seegongsik
Grade 5 (age 10-11)

grade 5 Reduction & Common Denominator

Simplifying a fraction means dividing its numerator and denominator by the same number, which keeps its size but makes it simpler; finding a common denominator means rewriting fractions with different denominators so that they share one. Take 6 slices of a pizza cut into 12: that is 6/12, and dividing top and bottom by 6 simplifies it to 1/2, the same amount. 1/3 and 1/4 are hard to compare directly, but rewrite both with the common denominator 12 and you only need to compare the numerators 4 and 3. In the first picture you change the numerator and denominator to watch a fraction simplify; in the second you change two denominators to rewrite two unit fractions over a common denominator.

What is simplifying?

✂️ Divide top and bottom by the same number
①The top bar in the first picture is 6/12, with 6 of its 12 parts colored
②Divide the numerator 6 and the denominator 12 by their GCD, 6, and you get 1/2, 1 of 2 parts, like the bottom bar, with the colored length unchanged
③Dividing the numerator and denominator by the same number without changing the size is called simplifying
④When the only number that divides both is 1, the fraction is in lowest terms. Dividing by the GCD gets you there in one step
6
12

What is a common denominator?

🔗 Make the denominators the same, then compare
①The two bars in the second picture are 1/3 and 1/4. Their parts are different sizes, so counting parts does not compare them
②Use 12, the LCM of 3 and 4, as the common denominator: 1/3 = 4/12 and 1/4 = 3/12
③With the same denominator you only compare numerators, so 4/12 is bigger and 1/3 is larger than 1/4
④It works when the numerator is not 1 too: 2/3 = 8/12 and 3/4 = 9/12, so 3/4 is larger
3
4

Summary

Simplifying
ab = a÷nb÷n
Dividing the numerator and denominator by the same number n keeps the fraction the same size
Common denominator
ab → a×mb×m
Multiply the numerator and denominator by the same number m to reach the common denominator; the LCM of the denominators is the simplest choice
🎯 At a glance
①Simplifying: divide the numerator and denominator by the same number
②Dividing by the GCD gives lowest terms in one step
③Common denominator: rewrite fractions so that their denominators match
④The LCM of the denominators is the simplest common denominator
⑤Neither simplifying nor finding a common denominator changes a fraction's size

Examples and Unit Test

Example 1
Write 4/8 in lowest terms.
1
The GCD of the numerator 4 and the denominator 8 is 4.
2
Divide both by 4: 4 ÷ 4 = 1 and 8 ÷ 4 = 2, so it is 1/2.
▸ 12
Dividing by the GCD gives lowest terms in one go; dividing by 2 only gives 2/4, which needs simplifying again.
Example 2
What is the simplest common denominator for 1/3 and 1/4?
1
Find the LCM of the denominators 3 and 4.
2
The LCM of 3 and 4 is 12, so the common denominator is 12: 1/3 = 4/12 and 1/4 = 3/12.
▸ 12
These are the starting fractions in the second picture, where 4 parts and 3 parts of a 12-part bar are colored.
Unit Test
What is 6/9 in lowest terms?
①39
②23
③69
④13
▸ ② 23
1
The GCD of 6 and 9 is 3.
2
6 ÷ 3 = 2 and 9 ÷ 3 = 3, so it is 2/3.
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Primes and Composites
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Fraction Multiplication & Division

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