Use the distributive property to calculate each value mentally. $123 \(\frac{5}{8}\) \(\cdot\) 1 \(\frac{1}{2}\) - 23 \(\frac{5}{8}\) \(\cdot\) 1 \(\frac{1}{2}\)$

Verified step by step guidance

1. First, convert the mixed numbers into improper fractions for easier multiplication. For example, convert \(123 \frac{5}{8}\) and \(1 \frac{1}{2}\) into improper fractions.
2. Express the problem as \(\left(123 \frac{5}{8}\right) \times \left(1 \frac{1}{2}\right) - \left(23 \frac{5}{8}\right) \times \left(1 \frac{1}{2}\right)\) using the improper fractions.
3. Notice that both terms share a common factor of \(1 \frac{1}{2}\), so apply the distributive property: factor out \(1 \frac{1}{2}\) from both terms.
4. Rewrite the expression as \(\left(123 \frac{5}{8} - 23 \frac{5}{8}\right) \times 1 \frac{1}{2}\), simplifying inside the parentheses first.
5. Finally, subtract the two mixed numbers inside the parentheses and then multiply the result by \(1 \frac{1}{2}\) to find the answer.

Key Concepts

Distributive Property

The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. It is expressed as a(b + c) = ab + ac. This property helps simplify expressions and perform mental calculations efficiently.

Mixed Numbers and Improper Fractions

Mixed numbers combine whole numbers and fractions, such as 1 1/2. Converting mixed numbers to improper fractions (e.g., 1 1/2 = 3/2) simplifies multiplication and subtraction. Understanding this conversion is essential for accurate calculations.

Mental Math Strategies

Mental math involves performing calculations in your head without paper. Using properties like distributive property and converting mixed numbers helps break down complex problems into simpler parts, enabling quicker and more accurate mental computation.