The 7th and 8th grades have enrollments of 520 and 650 respectively. The two grades have a total of 18 representatives on the Student Council. How many representatives should the 8th grade have if there is to be a fair representation of the two grades?
The problem asks for a fair representation of the two grades, which means that the ratio of representatives to students should be the same for both grades. Let x be the number of representatives for the 8th grade, and 18 - x be the number of representatives for the 7th grade. Then the ratio of representatives to students for the 8th grade is x/650, and the ratio for the 7th grade is (18 - x)/520. To make these ratios equal, I can set up a proportion and cross-multiply: x/650 = (18 - x)/520. Multiplying both sides by 650 and 520, I get 520x = 650(18 - x). Expanding the parentheses and simplifying, I get 520x = 11700 - 650x. Adding 650x to both sides, I get 1170x = 11700. Dividing both sides by 1170, I get x = 10. This means that the 8th grade should have 10 representatives, and the 7th grade should have 18 - 10 = 8 representatives. 