Grouping symbols, such as parentheses or brackets, can change the value of a numerical expression by altering the order of operations. In arithmetic, the order of operations dictates that we perform calculations in a specific sequence: parentheses first, then exponents (if any), followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). By adding grouping symbols, we can change which operations are performed together, thus changing the overall result of the expression.
Let's insert different grouping symbols into the expression \( 80 ÷ 8 \times 5 + 42 \) to show four different values.
-
First configuration: \[ (80 ÷ 8) × 5 + 42 \] Calculating this step-by-step: \[ 80 ÷ 8 = 10 \] \[ 10 × 5 = 50 \] \[ 50 + 42 = 92 \] Value: 92
-
Second configuration: \[ 80 ÷ (8 × 5) + 42 \] Calculating this step-by-step: \[ 8 × 5 = 40 \] \[ 80 ÷ 40 = 2 \] \[ 2 + 42 = 44 \] Value: 44
-
Third configuration: \[ 80 ÷ 8 × (5 + 42) \] Calculating this step-by-step: \[ 5 + 42 = 47 \] \[ 80 ÷ 8 = 10 \] \[ 10 × 47 = 470 \] Value: 470
-
Fourth configuration: \[ (80 ÷ 8 × 5) + 42 \] This is similar to the first configuration but changed in interpretation. Calculating this: \[ 80 ÷ 8 = 10 \] \[ 10 × 5 = 50 \] \[ 50 + 42 = 92 \] Value: 92 (This is a repetition, demonstrating alignment with the first grouping.)
-
Fifth configuration for variation: \[ 80 ÷ (8 × (5 + 42)) \] Calculating this step-by-step: \[ 5 + 42 = 47 \] \[ 8 × 47 = 376 \] \[ 80 ÷ 376 \approx 0.212766 (or \frac{80}{376} \text{ as a fraction}) \] Value: Approximately 0.21 (this is notably different from previous values due to the specific grouping used).
Thus, through the use of grouping symbols, we can generate different values from the same expression.