The Next 3 of AMC 12: Explained

Problem Source: AMC 12

Problem Level 3~4

Problem 4:

A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately? 

Problem 5:

The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers?

Problem 6:

Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

Solutions:

Problem 4:

A simple way to solve this would be to list the possibilities out:

At the first window, the price would be $100. Starting with the fifth, the price would be $400.  In total; both people will buy 15 windows. Because 5 windows cost $400, 15 windows cost $1200.  Dave needs (7) windows, so that would equal $600. Doug needs (8), and thus using the same equation, we would get $700.

600+700=1300

The difference of 1300 and 1200 is 100; which is also our answer.

Problem 5:

First we add the numbers together; 20 x 30 + 30 x 20.  Then we divide this by the total; 50.  The answer would be 24.

Problem 6:
Hint: d =rt  can be useful.
The answer is eight.