Equivalent Fractions
- Introduction
- What is an Equivalent Fraction?
- Example of Equivalent Fraction
- How to Find Equivalent Fractions?
- Methods to Determine Equivalent Fractions
- Equivalent Fractions Chart
- Solved Examples
- Practice Problem
- Frequently Asked Questions
Introduction
Fractions help us talk about parts of a whole or a group. A fraction has two parts: the numerator and the denominator. The numerator, which is the number on the top, tells you how many parts of the whole or group you're talking about. The denominator, the number on the bottom, tells you how many equal parts the whole is divided into. For example, if you have `3/8` of a cake, `3` is the numerator because you have `3` parts, and `8` is the denominator because the cake is divided into `8` equal parts.
What is an Equivalent Fraction?
Equivalent fractions are fractions that might look different but have the same value. Imagine you have a pizza and you cut it into four slices. If you take `2` slices, that's the same as taking half the pizza. Now, if you cut the same pizza into `6` slices and take `3`, you're still taking half of the pizza. That's because `2/4` and `3/6` are both equal to `1/2`. You can find equivalent fractions by multiplying a fraction's top and bottom numbers by the same number. For example, if you want to find another fraction equivalent to `2/3`, you can multiply both `2` and `3` by `3`, which gives you `6/9`.
Example of Equivalent Fraction
Example:
Consider `2/3`, `4/6`, `6/9`, and `8/12`. Let's visualize each fraction as circles with shaded parts. If we look closely, we can see that the shaded portions in all the circles represent the same portion when viewed as a whole.
So, even though the fractions have different numbers, they're equivalent because they represent the same portion of the whole when simplified.
How to Find Equivalent Fractions?
Equivalent fractions can be determined by either multiplying or dividing both the numerator and the denominator by the same number. Here's how these methods work:
Multiplying the Numerator and Denominator by the Same Number:
For example, take the fraction `3/8`. By multiplying both the numerator and the denominator by `2`, you get `6/16`. Similarly, multiplying by `3` yields `9/24`, and so on. All these fractions are equivalent to `3/8` because they represent the same proportion of a whole.
Dividing the Numerator and Denominator by the Same Number:
Consider the fraction `20/30`. To find its equivalent fraction, divide both `20` and `30` by their highest common factor, which is `10`. This gives you `2/3`, which is the simplified form of `20/30`.
Remember, you can only multiply or divide fractions by the same number to get equivalent fractions, ensuring both the numerator and denominator remain whole numbers.
Methods to Determine Equivalent Fractions
To determine if two fractions are equivalent, there are several methods we can use:
Method `1`: Making the Denominators the Same
By making the denominators the same, we can compare if two fractions are equivalent. For example, if we have the fractions `3/4` and `9/12`, we can make their denominators the same by finding their least common multiple (LCM). Multiplying `3/4` by `3/3` to make the denominator `12`, we get `9/12`. This shows that `3/4` and `9/12` are equivalent fractions.
Method `2`: Cross Multiplication
Another method is to cross-multiply the fractions. For instance, if we have `1/2` and `3/6`, when we cross multiply, we get `1 × 6 = 6` and `2 × 3 = 6`. Since both results are equal, `1/2` and `3/6` are equivalent fractions.
Method `3`: Convert to Decimals
Converting fractions to decimals can also help identify equivalent fractions. If the decimal forms of two fractions are the same, then they are equivalent. For example, `1/4` and `3/12` both result in `0.25` when converted to decimals, indicating that they are equivalent fractions.
Equivalent Fractions Chart
This chart displays equivalent fractions for various unit fractions. For instance, the unit fraction `1/2` has equivalent fractions such as `2/4`, `3/6`, `4/8`, and so on. Similarly, other unit fractions have their respective equivalent fractions listed in the chart.
Solved Examples
Example `1`. Convert the fraction `3/8` to a decimal.
Solution:
To convert the fraction `3/8` to a decimal, divide the numerator by the denominator:
`3 ÷ 8 = 0.375`
Therefore, `3/8` as a decimal is `0.375`.
Example `2`. Determine if `2/5` and `6/15` are equivalent fractions using cross multiplication.
Solution:
Cross multiplying the fractions `2/5` and `6/15`, we get:
`2 × 15 = 30`
`5 × 6 = 30`
Since both results are equal, `2/5` and `6/15` are equivalent fractions.
Example `3`. Find an equivalent fraction of `3/7` with a denominator of `14`.
Solution:
To find an equivalent fraction of `3/7` with a denominator of `14`, we multiply both the numerator and the denominator by `2`:
`3/7 × 2/2 = 6/14`
Therefore, an equivalent fraction of `3/7` with a denominator of `14` is `6/14`.
Example `4`. Find an equivalent fraction of `6/9` by dividing both the numerator and denominator by their greatest common factor.
Solution:
The greatest common factor of `6` and `9` is `3`. Dividing both the numerator and denominator by `3`:
`6 ÷ 3 = 2`
`9 ÷ 3 = 3`
Therefore, an equivalent fraction of `6/9` is `2/3`.
Example `5`. Find an equivalent fraction of `4/11` by multiplying both the numerator and denominator by `3`.
Solution:
To find an equivalent fraction of `4/11`, we multiply both the numerator and denominator by `3`:
`4/11 × 3/3 = 12/33`
Therefore, an equivalent fraction of `4/11` is `12/33`.
Example `6`. Are the two fractions `6/10` and `12/15` equivalent?
Solution:
The decimal equivalent of `6/10` is `0.6`.
The decimal equivalent of `12/15` is `0.8`.
As the decimal value for the two fractions is different, `6/10` and `12/15` are not equivalent fractions.
Practice Problems
Q`1`. Which of the following fractions is equivalent to `3/4`?
- `6/8`
- `4/5`
- `9/12`
- `2/3`
Answer: a
Q`2`. Which fraction is equivalent to `12/15`?
- `14/17`
- `5/10`
- `4/5`
- `6/7`
Answer: c
Q`3`. Are the fractions `17/9` and `51/27` equivalent?
- Yes
- No
Answer: a
Q`4`. Which of the following fractions is equivalent to `5/8`?
- `10/16`
- `4/6`
- `15/20`
- `7/12`
Answer: a
Q`5`. Which of the following sets represents equivalent fractions?
- `1/2`, `1/4`, `1/6`, `1/8`
- `2/3`, `4/5`, `6/7`, `8/9`
- `50/50`, `40/30`, `30/20`, `20/10`
- `6/7`, `12/14`, `18/21`, `24/28`
Answer: d
Frequently Asked Questions
Q`1`. What are equivalent fractions?
Answer: Equivalent fractions are fractions that represent the same value, even though they may look different. They may have different numerators and denominators, but when simplified, they result in the same fraction.
Q`2`. How do I know if two fractions are equivalent?
Answer: Two fractions are equivalent if they can be simplified to the same fraction. You can simplify fractions by dividing both the numerator and denominator by their greatest common factor until they cannot be simplified further.
Q`3`. What methods can I use to find equivalent fractions?
Answer:
There are several methods to find equivalent fractions. A few of them are listed below:
`1`. Making the denominators the same and comparing the numerators.
`2`. Using cross multiplication.
`3`. Converting fractions to decimals and comparing their values.
Q`4`. Why are equivalent fractions important?
Answer: Equivalent fractions are important because they help us compare and operate with fractions more easily. They allow us to express the same quantity in different ways, which is useful in various mathematical operations, such as addition, subtraction, multiplication, and division of fractions.
Q`5`. Can fractions with different denominators be equivalent?
Answer: Yes, fractions with different denominators can be equivalent. Equivalent fractions have different numerators and denominators, but they represent the same portion or value of a whole. By simplifying fractions, you can determine if they are equivalent, regardless of their initial denominators.