AMC 8 · 2019 · #1

Grade 5 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
multi-digit-arithmeticfraction-decimal-conversionmental-arithmetic identify-subproblemsguess-and-checkbound-inequality-then-enumerate ↑ Prerequisites: multi-digit-arithmeticfraction-decimal-conversion
📏 Medium solution 💡 2 insights
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Problem
Ike and Mike have 30.00</span>total.Eachsandwichcosts<spanclass="mk−c">30.00</span> total. Each sandwich costs <span class="mk-c">4.50 and each soft drink costs $1.00. They buy as many sandwiches as they can afford, then spend whatever is left on soft drinks. How many items (sandwiches + drinks) do they take home in all?

Pick an answer.

(A)
6
(B)
7
(C)
8
(D)
9
(E)
10

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The question has two natural subproblems (Tool #7): (a) find the largest whole number of sandwiches that fits in the 30budget,and(b)figureouthowmany30 budget, and (b) figure out how many1 drinks the leftover money can buy. Subproblem (a) is a perfect fit for Tool #6 (Guess and Check) on the small candidates 6 vs 7 sandwiches — much friendlier than dividing 30÷30 ÷4.50, which would force decimal division. Tool #6 also lets us verify directly that no other answer choice is reachable.

1STEP 1

Test near the budget: 6 × 4.50=4.50 =27.00 fits, but 7 × 4.50=4.50 =31.50 goes over — so the most is 6 sandwiches.

6 × 4.50=4.50 =27.00 ≤ 30✓7×30 ✓ 7 ×4.50 = 31.50＞31.50 ＞30 ×
2STEP 2

Subtract the sandwich cost from the budget: 30.00−30.00 −27.00 = $3.00 left over.

30.00−30.00 -27.00 = $3.00
3STEP 3

Each drink is 1.00,sotheleftover1.00, so the leftover3.00 buys 3 ÷ 1 = 3 drinks.

3.00÷3.00 ÷1.00 = 3 drinks
4STEP 4

Add the two counts: 6 sandwiches + 3 drinks = 9 items in all → (D).

6 sandwiches + 3 drinks = 9 items → (D)
Answer
9
Check the money: 6 sandwiches at 4.50cost4.50 cost27.00, plus 3 drinks at 1.00cost1.00 cost3.00, for a total of 30.00exactly—everydollarisspent.Thetotalof9itemsischoice(D),whichsitsinthemiddleoftheanswerchoicesandmatchestheeverydayfeelof"afewsandwichesandafewdrinks".Asmallsanitycheck:iftheyhadboughtonemoresandwich,theywouldhaveneeded30.00 exactly — every dollar is spent. The total of 9 items is choice (D), which sits in the middle of the answer choices and matches the everyday feel of "a few sandwiches and a few drinks". A small sanity check: if they had bought one more sandwich, they would have needed31.50 ＞ $30, so 6 is indeed the cap.
💡Key takeaway

This AMC 8 problem only needs Grade 5 decimal multiplication — like figuring out the cost of 6 sandwiches at $4.50 — that you already know!

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