1) For the second semester, we recognize that the number of actual students (75) is 15 times the ratio number (5) used to represent it. We also recognize that the total number of ratio units (5+4) for the second semester is the same as the total number (2+7) for the first semester.
If each ratio unit reprsents 15 students, in the first semester there were
... 2×15 = 30 students in art
... 7×15 = 105 students in gym
2) We can use the same sort of logic for the second problem as for the first. 6 ratio units represents $936 before money is moved, so each represents $156.
After the move, the account balances are
... savings: 4×$156 = $624
... checking: 3×$156 = $468
_____
Since the numbers of "ratio units" add to the same value in each case for both problems, we can solve this the way we have done—by determining the value a "ratio unit" stands for. If the problem had been posed differently, we might have been required to find the total number of students (dollars), then reapportion them according to the new ratios.
Essentially, when you have
... part1 : part2
you can find the total from
... part1 : total = part1 : (part1 + part2)
or
... (part1 + part2)/(part 1) × (actual value of part 1) = (actual value of total)
