Question

Evaluate each expression. Check your answers with a calculator. a. \(20(100-2)\) b. \(20 \cdot 100-2\) c. \(20 \cdot 100-20 \cdot 2\) d. \(100-20 \cdot 2\) e. Which two of the preceding arithmetic expressions have the same answer? f. State the property that produces the same answer for that pair of expressions.

Short answer

Short answer: Expressions a and c both have the same answer of 1960. This is due to the Distributive Property, which allows us to distribute the 20 across the addition or subtraction inside the parentheses, resulting in the same final answer.

Step-by-step solution

a. Evaluate the expression \(20(100-2)\)

First, we need to compute the value inside the parentheses, that is, subtract 2 from 100. Then we multiply the result by 20. 1. Compute \(100 - 2\). The result is 98. 2. Multiply 20 by 98. The result is \(20 \times 98 = 1960\). So, \(20(100 - 2) = 1960\).

b. Evaluate the expression \(20 \cdot 100-2\)

1. Multiply 20 by 100. The result is \(20 \cdot 100 = 2000\). 2. Subtract 2 from 2000. The result is \(2000 - 2 = 1998\). So, \(20 \cdot 100 - 2 = 1998\).

c. Evaluate the expression \(20 \cdot 100-20 \cdot 2\)

1. Multiply 20 by 100. The result is \(20 \cdot 100 = 2000\). 2. Multiply 20 by 2. The result is \(20 \cdot 2 = 40\). 3. Subtract 40 from 2000. The result is \(2000 - 40 = 1960\). So, \(20 \cdot 100 - 20 \cdot 2 = 1960\).

d. Evaluate the expression \(100 - 20 \cdot 2\)

1. Multiply 20 by 2. The result is \(20 \cdot 2 = 40\). 2. Subtract 40 from 100. The result is \(100 - 40 = 60\). So, \(100 - 20 \cdot 2 = 60\).

e. Determine which two expressions from a-d have the same answer

Compare the answers obtained for each expression: - \(20(100 - 2) = 1960\) - \(20 \cdot 100 - 2 = 1998\) - \(20 \cdot 100 - 20 \cdot 2 = 1960\) - \(100 - 20 \cdot 2 = 60\) Expressions a and c have the same answer: 1960.

f. State the property that produces the same answer for the pair of expressions with the same answer

The property that produces the same answer for expressions a and c is the Distributive Property because both expressions distribute the 20 evenly. In other words, when multiplying 20 by a sum or difference (inside the parentheses), it is the same as multiplying 20 by each term and then adding/subtracting the results.