AMC 10 · 2006 · #22

Grade 6 number-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration
linear-diophantineprime-factorizationfactors casework ↑ Prerequisites: linear-diophantineprime-factorization
📏 Long solution 💡 4 insights
Problem
Elmo makes N sandwiches. Each sandwich uses B globs of peanut butter at 4 cents per glob and J blobs of jam at 5 cents per blob. Making all N sandwiches costs $2.53 in peanut butter and jam. B, J, and N are positive integers with N greater than 1. What is the total cost of the jam Elmo uses?

Pick an answer.

(A)
1.05
(B)
1.25
(C)
1.45
(D)
1.65
(E)
1.85

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The costs are spelled out per glob and per blob, so Tool #4 (Introduce a Variable) turns the story into the single equation N(4B+5J)=253. Tool #8 (Analyze the Units) is what makes that equation clean: switching from dollars to cents turns $2.53 into the whole number 253, so everything is an integer. The winning idea is that 253 barely factors — 253=11 × 23 — so N has almost no choices. Tool #2 (Make a Systematic List) writes down those few divisors, and Tool #3 (Eliminate Possibilities) throws out the ones that leave no whole-number B and J, until only one case survives.

1STEP 1

Turn the story into one equation

Work in cents so everything is a whole number: one sandwich costs 4B+5J, and $2.53 total gives N(4B+5J)=253.

N(4B+5J)=253
2STEP 2

Factor 253 to pin down N

N and 4B+5J are whole numbers, so N divides 253=11×23, whose only divisors are 1, 11, 23, 253 — leaving N=11, 23, or 253.

253=11 × 23, N∈{11, 23, 253}
3STEP 3

Test each N and drop the dead ends

4B+5J must equal 253÷N with B,J at least 1. N=253 needs 1 (under the 9 minimum) and N=23 needs 11 (no fit), so only N=11 survives.

N=253 : 4B+5J=1 (no); N=23 : 4B+5J=11 (no); N=11 : 4B+5J=23
4STEP 4

Solve for B and J, then price the jam

Testing 4B+5J=23: J=3 forces 4B=8, so B=2. Jam costs 11×3×5=165 cents = $1.65, choice (D).

4B+5J=23→ B=2, J=3; jam=N · J · 5=11 · 3 · 5=165 cents=$1.65 → (D)
Answer
1.65
Check the numbers against the story: N=11 sandwiches, each with B=2 globs and J=3 blobs. Per sandwich that is 4 · 2+5 · 3=8+15=23 cents, and 11 × 23=253 cents =$2.53, matching exactly. The jam part is 11 × 3 × 5=165 cents =$1.65, and the peanut butter part is 11 × 2 × 4=88 cents, and 165+88=253 — the two pieces add back to the total, so $1.65, choice (D), holds up.
💡Key takeaway

When a total is a whole number of cents, factor it: 253=11 × 23 leaves almost no choices, so N must be 11 sandwiches — and the jam works out to $1.65.

  • Turn the story into one equation
  • Factor 253 to pin down N
  • Test each N and drop the dead ends
  • Solve for B and J, then price the jam

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