AMC 10 · 2011 · #4

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

LeRoy and Bernardo split a trip's costs equally. LeRoy paid A dollars and Bernardo paid B dollars, with A ＜ B. Find how many dollars LeRoy must hand to Bernardo so each ends up having paid the same amount.

Pick an answer.

(A)

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

(B)

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

(C)

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

(D)

B - A

(E)

A + B

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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

Together they spent A+B dollars, so an equal split makes each person's fair share (A+B)/2 dollars.

total=A+B → fair share=(A+B)/2

2STEP 2

Set up the transfer

LeRoy paid the smaller amount A, so let x be what he hands over; then A+x=(A+B)/2.

A + x = (A+B)/2

3STEP 3

Solve for the transfer

Subtract A: x=(A+B)/2-A=(A+B-2A)/2=(B-A)/2, positive since A ＜ B — 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

More like this