Question

Write the simplest form for numerator 21 and denominator 28?

Answer 1

Hint: Using the concept of GCF (Greatest Common Factor) which is the largest positive number that divides evenly and thus giving zero remainders. So to express the given numbers in the simplest fraction we have to find their GCF, divide the numerator and denominator with the GCF, and then rewrite the fraction.

Complete step by step answer:
Given
Numerator: $ 21 $
Denominator: $ 28 $
Therefore the resulting fraction would be: $ \dfrac{{21}}{{28}}..........................\left( i \right) $
Now we need to find its GCF.
The GCF for $ 21 $ and $ 28 $ is $ 7 $ according to the definition.
Now dividing the numerator and denominator with their GCF, such that (i) becomes:
\[
   \Rightarrow \dfrac{{21}}{{28}} = \dfrac{{\dfrac{{21}}{7}}}{{\dfrac{{28}}{7}}} \\
   \Rightarrow \dfrac{{\dfrac{{21}}{7}}}{{\dfrac{{28}}{7}}} = \dfrac{3}{4}..........................\left( {ii} \right) \\
 \]
Therefore the simplest form of fraction with numerator $ 21 $ and denominator $ 28 $ is: \[\dfrac{3}{4}\]using GCF method.
Alternative Method:
We can also solve this method using Prime Factorization Method:
So in Prime Factorization Method, we find the prime numbers of both numerator and denominator and then we rewrite the fraction using the prime numbers.
So Given
Given
Numerator: $ 21 $
Denominator: $ 28 $
Therefore the resulting fraction would be: $ \dfrac{{21}}{{28}}..........................\left( {iii} \right) $
Now Prime Factors of numerator and denominator is to be found:
Prime Factors of $ 21 = 3 \times 7 $
Prime Factors of $ 28 = 2 \times 2 \times 7 $
Now rewriting (iii) with the help of the prime factors, and solving it such that (iii) becomes:
 $ \dfrac{{21}}{{28}} = \dfrac{{3 \times 7}}{{2 \times 2 \times 7}} = \dfrac{3}{4}....................\left( {iv} \right) $
Therefore the simplest form of fraction with numerator $ 21 $ and denominator $ 28 $ is: \[\dfrac{3}{4}\]using prime factorization method.

Note:
Using GCF or prime factorization method the result that one obtains is the same and both are accurate methods.
During the prime factorization method, one should take prime numbers only and avoid composite numbers.
While finding the GCF one should proceed according to the definition and the answer should be compatible with the definition of GCF.