1. **State the problem:** Phillip wants to buy pencils and erasers so that the number of pencils equals the number of erasers, with pencils sold in packages of 10 and erasers in packages of 12. 2. **Identify the goal:** Find the smallest number of packages of pencils and erasers so that the total pencils equal total erasers. 3. **Set variables:** Let $x$ be the number of pencil packages and $y$ be the number of eraser packages. 4. **Write the equation:** Total pencils = $10x$, total erasers = $12y$. We want: $$10x = 12y$$ 5. **Simplify the equation:** Divide both sides by 2: $$\cancel{10}x = \cancel{12}y \Rightarrow 5x = 6y$$ 6. **Find integer solutions:** We want integers $x,y$ such that $5x = 6y$. 7. **Express $x$ in terms of $y$:** $$x = \frac{6}{5}y$$ 8. **Since $x$ must be integer, $y$ must be multiple of 5:** Let $y=5k$ for integer $k$. 9. **Then:** $$x = \frac{6}{5} \times 5k = 6k$$ 10. **Smallest positive integer solution:** $k=1$ gives $x=6$, $y=5$. 11. **Interpretation:** Phillip should buy 6 packages of pencils and 5 packages of erasers. 12. **Check total pencils and erasers:** $$10 \times 6 = 60$$ pencils $$12 \times 5 = 60$$ erasers They are equal, satisfying the condition. **Final answer:** Phillip should buy 6 packages of pencils and 5 packages of erasers.