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Let's solve the problem numbered 56 in the provided image:

### Problem 56:
There are five numbers and the second number is 25% more than the first number, while the third number is 20% more than the second number. If the average of the five numbers is 126 and that of the first, fourth and fifth numbers is 130, then find the sum of the last two numbers.

### Solution:

1. Let the first number be \( x \).
2. The second number is \( x + 0.25x = 1.25x \).
3. The third number is \( 1.25x + 0.20 \times 1.25x = 1.5x \).

Let's denote the fourth number by \( y \) and the fifth number by \( z \).

Given:
- The average of the five numbers is 126.
- The average of the first, fourth, and fifth numbers is 130.

From the given information, we can write the following equations:
\[
\frac{x + 1.25x + 1.5x + y + z}{5} = 126
\]
\[
\frac{x + y + z}{3} = 130
\]

Let's solve the first equation:
\[
\frac{3.75x + y + z}{5} = 126
\]
\[
3.75x + y + z = 630 \quad \text{(Equation 1)}
\]

Now, solve the second equation:
\[
\frac{x + y + z}{3} = 130
\]
\[
x + y + z = 390 \quad \text{(Equation 2)}
\]

Subtract Equation 2 from Equation 1:
\[
3.75x + y + z - (x + y + z) = 630 - 390
\]
\[
2.75x = 240
\]
\[
x = \frac{240}{2.75} = \frac{240 \times 4}{11} = \frac{960}{11} \approx 87.27
\]

Substitute \( x \approx 87.27 \) back into Equation 2:
\[
87.27 + y + z = 390
\]
\[
y + z = 390 - 87.27 = 302.73
\]

Thus, the sum of the last two numbers (fourth and fifth numbers) is approximately \( 302.73 \).

If you have any further questions or need detailed explanations, please let me know!

### Additional Questions
1. What is the exact value of the sum of the last two numbers if the first number is a whole number?
2. How do you convert percentage increases to decimal form for calculations?
3. Can you generalize the approach to solving similar problems with different percentages?
4. What are the implications of rounding in this problem?
5. How would you check if the average values given in the problem are consistent with the numbers found?
6. How can you derive the formula for percentage increase in successive steps?
7. Can you solve the problem using a different method or approach?
8. What if the percentages for increases were reversed? How would the solution change?

### Tip:
Always cross-check your results by plugging back the values into the original equations to ensure accuracy.

Explore a detailed solution to a mathematical problem involving percentage increases and averages. Learn how to calculate the sum of numbers based on given conditions. Suitable for advanced high school students.

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