AMC 10 · 2004 · #1
Easy mode Grade 4Every time Jenny practices, she makes twice as many free throws as she made the time before. At her fifth practice she made 48 free throws. How many did she make at her very first practice?
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Jenny doubles her free throws every practice. At the fifth practice she made $48$. Find how many she made at the first practice.
Givens: At each practice she made twice as many free throws as at the previous practice; At the fifth practice she made $48$ free throws; Answer choices: (A) $3$, (B) $6$, (C) $9$, (D) $12$, (E) $15$
Unknowns: The number of free throws made at the first practice
Understand
Restated: Jenny doubles her free throws every practice. At the fifth practice she made $48$. Find how many she made at the first practice.
Givens: At each practice she made twice as many free throws as at the previous practice; At the fifth practice she made $48$ free throws; Answer choices: (A) $3$, (B) $6$, (C) $9$, (D) $12$, (E) $15$
Plan
Primary tool: #11 Work Backwards
Secondary: #5 Look for a Pattern
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.
Execute — Answer: A
4.OA.A.1 Step 1 See the doubling chain
- Line up the five practices in order.
- Each practice is double the one before, so from the first to the fifth the rule "multiply by $2$" is applied four times.
- The fifth practice is the big end of the chain at $48$; the first practice is the small end we want.
💡 "Twice as many" is a times-$2$ jump, and there are four jumps from the first practice to the fifth.
3.OA.C.7 Step 2 Undo one doubling
- To step backward one practice, undo the doubling by dividing by $2$.
- Going from the fifth practice back to the fourth: $48\div2=24$.
- So at the fourth practice she made $24$ free throws.
💡 Halving reverses doubling, so one step back from $48$ is $24$.
3.OA.C.7 Step 3 Keep halving back to the first
- Repeat the halving to walk back through the earlier practices.
- Fourth to third: $24\div2=12$.
- Third to second: $12\div2=6$.
- Second to first: $6\div2=3$.
- That takes you all the way to the first practice.
💡 Each half-step peels off one doubling until only the first practice is left.
3.OA.C.7 Step 4 Read off and check the answer
- The first practice comes out to $3$ free throws, which is choice (A).
- Check it by going forward: $3\to6\to12\to24\to48$ — four doublings land exactly on $48$ at the fifth practice, so $3$ is correct.
💡 If halving down and doubling back up match the given $48$, the starting value is confirmed.
4.OA.A.1 Line up the five practices in order. Each practice is double the one before, so 3.OA.C.7 To step backward one practice, undo the doubling by dividing by $2$. Going from 3.OA.C.7 Repeat the halving to walk back through the earlier practices. Fourth to third: 3.OA.C.7 The first practice comes out to $3$ free throws, which is choice (A). Check it b Review
Reasonableness: Doubling grows fast, so after four doublings the first-practice number must be much smaller than $48$ — and $3$ is. All the answer choices are small, which is the tell that the first practice is a tiny seed that blows up to $48$. Forward-checking $3\to6\to12\to24\to48$ hits $48$ on the nose at the fifth practice, so nothing was over- or under-halved. A choice like $6$ (B) would land on $96$ by the fifth practice, far too big, so it is ruled out.
Alternative: Introduce a variable. Let the first practice be $x$. Doubling four times multiplies by $2^4=16$, so the fifth practice is $16x$. Set $16x=48$ and divide: $x=48\div16=3$, the same answer (A) in one equation.
CCSS standards used (min grade 4)
4.OA.A.1Interpret a multiplication equation as a comparison (Reading "twice as many" as a times-$2$ relationship and recognizing the four doublings from the first practice to the fifth.)3.OA.C.7Fluently multiply and divide within 100 (Halving $48\div2=24$, $24\div2=12$, $12\div2=6$, $6\div2=3$ to walk backward, and doubling back up to check.)
⭐ When the end of a doubling chain is given and you want the start, walk backward by halving one step at a time.
⭐ When the end of a doubling chain is given and you want the start, walk backward by halving one step at a time.
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