1. Problem: Misha and Masha eat together at a restaurant. The total cost including tax is 132000. The tax is 18000, and they split the food cost equally. Find how much each pays. Step 1: Calculate the food cost by subtracting tax from total cost. $$\text{Food cost} = 132000 - 18000 = 114000$$ Step 2: Since they split equally, each pays half of the food cost plus half of the tax. $$\text{Each pays} = \frac{114000}{2} + \frac{18000}{2} = 57000 + 9000 = 66000$$ Answer: Each pays 66000. 2. Problem: Two cars move towards each other. Car 1 speed = 60 km/h, Car 2 speed = 64 km/h, distance = 384 km. Find the time until they meet. Step 1: Relative speed when moving towards each other is sum of speeds. $$v_{rel} = 60 + 64 = 124 \text{ km/h}$$ Step 2: Time to meet is distance divided by relative speed. $$t = \frac{384}{124} = 3.0968 \text{ hours}$$ Answer: They meet after approximately 3.1 hours. 3. Problem: A person wants 50 million interest per year. He has 200 million in bank 1 with 9% interest. Find how much to deposit in bank 2 with 8% interest to get total 50 million interest. Step 1: Let amount in bank 2 be $x$. Step 2: Interest from bank 1: $$200000000 \times 0.09 = 18000000$$ Step 3: Interest from bank 2: $$x \times 0.08$$ Step 4: Total interest is 50 million: $$18000000 + 0.08x = 50000000$$ Step 5: Solve for $x$: $$0.08x = 50000000 - 18000000 = 32000000$$ $$x = \frac{32000000}{0.08} = 400000000$$ Answer: He must deposit 400 million in bank 2. 4. Problem: A climber descends 1 km/h faster than ascending. Time descending = 3 hours, ascending = 4.5 hours. Find speeds. Step 1: Let ascending speed be $v$ km/h, descending speed $v+1$ km/h. Step 2: Distance is same both ways: $$v \times 4.5 = (v+1) \times 3$$ Step 3: Expand: $$4.5v = 3v + 3$$ Step 4: Solve for $v$: $$4.5v - 3v = 3$$ $$1.5v = 3$$ $$v = 2 \text{ km/h}$$ Step 5: Descending speed: $$v + 1 = 3 \text{ km/h}$$ Answer: Ascending speed is 2 km/h, descending speed is 3 km/h. 5a. Problem: Price per juice bottle is 3500. Sarah has 120000. Let $x$ be number of bottles. Mathematical model: $$3500x \leq 120000$$ 5b. Find maximum bottles Sarah can buy. Step 1: Solve inequality: $$x \leq \frac{120000}{3500} = 34.2857$$ Step 2: Since $x$ must be whole number, maximum is 34. Answer: Sarah can buy at most 34 bottles.