To solve the problem using model diagrams, we'll follow these steps:

1. Define the Variables:

   • Let $A$ represent the total number of apple pies baked.
   • Let $B$ represent the total number of banana pies baked.

2. Total Number of Pies:

   • We know the total number of pies baked is 440: $A + B = 440$

3. Sold and Unsold Pies:

   • $\frac{3}{4}$ of the apple pies were sold. Therefore, $\frac{1}{4}$ of the apple pies were unsold: $\text{Unsold Apple Pies} = \frac{1}{4}A$
   • $\frac{2}{5}$ of the banana pies were sold. Therefore, $\frac{3}{5}$ of the banana pies were unsold: $\text{Unsold Banana Pies} = \frac{3}{5}B$

4. Relationship Between Unsold Pies:

   • The number of unsold banana pies is twice the number of unsold apple pies: $\frac{3}{5}B = 2 \left(\frac{1}{4}A\right)$ Simplify this equation: $\frac{3}{5}B = \frac{1}{2}A$ Multiply both sides by 10 to clear the fractions: $6B = 5A$

5. Solve the System of Equations:

   • We now have two equations: $A + B = 440$ $6B = 5A$
   • Solve the second equation for $B$: $B = \frac{5}{6}A$
   • Substitute $B$ in the first equation: $A + \frac{5}{6}A = 440$ Combine like terms: $\frac{11}{6}A = 440$ Multiply both sides by $\frac{6}{11}$: $A = 240$
   • Substitute $A$ back into $B = \frac{5}{6}A$: $B = \frac{5}{6} \times 240 = 200$

6. Conclusion:

   • The number of apple pies baked at first is $240$.
   • The number of banana pies baked at first is $200$.

Therefore, more apple pies were baked initially, and the difference is: $240 - 200 = 40$

Model Diagram Representation:

Total Pies Baked

$\boxed{440 \text{ pies}}$

Apple Pies

$\boxed{240 \text{ apple pies}}$

• Sold: $\frac{3}{4} \times 240 = 180$
• Unsold: $\frac{1}{4} \times 240 = 60$

Banana Pies

$\boxed{200 \text{ banana pies}}$

• Sold: $\frac{2}{5} \times 200 = 80$
• Unsold: $\frac{3}{5} \times 200 = 120$

The number of unsold banana pies (120) is twice the number of unsold apple pies (60), which confirms our solution.

Apple pies were baked more at first, and there were 40 more apple pies baked than banana pies.

Would you like more details or have any questions?

Relative Questions:

1. What if the total number of pies baked was different? How would that change the solution?
2. How would the problem change if the fractions of sold pies were different?
3. What if the unsold ratio was different, such as three times instead of twice?
4. How would you solve it if you only knew the difference in the number of pies baked?
5. What if the number of pies sold was given in actual numbers instead of fractions?

Tip:

When solving word problems involving fractions and ratios, clearly define your variables and equations step-by-step to avoid confusion and ensure accuracy.