What's the least common denominator (LCD) for the group of fractions: [tex]$\frac{3}{4}$[/tex] and [tex]$\frac{7}{10}$[/tex] ?
Select one of four
30
15
40
20
Answer (1)
Find the prime factorization of each denominator: 4 = 2 2 and 10 = 2 × 5 .
Identify the highest powers of all prime factors: 2 2 and 5 1 .
Multiply the highest powers together: 2 2 × 5 = 4 × 5 = 20 .
The least common denominator is 20 .
Explanation
Problem Analysis We are given two fractions, 4 3 and 10 7 , and we need to find the least common denominator (LCD). The LCD is the smallest multiple that both denominators share.
Prime Factorization First, let's find the prime factorization of each denominator.
The prime factorization of 4 is 2 × 2 = 2 2 .
The prime factorization of 10 is 2 × 5 .
Identifying Highest Powers Now, we identify the highest power of each prime factor present in the factorizations. The prime factors are 2 and 5.
The highest power of 2 is 2 2 = 4 .
The highest power of 5 is 5 1 = 5 .
Calculating the LCD To find the LCD, we multiply the highest powers of all prime factors together: 2 2 × 5 = 4 × 5 = 20 .
Final Answer Therefore, the least common denominator (LCD) of 4 3 and 10 7 is 20 .
Examples
When planning a party, you might want to serve pizza and cake. If you cut the pizza into slices that are 4 1 of the whole and the cake into slices that are 10 1 of the whole, the least common denominator (LCD) helps you determine a common size to easily compare and manage the portions. In this case, the LCD of 4 and 10 is 20, meaning you can think of the pizza as having 20 'mini-slices' (where each original slice is 5 mini-slices) and the cake as having 20 'mini-slices' (where each original slice is 2 mini-slices). This makes it easier to plan how many total slices you need to serve all your guests fairly.
