Find the least common denominator for these fractions. Enter your answer in the space provided.
[tex]\frac{4}{7}[/tex] and [tex]\frac{2}{8}[/tex]
Answer (1)
Identify the denominators of the fractions as 7 and 8.
Determine that the least common multiple (LCM) of 7 and 8 is needed to find the least common denominator (LCD).
Calculate the LCM of 7 and 8 by multiplying them since they have no common factors: 7 × 8 = 56 .
Conclude that the least common denominator (LCD) is 56 .
Explanation
Understanding the Problem We are given two fractions, 7 4 and 8 2 , and we need to find their least common denominator (LCD). The least common denominator is the least common multiple (LCM) of the denominators.
Identifying Denominators The denominators of the given fractions are 7 and 8. We need to find the least common multiple of 7 and 8.
Finding the LCM Since 7 is a prime number and 8 is a power of 2 ( 8 = 2 3 ), they do not share any common factors other than 1. Therefore, the least common multiple of 7 and 8 is simply their product.
Calculating the LCM To calculate the LCM, we multiply 7 and 8: 7 × 8 = 56 .
Determining the LCD The least common denominator (LCD) of the fractions 7 4 and 8 2 is the LCM of their denominators, which is 56.
Final Answer Therefore, the least common denominator for the fractions 7 4 and 8 2 is 56 .
Examples
Imagine you are baking a cake and need to measure ingredients using fractions of a cup. If one recipe calls for 7 4 of a cup of flour and another calls for 8 2 of a cup of sugar, finding the least common denominator (LCD) helps you compare and combine these measurements easily. In this case, the LCD of 7 and 8 is 56, allowing you to express both fractions with the same denominator: 7 4 = 56 32 and 8 2 = 56 14 . This makes it simple to add or compare the amounts of flour and sugar needed for your cake.
