https://file.aiquickdraw.com/mathbot/file/17267432888475093etcj.jpg

From the image, we can see the following problem:

### Problem statement:
- **Each plate** costs **$5.90**.
- **Each mug** costs **$3.65**.
- Kate bought **67 more mugs** than plates.
- She spent **$80.30 more** on the mugs than the plates.

### Questions:
1. **(a)** What was the difference between the cost of a plate and a mug?
2. **(b)** How much did she spend on the mugs in total?

---

#### (a) What was the difference between the cost of a plate and a mug?
The solution begins by finding the difference between the cost of a plate and a mug.

\[
\text{Difference} = 5.90 - 3.65 = 2.25
\]

So, the difference between the cost of a plate and a mug is **$2.25**.

#### (b) How much did she spend on the mugs in total?
We are not provided with the exact number of plates or mugs, but we know that:
- She bought 67 more mugs than plates.
- The total spending on mugs exceeds that on plates by $80.30.

Let the number of plates be denoted as $ x $. Then the number of mugs is $ x + 67 $.

- Cost of plates = $ 5.90x $
- Cost of mugs = $ 3.65(x + 67) $

According to the problem, the difference between the cost of the mugs and plates is $80.30:
\[
3.65(x + 67) - 5.90x = 80.30
\]
Let's solve for $ x $ to determine the number of plates and mugs.

I'll calculate this step now.The number of plates Kate bought is **73**.

Now, we can calculate the total cost of the mugs. Since she bought **67 more mugs** than plates, the number of mugs is:

\[
x + 67 = 73 + 67 = 140
\]

The total cost for the mugs is:

\[
\text{Cost of mugs} = 140 \times 3.65
\]

I'll calculate the total cost of the mugs now.The total amount Kate spent on the mugs is **$511.00**.

### Summary:
- (a) The difference between the cost of a plate and a mug is **$2.25**.
- (b) The total amount Kate spent on the mugs is **$511.00**.

Would you like further explanation or details on any part of this problem? 

Here are 5 related questions you could explore:
1. How much did Kate spend on the plates?
2. What is the total amount Kate spent on both the mugs and the plates?
3. How many plates would Kate need to buy for the cost of mugs to exceed the cost of plates by $100?
4. What would be the total cost if the price of a plate increased by 10%?
5. If she bought 5 more plates, how would that affect her total cost?

**Tip:** For problems involving two related quantities, setting up an equation with variables is crucial for finding the solution efficiently!

Kate bought some plates and mugs for an event. Each plate costs $5.90, and each mug costs $3.65. She bought 67 more mugs than plates and spent $80.30 more on the mugs than on the plates.

(a) What was the difference between the cost of a plate and a mug?
(b) How much did she spend on the mugs in total?

Solve an algebra problem involving the cost comparison of plates and mugs. Learn how to set up equations to find differences in cost and calculate total spending. Suitable for middle school students in Grades 6-8.

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