PROLOGUE: Principles of Problem Solving 1. Let r be the rate of the descent. We use the formula time  distance rate ; the ascent takes 1 15 h, the descent takes 1 r h, and the total trip should take 2 30  1 15 h. Thus we have 1 15  1 r  1 15  1 r  0, which is impossible. So the car cannot go fast enough to average 30 mi/h for the 2-mile trip. 2. Let us start with a given price P. After a discount of 40%, the price decreases to 06P. After a discount of 20%, the price decreases to 08P, and after another 20% discount, it becomes 08 08P  064P. Since 06P  064P, a 40% discount is better. 3. We continue the pattern. Three parallel cuts produce 10 pieces. Thus, each new cut produces an additional 3 pieces. Since the first cut produces 4 pieces, we get the formula f n  4  3 n  1, n  1. Since f 142  4  3 141  427, we see that 142 parallel cuts produce 427 pieces. 4. By placing two amoebas into the vessel, we skip the first simple division which took 3 minutes. Thus when we place two amoebas into the vessel, it will take 60  3  57 minutes for the vessel to be full of amoebas. 5. The statement is false. Here is one particular counterexample: Player A Player B First half 1 hit in 99 at-bats: average  1 99 0 hit in 1 at-bat: average  0 1 Second half 1 hit in 1 at-bat: average  1 1 98 hits in 99 at-bats: average  98 99 Entire season 2 hits in 100 at-bats: average  2 100 99 hits in 100 at-bats: average  99 100 6. Method 1: After the exchanges, the volume of liquid in the pitcher and in the cup is the same as it was to begin with. Thus, any coffee in the pitcher of cream must be replacing an equal amount of cream that has ended up in the coffee cup. Method 2: Alternatively, look at the drawing of the spoonful of coffee and cream mixture being returned to the pitcher of cream. Suppose it is possible to separate the cream and the coffee, as shown. Then you can see that the coffee going into the cream occupies the same volume as the cream that was left in the coffee. coffee cream Method 3 (an algebraic approach): Suppose the cup of coffee has y spoonfuls of coffee. When one spoonful of cream is added to the coffee cup, the resulting mixture has the following ratios: cream mixture  1 y  1 and coffee mixture  y y  1 . So, when we remove a spoonful of the mixture and put it into the pitcher of cream, we are really removing 1 y  1 of a spoonful of cream and y y  1 spoonful of coffee. Thus the amount of cream left in the mixture (cream in the coffee) is