AMC 10 · 2011 · #3

Grade 7 arithmetic

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

linear-equations-one-varfraction-arithmetic convert-to-algebra ↑ Prerequisites: linear-equations-one-var

📏 Short solution 💡 2 insights

Problem

Two people paid unequal amounts for a shared trip and must end up even. Find the transfer.

Pick an answer.

(A)

$\frac{A+B}{2}$

(B)

$\frac{A-B}{2}$

(C)

$\frac{B-A}{2}$

(D)

B-A

(E)

A+B

How to solve

Strategy Convert to Algebra

The answer choices are all expressions in A and B, so Tool #13 (Convert to Algebra) turns the words into an equation. The fair share is half the total, (A+B)/2. Tool #4 (Introduce a Variable) names the transfer amount x and writes A+x=(A+B)/2, and Tool #6 (Guess and Check) confirms the simplified answer against real numbers like A=30, B=60.

1STEP 1

Write the fair share

The fair share is half the combined total.

total=A+B → fair share=(A+B)/2

2STEP 2

Set up the transfer

The transfer lifts the smaller payer up to it.

A + x = (A+B)/2

3STEP 3

Solve for the transfer

Solving gives half the gap, choice (C).

x=(A+B)/2-A=(A+B-2A)/2=(B-A)/2 → (C)

Answer

(B-A)/2

The transfer should be positive (LeRoy underpaid) and smaller than the whole gap B-A, since handing over the full gap would overshoot and flip who paid more. The answer (B-A)/2 is half the gap, so it lands exactly in the middle — positive and less than B-A. Check with numbers: if A=30 and B=60, the total is 90 and each owes 45; LeRoy gives (60-30)/2=15, making his total 45 and Bernardo's 60-15=45. Equal, as required.

💡Key takeaway

To even up who paid what, the person who paid less hands over half of the gap between the two payments.

Write the fair share
Set up the transfer
Solve for the transfer

Write the fair share

Set up the transfer

Solve for the transfer

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