AMC 10 · 2014 · #6
Grade 6 arithmeticOrvin went to the store with just enough money to buy 30 balloons. When he arrived, he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at 31 off the regular price. What is the greatest number of balloons Orvin could buy?
Pick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Orvin brings exactly enough money to buy $30$ balloons at the regular price. A sale lets him buy one balloon at the regular price and a second balloon at $\frac{1}{3}$ off. Using all his money and the deal as often as possible, find the greatest number of balloons he can buy.
Givens: His money equals the cost of $30$ balloons at the regular price; Deal: pay full price for one balloon, then get a second balloon at $\frac{1}{3}$ off; He wants the greatest number of balloons for the money he has; Answer choices: (A) $33$, (B) $34$, (C) $36$, (D) $38$, (E) $39$
Unknowns: The greatest number of balloons Orvin can buy under the sale
Understand
Restated: Orvin brings exactly enough money to buy $30$ balloons at the regular price. A sale lets him buy one balloon at the regular price and a second balloon at $\frac{1}{3}$ off. Using all his money and the deal as often as possible, find the greatest number of balloons he can buy.
Givens: His money equals the cost of $30$ balloons at the regular price; Deal: pay full price for one balloon, then get a second balloon at $\frac{1}{3}$ off; He wants the greatest number of balloons for the money he has; Answer choices: (A) $33$, (B) $34$, (C) $36$, (D) $38$, (E) $39$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #8 Analyze the Units, #7 Identify Subproblems
No dollar amount is given, so Tool #4 (Introduce a Variable) fixes a convenient regular price; because the count of balloons cannot depend on the price, any handy value works, and $3 makes the $\frac{1}{3}$ discount a whole number. Tool #8 (Analyze the Units) tracks dollars per balloon and dollars per pair. Tool #7 (Identify Subproblems) turns the sale into one repeatable unit — a pair of balloons at a fixed cost — so the whole purchase is just 'how many pairs fit in the money.'
Execute — Answer: C
6.RP.A.3 Step 1 Pick a convenient price
- The regular price is never given, and the number of balloons cannot depend on what it is, so pick a value that makes the discount tidy.
- Let the regular price be $3$ dollars per balloon.
- Then Orvin's money, which buys exactly $30$ balloons at full price, is $30 \times 3 = 90$ dollars.
💡 When only ratios matter, you are free to choose the actual price; a clean number just makes the arithmetic easy.
5.NF.B.6 Step 2 Price the discounted balloon
- In each pair the second balloon is $\frac{1}{3}$ off.
- One third of $3$ dollars is $1$ dollar, so the discount is $1$ dollar and the second balloon costs $3 - 1 = 2$ dollars.
💡 Taking a fraction off means paying the rest: $\frac{1}{3}$ off leaves $\frac{2}{3}$ of the price.
4.OA.A.3 Step 3 Cost of one pair
- Buy the balloons two at a time so the deal is used on every second balloon.
- Each pair is one full-price balloon plus one discounted balloon: $3 + 2 = 5$ dollars for $2$ balloons.
💡 The sale repeats in blocks of two, so the natural unit to count is the pair, not the single balloon.
4.OA.A.3 Step 4 Fit the pairs into the money
- Each pair costs $5$ dollars, and Orvin has $90$ dollars.
- The number of whole pairs he can afford is $90 \div 5 = 18$ pairs, using up all $90$ dollars with nothing left over.
💡 Once each pair has a fixed price, the whole budget question becomes a single division.
4.OA.A.3 Step 5 Count the balloons
- Each of the $18$ pairs holds $2$ balloons, so Orvin gets $18 \times 2 = 36$ balloons.
- His money is spent exactly, so he cannot buy even one more.
- The greatest number of balloons is $36$, which is choice (C).
💡 The saved money — one third off every second balloon — is exactly what buys the extra six balloons beyond the original thirty.
6.RP.A.3 The regular price is never given, and the number of balloons cannot depend on wh 5.NF.B.6 In each pair the second balloon is $\frac{1}{3}$ off. One third of $3$ dollars i 4.OA.A.3 Buy the balloons two at a time so the deal is used on every second balloon. Each 4.OA.A.3 Each pair costs $5$ dollars, and Orvin has $90$ dollars. The number of whole pai 4.OA.A.3 Each of the $18$ pairs holds $2$ balloons, so Orvin gets $18 \times 2 = 36$ ball Review
Reasonableness: Without the sale, $90$ dollars buys $30$ balloons, so the answer must be more than $30$; $36$ is a believable bump. Check the saving directly: each pair costs $5$ instead of $6$, a $\frac{1}{6}$ saving, so the money stretches to $30 \times \frac{6}{5} = 36$ balloons. Both routes give $36$, and the money is used up exactly, matching choice (C).
Alternative: Keep the price as a symbol $p$ and watch it cancel. Money is $30p$; a pair costs $p + \frac{2}{3}p = \frac{5}{3}p$. Number of pairs is $30p \div \frac{5}{3}p = 30 \times \frac{3}{5} = 18$, and the $p$ disappears, confirming the count is independent of the price. Then $18 \times 2 = 36$ balloons, choice (C).
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Recognizing the balloon count is independent of the price, so a convenient regular price of $3 can be chosen, and treating the sale as a fixed cost per pair.)5.NF.B.6Solve real-world problems involving multiplication of fractions and mixed numbers (Computing $\frac{1}{3}$ off a $3 balloon: $\frac{1}{3}$ of $3$ is $1$, so the discounted balloon costs $2$.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Adding the pair cost $3+2=5$, dividing the budget $90 \div 5 = 18$ pairs, and multiplying $18 \times 2 = 36$ balloons.)
⭐ Buy balloons two at a time: every second one is a third off, so five dollars' worth of deal gets you six dollars' worth of balloons, stretching money for 30 into 36.
⭐ Buy balloons two at a time: every second one is a third off, so five dollars' worth of deal gets you six dollars' worth of balloons, stretching money for 30 into 36.
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