We can solve this problem using a system of linear equations. Let the cost of one party hat be $x$ and the cost of one balloon be $y$.

From the problem:

1. The cost of 5 party hats and 3 balloons is 35p: $5x + 3y = 35$
2. The cost of 3 party hats and 2 balloons is 22p: $3x + 2y = 22$

We are tasked with finding the cost of 8 party hats and 5 balloons: $8x + 5y = ?$

Solving the system of equations:

Step 1: Eliminate $y$

To eliminate $y$, multiply the first equation by 2 and the second equation by 3: $2(5x + 3y) = 2(35) \implies 10x + 6y = 70$ $3(3x + 2y) = 3(22) \implies 9x + 6y = 66$

Subtract the second equation from the first: $(10x + 6y) - (9x + 6y) = 70 - 66$ $x = 4$

Step 2: Solve for $y$

Substitute $x = 4$ into the second equation: $3(4) + 2y = 22$ $12 + 2y = 22$ $2y = 10 \implies y = 5$

Step 3: Find $8x + 5y$

Substitute $x = 4$ and $y = 5$: $8x + 5y = 8(4) + 5(5)$ $8x + 5y = 32 + 25 = 57$

Final Answer:

The cost of 8 party hats and 5 balloons is 57p.

Would you like further clarification or additional steps explained? Here are some related questions for practice:

1. What is the cost of 10 party hats and 6 balloons?
2. What is the cost of 2 party hats and 1 balloon?
3. If the total budget is 100p, how many party hats and balloons can you buy?
4. What is the cost difference between 6 balloons and 6 party hats?
5. How would the equations change if the cost of 4 party hats and 1 balloon was 18p?

Tip: Always double-check your substitutions in a system of equations to avoid calculation errors!