Jennifer went to the post office for stamps. She bought the same number of 8-cent stamps and 10-cent stamps. She also bought as many 2-cent stamps as both of the other two kinds combined. How many of each kind did she get if she paid a total of $4.40 for them all?
The correct answer and explanation is:
Let’s break down the problem step by step to find out how many stamps Jennifer bought of each kind.
Step 1: Define the variables
Let xx be the number of 8-cent stamps she bought.
Then she also bought x 10-cent stamps, since it says the same number.
She bought as many 2-cent stamps as the total of the 8-cent and 10-cent stamps, which means she bought 2x2x 2-cent stamps.
Step 2: Total cost equation
Now we calculate the cost of each type of stamp:
- 8-cent stamps: 8x8x cents
- 10-cent stamps: 10x10x cents
- 2-cent stamps: 2⋅2x=4×2 \cdot 2x = 4x cents
Add them up for total cost in cents: 8x+10x+4x=22x cents8x + 10x + 4x = 22x \text{ cents}
We are told the total cost is $4.40, which is 440 cents.
So we set up the equation: 22x=44022x = 440
Step 3: Solve for xx
x=44022=20x = \frac{440}{22} = 20
Step 4: Final Answer
- 8-cent stamps: x=20x = 20
- 10-cent stamps: x=20x = 20
- 2-cent stamps: 2x=402x = 40
Explanation (300 words)
This problem involves simple algebra and a good understanding of how to translate words into mathematical expressions. Jennifer bought three types of stamps with different costs and in quantities related to each other. By carefully assigning a variable to the number of one type of stamp (8-cent), we were able to express the other two types in terms of that same variable. The 10-cent stamps were bought in the same quantity as the 8-cent stamps, so both were represented by the variable xx. The 2-cent stamps were bought in an amount equal to the total of the other two kinds, meaning she got x+x=2xx + x = 2x of those.
Next, we calculated the cost of each type of stamp. Since costs were in cents, we kept all calculations in cents to match the $4.40 total, which equals 440 cents. Adding all the costs together gave us a simple equation, 22x=44022x = 440, which we solved by dividing both sides by 22. The result, x=20x = 20, tells us she bought 20 of each of the 8- and 10-cent stamps, and 2x=402x = 40 of the 2-cent stamps.
This type of problem is a classic example of using variables and equations to model a real-life situation. It teaches the importance of organizing information and being careful with units (like cents vs. dollars).