Divide.
$\frac{2 x-10}{12} \div \frac{4 x-20}{8}$
Answer (1)
Divide the expressions by multiplying by the reciprocal: 12 2 x − 10 ÷ 8 4 x − 20 = 12 2 x − 10 ⋅ 4 x − 20 8 .
Factor and cancel common terms: 12 2 ( x − 5 ) ⋅ 4 ( x − 5 ) 8 = 3 1 .
Add the expressions by finding a common denominator: 12 2 x − 10 + 8 4 x − 20 = 24 2 ( 2 x − 10 ) + 24 3 ( 4 x − 20 ) .
Simplify the sum: 24 16 x − 80 = 3 2 ( x − 5 ) .
The simplified expressions are 3 1 and 3 2 ( x − 5 ) .
Explanation
Dividing Fractions First, we need to simplify the division expression: 12 2 x − 10 ÷ 8 4 x − 20 . To divide fractions, we multiply by the reciprocal of the second fraction.
Multiply by Reciprocal So, 12 2 x − 10 ÷ 8 4 x − 20 = 12 2 x − 10 ⋅ 4 x − 20 8 .
Factoring Next, we factor out constants from the numerators: 12 2 ( x − 5 ) ⋅ 4 ( x − 5 ) 8 .
Canceling Common Factors Now, we cancel common factors: 12 2 ( x − 5 ) ⋅ 4 ( x − 5 ) 8 = 12 2 ⋅ 4 8 ⋅ x − 5 x − 5 . Assuming x = 5 , we can cancel ( x − 5 ) from the numerator and denominator.
Simplifying the Division This simplifies to 12 2 ⋅ 4 8 = 6 1 ⋅ 2 = 3 1 .
Adding Fractions Now, we simplify the addition expression: 12 2 x − 10 + 8 4 x − 20 . To add fractions, we need a common denominator. The least common multiple of 12 and 8 is 24.
Common Denominator We rewrite the fractions with the common denominator: 24 2 ( 2 x − 10 ) + 24 3 ( 4 x − 20 ) .
Combining Fractions Then, we combine the fractions: 24 4 x − 20 + 12 x − 60 .
Simplifying Numerator Simplifying the numerator gives us 24 16 x − 80 .
Factoring We factor out a common factor from the numerator: 24 16 ( x − 5 ) .
Simplifying the Addition Finally, we simplify the fraction: 24 16 ( x − 5 ) = 3 2 ( x − 5 ) .
Final Answer Therefore, the simplified division expression is 3 1 and the simplified addition expression is 3 2 ( x − 5 ) .
Examples
Understanding how to simplify algebraic expressions through division and addition is crucial in many real-world applications. For instance, when calculating the efficiency of a machine, you might need to divide one algebraic expression representing output by another representing input. Similarly, in financial analysis, you might add algebraic expressions representing different revenue streams to find the total revenue. These skills are fundamental in engineering, economics, and computer science, allowing for precise modeling and problem-solving.
