AMC 10 · 2006 · #14

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

Sandwiches use whole numbers of two ingredients at fixed prices, and the total ingredient cost is given. More than one sandwich is made. Find how much one ingredient alone costs.

Pick an answer.

(A)

1.05

(B)

1.25

(C)

1.45

(D)

1.65

(E)

1.85

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

The story becomes one equation whose left side is a product.

N(4B+5J)=253

2STEP 2

Factor 253 to pin down N

Factoring the total leaves only three candidate counts.

253=11 × 23, N∈{11, 23, 253}

3STEP 3

Test each N and drop the dead ends

Two leave no room for positive ingredient counts, so only one 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

Solving and pricing that ingredient gives 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

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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