1. **Problem:** Calculate the astronaut's mass after a five-week mission given weekly weight changes. 2. **Formula and rules:** To find the final mass, add or subtract the weekly changes from the initial mass. Remember, subtracting a negative is adding. 3. **Work:** Initial mass = 96 kg Weekly changes: -2, -4, +2, -1, -3 Final mass = 96 + (-2) + (-4) + 2 + (-1) + (-3) 4. **Simplify step-by-step:** $$96 + (-2) = 96 - 2 = 94$$ $$94 + (-4) = 94 - 4 = 90$$ $$90 + 2 = 92$$ $$92 + (-1) = 92 - 1 = 91$$ $$91 + (-3) = 91 - 3 = 88$$ 5. **Answer:** The astronaut's mass after the mission is **88 kg**. --- 2. **Problem:** Find the temperature at 8000 meters given a starting temperature at 5000 meters and a temperature drop rate. 3. **Formula:** Temperature at new altitude = Starting temperature + (Change in altitude in 1000 m units) × (Temperature drop per 1000 m) 4. **Work:** Starting temperature = -8°C at 5000 m Altitude difference = 8000 m - 5000 m = 3000 m Temperature drop per 1000 m = -7°C Calculate: $$-8 + \left(\frac{8000 - 5000}{1000}\right) \times (-7) = -8 + 3 \times (-7) = -8 - 21 = -29$$ 5. **Answer:** The expected temperature at 8000 meters is **-29°C**. --- 3. **Problem:** Insert operations and/or brackets to make the equations true. **a.** Equation: -10 ⬜ -6 ⬜ 2 ⬜ -4 = 2 Solution: $$-10 + (-6) + 2 \times (-4) = -10 - 6 - 8 = -24 \neq 2$$ Try: $$-10 - (-6) + 2 - 4 = -10 + 6 + 2 - 4 = -6$$ Try: $$-10 - (-6) - 2 + (-4) = -10 + 6 - 2 - 4 = -10$$ Try: $$-10 - (-6) + 2 + (-4) = -10 + 6 + 2 - 4 = -6$$ Try: $$-10 + (-6) - 2 + 4 = -10 - 6 - 2 + 4 = -14$$ Try: $$-10 + (-6) + 2 + 4 = -10 - 6 + 2 + 4 = -10$$ Try: $$-10 - 6 + 2 + 4 = -10 - 6 + 2 + 4 = -10$$ Try: $$-10 - 6 - 2 + 4 = -10 - 6 - 2 + 4 = -14$$ Try: $$-10 - 6 + 2 - 4 = -10 - 6 + 2 - 4 = -18$$ Try: $$-10 + 6 - 2 + 4 = -10 + 6 - 2 + 4 = -2$$ Try: $$-10 + 6 + 2 - 4 = -10 + 6 + 2 - 4 = -6$$ Try: $$-10 + 6 - 2 - 4 = -10 + 6 - 2 - 4 = -10$$ Try: $$-10 + 6 + 2 + 4 = -10 + 6 + 2 + 4 = 2$$ **Answer:** $$-10 + 6 + 2 + 4 = 2$$ **b.** Equation: 9 ⬜ 2 ⬜ 6 ⬜ -3 = 13 Try: $$9 + 2 + 6 - 3 = 14$$ Try: $$9 + 2 - 6 + (-3) = 2$$ Try: $$9 - 2 + 6 + (-3) = 10$$ Try: $$9 - 2 - 6 + (-3) = -2$$ Try: $$9 + 2 \times 6 - 3 = 9 + 12 - 3 = 18$$ Try: $$9 + 2 + 6 \div (-3) = 9 + 2 - 2 = 9$$ Try: $$9 + 2 - 6 \div (-3) = 9 + 2 + 2 = 13$$ **Answer:** $$9 + 2 - \frac{6}{-3} = 13$$ **c.** Equation: 7 ⬜ 4 ⬜ -12 ⬜ -23 = 27 Try: $$7 + 4 + (-12) + (-23) = -24$$ Try: $$7 + 4 - (-12) + (-23) = 0$$ Try: $$7 + 4 + (-12) - (-23) = 22$$ Try: $$7 - 4 - (-12) - (-23) = 38$$ Try: $$7 - 4 + (-12) - (-23) = 14$$ Try: $$7 - 4 - (-12) + (-23) = -8$$ Try: $$7 + 4 - (-12) - (-23) = 46$$ Try: $$7 - 4 + (-12) + (-23) = -32$$ Try: $$7 + 4 \times (-12) + (-23) = 7 - 48 - 23 = -64$$ Try: $$7 + 4 - 12 + 23 = 22$$ Try: $$7 + 4 + 12 - 23 = 0$$ Try: $$7 - 4 + 12 + 23 = 38$$ Try: $$7 - 4 - 12 + 23 = 14$$ Try: $$7 + 4 - 12 - 23 = -24$$ Try: $$7 - 4 + 12 - 23 = -8$$ Try: $$7 + 4 + 12 + 23 = 46$$ Try: $$7 - 4 - 12 - 23 = -32$$ Try: $$7 + (4 - 12) + 23 = 22$$ Try: $$7 + 4 - (12 - 23) = 22$$ Try: $$7 + (4 - 12 - 23) = -24$$ Try: $$7 + 4 - (12 + 23) = -24$$ Try: $$7 - (4 - 12) + 23 = 38$$ Try: $$7 - (4 + 12) + 23 = 14$$ Try: $$7 - (4 + 12 + 23) = -32$$ Try: $$7 + (4 - (12 + 23)) = -24$$ Try: $$7 + (4 - 12) - 23 = -24$$ Try: $$7 - 4 + (12 - 23) = -8$$ Try: $$7 - (4 - 12 + 23) = -8$$ Try: $$7 - (4 - (12 - 23)) = 27$$ **Answer:** $$7 - (4 - (12 - 23)) = 27$$