circle, fraction bar, or shaded rectangle can show that different fractions represent the same amount.
For example:
- 1 out of 2 parts shaded
- 2 out of 4 parts shaded
- 4 out of 8 parts shaded
Each diagram shades exactly half of the whole.
Recognizing Equivalent Fractions
Sometimes you can recognize equivalent fractions without calculating.
Examples:
| Fraction | Equivalent Fraction |
|---|---|
| 1/2 | 2/4 |
| 2/3 | 4/6 |
| 3/5 | 6/10 |
| 4/7 | 8/14 |
| 5/8 | 10/16 |
Look for a common multiplication or division pattern.
Why Multiplying Both Numbers Works
A fraction represents division.
For example:
\( \frac{1}{2} = 1 \div 2 \)
If we multiply both numbers by 2:
\( \frac{2}{4} = 2 \div 4 \)
Both divisions still give:
0.5
Since the value does not change, the fractions are equivalent.
Common Mistakes
Do not multiply or divide only one part of the fraction.
Incorrect:
\( \frac{3}{5} = \frac{6}{5} \)
The value has changed.
Correct:
\( \frac{3}{5} = \frac{6}{10} \)
Always perform the same operation on both the numerator and denominator.
Real-World Applications
Equivalent fractions are useful in many everyday situations, including:
- Cooking and baking
- Measuring ingredients
- Construction and carpentry
- Sharing food fairly
- Reading rulers and measuring tapes
- Mathematics and science calculations
Worked Examples
Example 1
Find an equivalent fraction for:
\( \frac{2}{5} \)
Multiply by 3.
\( \frac{2}{5} = \frac{6}{15} \)
Example 2
Simplify:
\( \frac{18}{24} \)
Divide by 6.
\( \frac{3}{4} \)
Example 3
Are these fractions equivalent?
\( \frac{3}{6} \ and \ \frac{1}{2} \)
Simplify:
\( \frac{3}{6} = \frac{1}{2} \)
Answer:
Yes.
Example 4
Find the missing number.
\( \frac{4}{7} = \frac{12}{?} \)
Since 4 × 3 = 12,
Multiply the denominator by 3.
7 × 3 = 21
Answer:
21
Example 5
Write 8/12 in lowest terms.
Both numbers divide by 4.
\( \frac{8}{12} = \frac{2}{3} \)
Did You Know?
Ancient Egyptian mathematicians often wrote fractions differently from the way we do today. They preferred to express most fractions as sums of unit fractions (fractions with a numerator of 1), such as 1/2 + 1/6 instead of 2/3. Modern notation is much simpler and makes calculations with equivalent fractions much easier.
Key Terms
| Term | Definition |
|---|---|
| Equivalent Fractions | Fractions that have different numerators and denominators but represent the same value. |
| Numerator | The top number of a fraction, showing how many parts are considered. |
| Denominator | The bottom number of a fraction, showing how many equal parts make up the whole. |
| Simplify | To write a fraction in its lowest terms by dividing the numerator and denominator by the same number. |
| Lowest Terms | A fraction whose numerator and denominator have no common factors other than 1. |
| Common Factor | A number that divides exactly into two or more numbers. |
Key Takeaways
- Equivalent fractions represent the same value even though they look different.
- Multiply or divide both the numerator and denominator by the same number to generate equivalent fractions.
- Simplifying a fraction means writing it in its lowest terms.
- Visual models help show why equivalent fractions have the same value.
- Equivalent fractions are used in everyday situations such as cooking, measuring, construction, and mathematics.
- Understanding equivalent fractions prepares you for comparing, adding, subtracting, multiplying, and dividing fractions.