1. **State the problem:** We have a piggy bank with a total amount $T$. The problem states that one quarter of the total amount plus 5 equals one third of the total amount. 2. **Write the equation:** $$\frac{1}{4}T + 5 = \frac{1}{3}T$$ 3. **Isolate the variable $T$:** Subtract $\frac{1}{4}T$ from both sides: $$\frac{1}{4}T + 5 - \frac{1}{4}T = \frac{1}{3}T - \frac{1}{4}T$$ which simplifies to: $$5 = \frac{1}{3}T - \frac{1}{4}T$$ 4. **Find common denominator and simplify the right side:** $$5 = \frac{4}{12}T - \frac{3}{12}T = \frac{1}{12}T$$ 5. **Solve for $T$ by multiplying both sides by 12:** $$12 \times 5 = 12 \times \frac{1}{12}T$$ $$60 = \cancel{12} \times 5 = \cancel{12} \times \frac{1}{12}T = T$$ 6. **Final answer:** The total amount in the piggy bank is $\boxed{60}$.