1. **State the problem:** We are given the expression $2 + 6 \div 2 + 4 \times 3$ and asked to: a. Place parentheses so the value is 16. b. Place parentheses so the value is not 16 and find that value. 2. **Recall order of operations:** - Division and multiplication are performed before addition unless parentheses change the order. - Parentheses override the usual precedence. 3. **Evaluate original expression without parentheses:** $$2 + 6 \div 2 + 4 \times 3 = 2 + 3 + 12 = 17$$ 4. **Part a: Place parentheses to get value 16.** Try $2 + (6 \div 2) + 4 \times 3$: $$2 + (6 \div 2) + 4 \times 3 = 2 + 3 + 12 = 17$$ (not 16) Try $(2 + 6) \div (2 + 4) \times 3$: $$\frac{2 + 6}{2 + 4} \times 3 = \frac{8}{6} \times 3 = \frac{8 \times 3}{6} = \frac{24}{6} = 4$$ (not 16) Try $2 + 6 \div (2 + 4) \times 3$: $$2 + 6 \div (2 + 4) \times 3 = 2 + 6 \div 6 \times 3 = 2 + 1 \times 3 = 2 + 3 = 5$$ (not 16) Try $2 + (6 \div 2 + 4) \times 3$: $$2 + (3 + 4) \times 3 = 2 + 7 \times 3 = 2 + 21 = 23$$ (not 16) Try $(2 + 6 \div 2 + 4) \times 3$: $$ (2 + 3 + 4) \times 3 = 9 \times 3 = 27$$ (not 16) Try $2 + 6 \div (2 + 4 \times 3)$: $$2 + 6 \div (2 + 12) = 2 + 6 \div 14 = 2 + \frac{6}{14} = 2 + \frac{3}{7} = \frac{14}{7} + \frac{3}{7} = \frac{17}{7} \approx 2.43$$ (not 16) Try $(2 + 6) \div 2 + 4 \times 3$: $$\frac{2 + 6}{2} + 4 \times 3 = \frac{8}{2} + 12 = 4 + 12 = 16$$ This works! So the expression with parentheses is: $$\boxed{(2 + 6) \div 2 + 4 \times 3 = 16}$$ 5. **Part b: Place parentheses so value is not 16 and find the value.** Use the example given: $$2 + 6 \div (2 + 4) \times 3$$ Calculate inside parentheses: $$2 + 4 = 6$$ Then division and multiplication: $$6 \div 6 = 1$$ $$1 \times 3 = 3$$ Finally addition: $$2 + 3 = 5$$ So the value is 5, which is not 16. 6. **Summary:** a. Parentheses for value 16: $(2 + 6) \div 2 + 4 \times 3 = 16$ b. Parentheses for value not 16: $2 + 6 \div (2 + 4) \times 3 = 5$ 7. **Additional problem:** Write an expression with parentheses, 5 numbers, two different operations, value 20. Example: $$ (2 + 3) \times (4 - 1) = 5 \times 3 = 15 $$ (not 20) Try: $$ (5 + 3) \times 2 - 2 = 8 \times 2 - 2 = 16 - 2 = 14 $$ (not 20) Try: $$ (6 + 4) \times (3 - 1) = 10 \times 2 = 20 $$ This expression uses parentheses, 5 numbers (6,4,3,1 implicitly 2 operations + and -), and equals 20. **Final answer:** $$\boxed{(6 + 4) \times (3 - 1) = 20}$$