Question

The least common denominator of addition of the fractions $\dfrac{4}{{15}},\dfrac{1}{{12}},\dfrac{3}{8}$ is
A) 40
B) 120
C) 180
D) 480
E) 1440

Answer 1

Hint: We are familiar with the addition of two or three fractions, and while performing so, we know that the common denominator required for the operation is the LCM of respective denominators of given fractions that are to be added.

Complete step by step solution:
For the addition of fractions, we simply take a common denominator that divides the denominator of each and every fraction that is to be added.
Now, for two or more numbers, a number that divides all the given numbers is a common multiple of the given numbers.
And, the least number that divides all the given numbers is called the ‘Least Common Multiple’ of the given numbers.
So, the least common denominator of the addition of the fractions is ‘Least Common Multiple’ of the denominators of the respective fractions.
Thus, the least common denominator of addition of the fractions $\dfrac{4}{{15}},\dfrac{1}{{12}},\dfrac{3}{8}$will be the ‘Least Common Multiple’ of their denominators, i.e. 15, 12, 8
Now, an easy method of finding the Least Common Multiple of numbers is the Prime Factorization Method.
In this method, we find the prime factorization of each number, and then collect all the factors of all the numbers without repetition.
Now, prime factorization of 15 is:
$15 = 3 \times 5$
prime factorization of 8 is:
$8 = 2 \times 2 \times 2$
Prime factorization of 12 is:
$12 = 2 \times 2 \times 3$
Now, collecting all the primes to get the Least Common Multiple of 15, 12, and 8
LCM: $2 \times 2 \times 2 \times 3 \times 5$
LCM$ = 120$
Thus, the Least Common Multiple of 15, 12, and 8 is 120
Hence, the least common denominator of addition of the fractions $\dfrac{4}{{15}},\dfrac{1}{{12}},\dfrac{3}{8}$ is 120, so option B is correct.

Note: Students must not get confused between HCF and LCM. LCM is a number that is a multiple of the given numbers, or we can say, LCM is the smallest number when divided by each number gives a whole number. HCF is a number that is a common factor of the given numbers, or we can say that HCF is the highest number which divides all the given numbers.
LCM is always greater than the numbers or equal to the one of the numbers.
HCF is always smaller than the numbers or equal to the one of the numbers.