Equivalent Fractions
Equivalent fractions look different but have the same value, like 1/2 = 8/16. To find them, multiply top and bottom by the same number to expand, or divide to simplify. Either way, the value stays the same.
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Expanding Fractions
- To expand a fraction, multiply the numerator and denominator by the same number.
- This changes how the fraction looks, but the value stays the same.
Simplifying Fractions
- To simplify a fraction, divide the numerator and denominator by the same number.
- This keeps the value the same, but makes the fraction simpler.
How to Expand Fractions
- Find the number that both the numerator and denominator are multiplied by.
- For example, , so .
- So, 2/7 can be expanded by 9 to make 18/63.
How to Simplify Fractions
- Find the number that both the numerator and denominator are divided by.
- For example, and .
- So, 72/120 can be simplified by 6 to make 12/20.
Fully Simplified Fractions
- A fraction is fully simplified when the numerator and denominator have no common factors other than 1.
- For example, 6/8 is not fully simplified, but 3/4 is.
Practice Questions
Test your understanding
Simplify the fraction . What is the result?
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Dividing the numerator and denominator by their greatest common divisor, 5, gives and , resulting in .
Simplify the fraction . What is the result?
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Dividing the numerator and denominator by their greatest common divisor, 9, gives and , resulting in .
Which fraction represents a fully simplified form?
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is already in its simplest form because 5 and 7 have no common divisors other than 1.
If you expand by a factor of 6, what is the resulting fraction?
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Expanding by a factor of 6 means multiplying both the numerator and denominator by 6: and , giving .
Fully simplify the fraction . What is the result?
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The greatest common divisor of 105 and 150 is 15. Dividing both by 15 gives and , so the simplified fraction is .
A fraction was expanded by a factor of 8 to become . What was the fraction in its simplest form?
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Divide both the numerator and denominator by 8 to undo the expansion: and . Simplifying by dividing both parts by 6 gives .
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Interactive Activity
Practice expanding and simplifying equivalent fractions
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Students Also Ask
The questions students bump into most on this topic
You find an equivalent fraction by multiplying or dividing the numerator and denominator by the same non-zero number. Multiplying is called expanding, and dividing is called simplifying. Both keep the value the same, so the new fraction represents exactly the same proportion as the original one.
Yes. Equivalent fractions always have the same value, even though they look different. Expanding multiplies the numerator and denominator, and simplifying divides them, but neither changes the value. For example, one half and two quarters cover the same proportion of a shape, so they are equal.
A fraction is fully simplified when its numerator and denominator share no common factor other than 1. If a number still divides into both, you can simplify again. Once the only number that divides both is 1, the fraction is in its simplest form and cannot be reduced further.
Expanding and simplifying are opposite processes. To expand a fraction, you multiply the numerator and denominator by the same non-zero number, making the terms larger. To simplify, you divide them by the same non-zero number, making the terms smaller. Both keep the value of the fraction unchanged.
Yes. You can always swap the left and right sides of an equivalent fractions equation. This is useful when you are finding a missing number, because it lets you multiply instead of divide, or divide instead of multiply, whichever is easier to work out.