Simon purchased 1 bat, 3 balls and 4 rackets at a cost of $276.00. Leo purchased 1 bat, 5
balls and 6 rackets at a cost of $382.00. Find the cost of 3 balls and 2 rackets if a ball
costs $5.00 more than the racket.

Let's denote the cost of a bat as $ B $, the cost of a ball as $ b $, and the cost of a racket as $ r $.

From the given problem, we have the following information:

1. Simon purchased 1 bat, 3 balls, and 4 rackets for $276:
   \[
   B + 3b + 4r = 276
   \]

2. Leo purchased 1 bat, 5 balls, and 6 rackets for $382:
   \[
   B + 5b + 6r = 382
   \]

3. A ball costs $5 more than a racket:
   \[
   b = r + 5
   \]

We have three equations:

1. $ B + 3b + 4r = 276 $
2. $ B + 5b + 6r = 382 $
3. $ b = r + 5 $

Let's substitute $ b $ in terms of $ r $ from the third equation into the first two equations:

First equation:
\[
B + 3(r + 5) + 4r = 276
\]
\[
B + 3r + 15 + 4r = 276
\]
\[
B + 7r + 15 = 276
\]
\[
B + 7r = 261 \quad \text{(Equation 4)}
\]

Second equation:
\[
B + 5(r + 5) + 6r = 382
\]
\[
B + 5r + 25 + 6r = 382
\]
\[
B + 11r + 25 = 382
\]
\[
B + 11r = 357 \quad \text{(Equation 5)}
\]

Now we have:
1. $ B + 7r = 261 $
2. $ B + 11r = 357 $

Let's solve these equations simultaneously by subtracting the first equation from the second:

\[
(B + 11r) - (B + 7r) = 357 - 261
\]
\[
4r = 96
\]
\[
r = 24
\]

Substitute $ r = 24 $ back into Equation 4:
\[
B + 7(24) = 261
\]
\[
B + 168 = 261
\]
\[
B = 93
\]

Now substitute $ r = 24 $ into the third equation to find $ b $:
\[
b = r + 5
\]
\[
b = 24 + 5
\]
\[
b = 29
\]

We have found the costs:
- Cost of a racket ($ r $) is $24.
- Cost of a ball ($ b $) is $29.
- Cost of a bat ($ B $) is $93.

Now, we need to find the cost of 3 balls and 2 rackets:
\[
3b + 2r = 3(29) + 2(24)
\]
\[
= 87 + 48
\]
\[
= 135
\]

Thus, the cost of 3 balls and 2 rackets is $135.

Would you like further details or have any questions?

Here are 8 related questions:
1. How do you derive the system of equations from word problems?
2. What methods can be used to solve simultaneous equations?
3. How can substitution help in solving equations?
4. What is the importance of checking your solution in context?
5. Can you solve a similar problem with different costs for items?
6. How do you identify and define variables in algebraic problems?
7. What is the process to verify the consistency of given conditions in a problem?
8. How can you apply this method to other real-life budgeting problems?

**Tip:** Always recheck your substitution steps to ensure accuracy in solving algebraic equations.

Learn how to solve a math problem involving the cost of balls, rackets, and bats. This problem uses linear equations and the substitution method to find costs. Suitable for students in Grades 7-9.

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