Let's solve the problem step by step.

Step 1: Establish the initial ratio and the number of cookies each had.

Let the total number of cookies in the packet be $x$.

Jane and Raju shared these cookies in the ratio 2:3. Therefore, the number of cookies Jane had is $\frac{2}{5}x$ and the number of cookies Raju had is $\frac{3}{5}x$.

Step 2: Account for the changes in the number of cookies.

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

So, the number of cookies Jane has now: $\frac{2}{5}x + 35$

The number of cookies Raju has after eating $\frac{1}{10}$ of his cookies: $\frac{3}{5}x - \frac{1}{10} \left( \frac{3}{5}x \right) = \frac{3}{5}x - \frac{3}{50}x = \frac{30}{50}x - \frac{3}{50}x = \frac{27}{50}x$

Step 3: Set up the equation given they have equal number of cookies now.

$\frac{2}{5}x + 35 = \frac{27}{50}x$

Step 4: Solve for $x$.

First, make the fractions have a common denominator: $\frac{20}{50}x + 35 = \frac{27}{50}x$

Subtract $\frac{20}{50}x$ from both sides: $35 = \frac{27}{50}x - \frac{20}{50}x$ $35 = \frac{7}{50}x$

Multiply both sides by 50 to solve for $x$: $35 \times 50 = 7x$ $1750 = 7x$ $x = 250$

So, the total number of cookies in the packet initially is 250.

Step 5: Calculate the number of cookies Raju ate.

Initially, Raju had: $\frac{3}{5}x = \frac{3}{5} \times 250 = 150$

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

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 simplify algebraic fractions?
2. What are some methods to solve ratio problems?
3. Can you explain more about proportional reasoning?
4. How do you solve equations involving fractions?
5. What are common mistakes to avoid in ratio problems?

Tip: Always double-check your calculations and ensure each step logically follows from the previous one to avoid errors in complex problems.