AMC 10 · 2009 · #8

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

percentagefraction-arithmetic work-backwardsidentify-subproblems ↑ Prerequisites: percentage

📏 Medium solution 💡 2 insights

Problem

Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$ % discount as children. The two members of the oldest generation receive a $25\%$ discount as senior citizens. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $$6.00$, is paying for everyone. How many dollars must he pay?

Pick an answer.

(A)

$\ 34$

(B)

$\ 36$

(C)

$\ 42$

(D)

$\ 46$

(E)

$\ 48$

AMC 10 2009 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.

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Toolkit + CCSS Solution

Understand

Restated: Six people go to the movies: two children, two middle-generation adults, and two seniors. Children pay 50% less than the regular price, seniors pay 25% less, and the middle generation pays the full regular price. One senior ticket costs $6.00. Find the total cost of all six tickets.

Givens: There are two people in each of three generations, so six tickets in all.; A child ticket is 50% off the regular price.; A senior ticket is 25% off the regular price.; A middle-generation ticket has no discount.; One senior ticket costs $6.00.

Unknowns: The regular (full) ticket price.; The total amount paid for all six tickets.

Understand

Plan

Primary tool: #11 Work Backwards

Secondary: #7 Identify Subproblems, #8 Analyze the Units

Every price is tied to one hidden number: the regular ticket price. We are handed a discounted price (the senior's $6.00) instead of the regular price, so we work backwards through the 25% discount to recover the regular price first. Once the regular price is known, each generation's ticket is a separate subproblem, and adding them gives the total.

Execute — Answer: B

#11 Work Backwards 6.RP.A.3 Step 1

Recover the regular price

A 25% discount means the senior pays the other 75% of the regular price.
So $6.00 is 75% of the regular price. Undo the discount by dividing by 0.75 to find the regular price is $8.00.

$$0.75 \times R = 6.00 \;\Rightarrow\; R = \frac{6.00}{0.75} = 8.00$$

💡 If $6 is only three-quarters of the price, the whole price must be a bit more, and dividing by 0.75 walks back to it.

#8 Analyze the Units 6.RP.A.3 Step 2

Price a child ticket

A child gets 50% off, so a child pays half of the regular $8.00 price. Half of 8 is 4, so each child ticket costs $4.00.

$$0.50 \times 8.00 = 4.00$$

💡 Fifty percent off just means pay half, and half of 8 is 4.

#8 Analyze the Units 6.RP.A.3 Step 3

Price a middle-generation ticket

The middle generation gets no discount, so each of them pays the full regular price of $8.00.

$$1.00 \times 8.00 = 8.00$$

💡 No discount means the ticket stays at the regular price.

#7 Identify Subproblems 4.OA.A.3 Step 4

Add up all six tickets

There are two of each ticket type.
Two children cost 2 times $4.00 = $8.00, two middle adults cost 2 times $8.00 = $16.00, and two seniors cost 2 times $6.00 = $12.00.
Adding the three groups gives 8 + 16 + 12 = 36.
Grandfather Wen must pay $36, so the answer is (B).

$$2(4) + 2(8) + 2(6) = 8 + 16 + 12 = 36$$

💡 Group the tickets by generation, double each price, then add the three totals.

[1] #11 6.RP.A.3 A 25% discount means the senior pays the other 75% of the regular price. So $6.0

[2] #8 6.RP.A.3 A child gets 50% off, so a child pays half of the regular $8.00 price. Half of 8

[3] #8 6.RP.A.3 The middle generation gets no discount, so each of them pays the full regular pr

[4] #7 4.OA.A.3 There are two of each ticket type. Two children cost 2 times $4.00 = $8.00, two

Review

Reasonableness: Six tickets at roughly a full price of $8 would top out near $48, and every discount pulls the total below that, so a total of $36 sits sensibly under the no-discount ceiling. Choice (E) 48 is the no-discount total and (A) 34 is too low once you notice the middle pair alone already costs $16; $36 lands right where two full, two three-quarter, and two half tickets should.

Alternative: Add the discount fractions instead of the dollars: the six tickets cost R(1 + 1 + 0.75 + 0.75 + 0.5 + 0.5) = 4.5R of the regular price. Since a senior ticket is 0.75R = 6, the regular price R = 8, and 4.5 times 8 = 36.

CCSS standards used (min grade 6)

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Interpreting the percent discounts, recovering the regular price from the senior's 75% price, and computing the discounted child price.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Doubling each generation's price and summing the three groups to get the $36 total.)

⭐ When you are given the discounted price, undo the discount first to find the full price, then everything else follows.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Recover the regular price

Price a child ticket

Price a middle-generation ticket

Add up all six tickets

Review

CCSS standards used (min grade 6)

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