A fraction, then a fraction of the rest
A jar holds 24 marbles. of them are red, and of the rest are blue. How many marbles are blue? Two fractions act here, one after the other, and each one has its own whole.
Start with the red marbles. Their whole is all 24 marbles, so cut the 24 into 3 equal groups: 24 ÷ 3 = 8. One group is red, so 8 marbles are red. That leaves 24 − 8 = 16 marbles that are not red.
The 24 marbles in 3 equal groups of 8. The colored group is the 8 red marbles, and the other 2 groups hold the 16 that are left.
The next fraction has a new whole
The words "of the rest" change the whole. The blue marbles are of the 16 marbles that are left, not of the 24. So cut the 16 into 4 equal groups: 16 ÷ 4 = 4. 4 marbles are blue.
Take the blue marbles out too, and 16 − 4 = 12 marbles are left.
Now the whole is the 16 marbles that are left, in 4 equal groups of 4. The colored group is the 4 blue marbles.
The whole shrinks at every step
Follow the whole through the steps: 24 marbles at the start, 16 after the red ones, and 12 after the blue ones. Each fraction acts on the number that the step before it left behind.
12 is half of 24, so at the end half of the marbles are neither red nor blue.
One piece is one marble. The top row is the 24 at the start, with the 8 red marbles colored. The middle row is the 16 that are left, with the 4 blue marbles colored. The bottom row is the 12 left at the end.
A fraction of what?
The usual mistake is to take both fractions of the start. A quarter of 24 is 6, so that way gives 8 red and 6 blue, which leaves 24 − 8 − 6 = 10. But the blue marbles are a quarter of the rest, and a quarter of 16 is 4, so 12 marbles are left, not 10.
Before working out each fraction, ask: a fraction of what? When a question says "of the rest" or "of the remainder", the whole is what is left after the step before, not the amount at the start.
Money spent in two steps
Sam has 40 dollars. He spends of his money on a book and of the rest on lunch. How much money does he have left?
The book costs of 40 dollars, which is 40 ÷ 4 = 10 dollars. That leaves 40 − 10 = 30 dollars. Lunch costs of the rest, so of 30. One third of 30 is 10, so two thirds is 2 × 10 = 20 dollars. Sam has 30 − 20 = 10 dollars left.
On a bar, cut the 40 dollars into 4 equal parts. The book takes 1 part, and the rest is 3 parts. Two thirds of those 3 parts is 2 parts for lunch, and the last part is what Sam has left.
Each part is 10 dollars: 1 part for the book, 2 parts for lunch, and 1 part left.