Which number can each term of the equation be multiplied by to eliminate the fractions before solving?
[tex]6-\frac{3}{4} x+\frac{1}{3}=\frac{1}{2} x+5[/tex]
A. 2
B. 3
C. 6
D. 12
Answer (1)
Identify the fractions in the equation and their denominators: 4, 3, and 2.
Find the least common multiple (LCM) of the denominators.
Calculate the LCM of 4, 3, and 2, which is 12.
Multiplying each term of the equation by 12 eliminates the fractions: 12 .
Explanation
Understanding the Problem We are given the equation 6 − 4 3 x + 3 1 = 2 1 x + 5 . Our goal is to find a number that we can multiply each term of the equation by to eliminate the fractions. This number should be the least common multiple (LCM) of the denominators of the fractions.
Identifying the Denominators The denominators of the fractions in the equation are 4, 3, and 2. We need to find the LCM of these numbers.
Finding the LCM To find the LCM of 4, 3, and 2, we can list the multiples of each number until we find a common multiple. Multiples of 4: 4, 8, 12, 16, ... Multiples of 3: 3, 6, 9, 12, 15, ... Multiples of 2: 2, 4, 6, 8, 10, 12, ... The smallest common multiple of 4, 3, and 2 is 12.
Prime Factorization Method Alternatively, we can use prime factorization to find the LCM. 4 = 2 2 3 = 3 1 2 = 2 1 The LCM is the product of the highest powers of all prime factors: 2 2 × 3 1 = 4 × 3 = 12 .
Verification Therefore, multiplying each term of the equation by 12 will eliminate the fractions. Let's verify this: 12 ( 6 − 4 3 x + 3 1 ) = 12 ( 2 1 x + 5 ) 12 ( 6 ) − 12 ( 4 3 x ) + 12 ( 3 1 ) = 12 ( 2 1 x ) + 12 ( 5 ) 72 − 9 x + 4 = 6 x + 60 As we can see, multiplying each term by 12 eliminates the fractions.
Final Answer The number that can be multiplied by each term of the equation to eliminate the fractions is 12.
Examples
When cooking, you might need to adjust a recipe that uses fractions of cups or spoons. Finding the least common multiple helps you scale the recipe up or down while keeping the ingredient ratios correct. For example, if a recipe calls for 2 1 cup of flour, 3 1 cup of sugar, and 4 1 cup of butter, you can multiply all the ingredients by the LCM of 2, 3, and 4 (which is 12) to get whole numbers: 6 cups of flour, 4 cups of sugar, and 3 cups of butter. This makes it easier to measure and adjust the recipe without dealing with fractions.
