Sure, let's solve the problem using the P5 method, which involves understanding the problem through proportions and logical steps without heavy algebra.

Step-by-Step Solution:

Step 1: Understand the problem.

• Jane and Raju share cookies in the ratio 2:3.
• After some changes, they both have the same number of cookies.

Step 2: Use the ratio to determine initial shares.

Since the total ratio parts are $2 + 3 = 5$:

• Jane has $2$ parts.
• Raju has $3$ parts.

Let the total number of cookies be $5$ parts.

Step 3: Determine the initial number of cookies each had.

Let's assume the total number of cookies is $5$ parts.

• Jane's share: $2$ parts.
• Raju's share: $3$ parts.

Step 4: Account for the changes.

• Jane buys 35 more cookies.
• Raju eats $\frac{1}{10}$ of his cookies.

Step 5: Translate changes into parts and equate their final amounts.

Let's denote the initial number of cookies each had in terms of parts.

• Jane's initial: $2$ parts.
• Raju's initial: $3$ parts.

After the changes:

• Jane's new number of cookies: $2$ parts + 35.
• Raju's new number of cookies: $3$ parts - $\frac{1}{10}$ of $3$ parts.

Since $\frac{1}{10}$ of $3$ parts is $0.3$ parts, Raju's new number of cookies is: $3 \text{ parts} - 0.3 \text{ parts} = 2.7 \text{ parts}$

Given Jane and Raju now have an equal number of cookies, we set their quantities equal: $2 \text{ parts} + 35 = 2.7 \text{ parts}$

Step 6: Solve for the number of cookies in one part.

The difference between $2.7$ parts and $2$ parts is: $2.7 \text{ parts} - 2 \text{ parts} = 0.7 \text{ parts}$

This difference of $0.7$ parts corresponds to the $35$ cookies Jane bought: $0.7 \text{ parts} = 35 \text{ cookies}$

To find the number of cookies in one part, we divide: $1 \text{ part} = \frac{35 \text{ cookies}}{0.7} = 50 \text{ cookies}$

Step 7: Calculate the total number of cookies initially.

Since there were $5$ parts initially: $5 \text{ parts} \times 50 \text{ cookies per part} = 250 \text{ cookies}$

Step 8: Determine how many cookies Raju ate.

Initially, Raju had: $3 \text{ parts} \times 50 \text{ cookies per part} = 150 \text{ cookies}$

Raju ate $\frac{1}{10}$ of his cookies: $\frac{1}{10} \times 150 = 15 \text{ cookies}$

Summary: a) Raju ate 15 cookies. b) The packet initially had 250 cookies.

Would you like more details or have any questions? Here are some other questions you might find interesting:

1. How do you understand ratios without algebra?
2. Can you explain the concept of parts in proportional reasoning?
3. How do you solve problems involving equal distribution changes?
4. What are practical examples of ratios in everyday life?
5. Can you explain how to break down a problem using logical steps?

Tip: When working with ratios and proportional problems, visualizing the problem with parts or shares can make it easier to understand and solve without needing complex algebra.