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## Problem

Suppose there are three individuals A, B, and C each with an unknown number of coins.

A says: “If I take 2/3 of B's coins and 1/3 of C's coins then I hold 100.”

B says: “If I take 2/3 of A's coins and 1/2 of C's coins then I hold 100.”

C says: “If I take 2/3 of A's coins and 2/3 of B's coins, then I hold 100.”

Question: How many coins do A, B, and C each hold?

## Solution

Construct the matrix

Zhang Qiujian constructed the equivalent matrix.

Use the MATLAB/OCTAVE command rref(A) to row reduce the matrix. Click here for an online Octave shell. For the full solution all written out, use this site: Linear Algebra Toolkit.

A has 60 coins.

B has 45 coins.

C has 30 coins.

## Step-by-Step Solution

Step 1: Subtract row 1 by row 3

Step 2: Subtract row 2 by two times row 3

Step 3: Subtract row 3 by two times row 1

Step 4: Subtract row 3 by row 2

Step 5: Divide row 3 by ten

Step 6: Add two times row 3 to row 1

Step 7: Add three times row 3 to row 1

Step 8: Divide row 2 by two