Question

How can knowing the GCF and LCM help you when you add, subtract, and multiply fractions?

Answer 1

Hint: To answer this question, we first define what GCF (Greatest Common Factor) and LCM (Least Common Multiple) are. Then we show with the help of an example how each of these help us in simplifying fractions.

Complete step by step solution:
The given question requires us to determine how knowing the GCF and LCM helps us to add, subtract and multiply fractions. We first explain what each of the terms are and their functions.
GCF (Greatest Common Factor) of two numbers is the highest number which can divide both the numbers completely. For example, we know the factors of 6 are 1, 2, 3, 6. The factors of 15 are 1, 3, 5, 15. The highest number which divides both the numbers completely is the greatest common factor and, in this case, it is 3.
$\Rightarrow GCF\left( 6,15 \right)=3$
LCM (Least Common Multiple) of two numbers is the smallest or least number that is divisible by both the numbers. Multiples of 4 are 4, 8, 12, 16, 20, 24… Multiples of 6 are 6, 12, 18, 24… The common multiples for 4 and 6 are 12, 24… The least common multiple is 12.
$\Rightarrow LCM\left( 4,6 \right)=12$
Now let us take an example of two fractions $\dfrac{3}{4}$ and $\dfrac{1}{6}$ ,and perform addition, subtraction and multiplication.
To add these two fractions, we need to first find the LCM of the denominators. We have already shown that the LCM of 4 and 6 are 12. Now we need to make both the denominators 12. For this we multiply the numerator and denominator of the first term by 3. We need multiply the numerator and denominator of the second term by 2.
$\Rightarrow \dfrac{3\times 3}{4\times 3}+\dfrac{1\times 2}{6\times 2}$
Multiplying the terms,
$\Rightarrow \dfrac{9}{12}+\dfrac{2}{12}$
Adding both the terms since they have the same denominator,
$\Rightarrow \dfrac{9+2}{12}=\dfrac{11}{12}$
Now we use the GCD to cancel out any common factors between the numerator and denominator. The factors of 11 are 1 and 11. Factors of 12 are 1, 2, 3, 4, 6, 12. There are no common factors apart from 1 therefore, we cannot cancel them.
Subtraction is performed in a similar manner with the steps shown above with the only difference being the minus sign instead of the plus sign.
Now consider the same two fractions and we multiply them,
$\Rightarrow \dfrac{3}{4}\times \dfrac{1}{6}$
Here we do not need the LCM. We multiply the numerators and denominators separately and write them respectively.
$\Rightarrow \dfrac{3}{24}$
We now know that the factors of 3 are 1 and 3. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Now we check for the greatest common factor between the two which is 3.
Hence, we divide the numerator and denominator by the factor of 3 to simplify and bring the fraction to its simplest terms.
$\Rightarrow \dfrac{3\div 3}{24\div 3}=\dfrac{1}{8}$
Hence, we have shown that knowing the GCF and LCM helps us when we add, subtract, and multiply fractions.

Note: Students need to have a good understanding of what GCD and LCM are for them to solve such questions easily. GCD or greatest common divisor is also known by another name called HCF which stands for highest common factor.