AMC 10 · 2004 · #1
Grade 4 arithmeticPick an answer.
The problem hands you the END of the chain (the fifth practice, 48) and asks for the START (the first practice). Tool #11 (Work Backwards) fits exactly: instead of building up, undo the rule one step at a time. Forward the rule is "multiply by 2," so backward it is "divide by 2." Tool #5 (Look for a Pattern) names the structure — a doubling chain — so you know there are exactly four backward steps from the fifth practice to the first.
See the doubling chain
From the first to the fifth practice the doubling is applied four times.
"Twice as many" is a times-2 jump, and there are four jumps from the first practice to the fifth.
Doubling four times over is one repeated multiplication, so undoing it is one repeated division.
▸ Why?
Twice as many is one equal group repeated twice, so each practice multiplies the count by the same fixed factor.
▸ Why?
Halving undoes doubling exactly, so stepping backwards from a known later value recovers the earlier one.
Undo one doubling
Undoing one doubling means halving: the fourth practice is 24.
Halving reverses doubling, so one step back from 48 is 24.
3.OA.C.7Work BackwardsKeep halving back to the first
Halving on back gives 12, then 6, then 3.
Each half-step peels off one doubling until only the first practice is left.
3.OA.C.7Work BackwardsRead off and check the answer
Walking forward lands exactly on 48, so the answer is 3, choice (A).
If halving down and doubling back up match the given 48, the starting value is confirmed.
3.OA.C.7Work BackwardsWhen the end of a doubling chain is given and you want the start, walk backward by halving one step at a time.
- See the doubling chain
- Undo one doubling
- Keep halving back to the first
- Read off and check the answer