Julia had some beads in a jar. 1/4 of the beads were red and the rest were green. After Julia put another 318 red beads into the jar, the fraction of green beads became 3/7. What was the total number of beads in the jar at first?
Nobody touched the green beads — they just sat there while more red ones went in. That one fact is the whole trick: the GREEN amount is the same number before and after, even though the jar's fraction of green changes.
The green stripe is the same width in real beads both times — it is just a smaller slice of a bigger jar. That lets us use the green beads as the one fixed number to measure everything else against.
Call the number of green beads G. Before, green was 3/4 of the jar, so the
jar held G ÷ 3 × 4 beads. After, green is 3/7 of the (bigger) jar, so it then held
G ÷ 3 × 7 beads. The jar only grew because 318 red beads were poured in — so:
(G ÷ 3 × 7) − (G ÷ 3 × 4) = 318
G ÷ 3 × 3 = 318 → G = 318
318 green beads were 3/4 of the jar at first, so the first total was
318 ÷ 3 × 4 = 424.
were in the jar at first.
The trap: trying to track the red beads, which change. Spotting the quantity that STAYS THE SAME — the green ones — is what makes the two fractions comparable at all.