Betty Makes Scarves And Hats. Each Scarf Sells For $ 15 \$15 $15 And Each Hat Sells For $ 25 \$25 $25 . If She Earned $ 180 \$180 $180 From Selling 8 Items, How Many Of Each Kind Of Item Were Sold?Complete The Equations. S = S = S = Scarves

Introduction

Betty is a skilled craftswoman who specializes in creating beautiful scarves and hats. Each of her scarves sells for $\$15$, while her hats are priced at $\$25$. One day, she had a successful sale, earning a total of $\$180$ from selling 8 items. However, she didn't keep track of how many of each item were sold. Can we help Betty figure out how many scarves and hats she sold?

The Problem

Let's denote the number of scarves sold as $s$ and the number of hats sold as $h$. We know that each scarf sells for $\$15$ and each hat sells for $\$25$. If Betty earned $\$180$ from selling 8 items, we can set up the following equation:

$$15s + 25h = 180$$

We also know that the total number of items sold is 8, so we can set up another equation:

$$s + h = 8$$

Solving the Equations

To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method to eliminate one of the variables.

First, we can multiply the second equation by 15 to get:

$$15s + 15h = 120$$

Now, we can subtract this equation from the first equation to eliminate the $s$ term:

$$10h = 60$$

Dividing both sides by 10, we get:

$$h = 6$$

Now that we have found the value of $h$, we can substitute it into the second equation to find the value of $s$:

$$s + 6 = 8$$

Subtracting 6 from both sides, we get:

$$s = 2$$

Conclusion

So, Betty sold 2 scarves and 6 hats. This means that she earned $\$30$ from selling the 2 scarves and $\$150$ from selling the 6 hats, for a total of $\$180$.

Discussion

This problem is a classic example of a system of linear equations. We can use various methods, such as substitution or elimination, to solve these equations. In this case, we used the elimination method to eliminate one of the variables and solve for the other.

Tips and Variations

• To make this problem more challenging, we can add more variables or equations. For example, we can introduce a third item, such as a scarf-hat combination, that sells for $\$20$.
• We can also change the prices of the scarves and hats. For example, if the price of a scarf increases to $\$20$ and the price of a hat increases to $\$30$, we need to update the equations accordingly.
• To make this problem more realistic, we can add constraints, such as the availability of materials or the time it takes to produce each item.

Practice Problems

1. A bakery sells two types of bread: whole wheat and white bread. The whole wheat bread sells for $\$2$ per loaf, while the white bread sells for $\$3$ per loaf. If the bakery earns $\$120$ from selling 12 loaves, how many of each type of bread were sold?
2. A clothing store sells two types of shirts: short-sleeve and long-sleeve. The short-sleeve shirts sell for $\$15$ each, while the long-sleeve shirts sell for $\$25$ each. If the store earns $\$180$ from selling 8 shirts, how many of each type of shirt were sold?

Solutions

1. Let $s$ be the number of whole wheat loaves sold and $w$ be the number of white loaves sold. We can set up the following equations:

   $$2s + 3w = 120$$

   $$s + w = 12$$

   Solving these equations, we get $s = 6$ and $w = 6$.

2. Let $s$ be the number of short-sleeve shirts sold and $l$ be the number of long-sleeve shirts sold. We can set up the following equations:

   $$15s + 25l = 180$$

   $$s + l = 8$$

   Solving these equations, we get $s = 2$ and $l = 6$.
   Betty's Scarf and Hat Sales Mystery: Q&A =============================================

Q: What is the main problem that Betty is facing?

A: Betty is facing a problem where she earned $\$180$ from selling 8 items, but she doesn't know how many of each item (scarves and hats) were sold.

Q: What are the prices of the scarves and hats?

A: The price of each scarf is $\$15$, and the price of each hat is $\$25$.

Q: What are the two equations that we need to solve?

A: The two equations are:

$$15s + 25h = 180$$

$$s + h = 8$$

where $s$ is the number of scarves sold and $h$ is the number of hats sold.

Q: How did we solve the equations?

A: We used the elimination method to eliminate one of the variables. We multiplied the second equation by 15 to get:

$$15s + 15h = 120$$

Then, we subtracted this equation from the first equation to eliminate the $s$ term:

$$10h = 60$$

Dividing both sides by 10, we got:

$$h = 6$$

Q: What is the value of $s$?

A: Now that we have found the value of $h$, we can substitute it into the second equation to find the value of $s$:

$$s + 6 = 8$$

Subtracting 6 from both sides, we get:

$$s = 2$$

Q: How many scarves and hats were sold?

A: Betty sold 2 scarves and 6 hats.

Q: What is the total amount of money earned from selling the scarves and hats?

A: The total amount of money earned from selling the scarves is $\$30$ (2 scarves * $\$15$ each), and the total amount of money earned from selling the hats is $\$150$ (6 hats * $\$25$ each). The total amount of money earned is $\$180$.

Q: What are some tips and variations for this problem?

A: To make this problem more challenging, we can add more variables or equations. For example, we can introduce a third item, such as a scarf-hat combination, that sells for $\$20$. We can also change the prices of the scarves and hats. For example, if the price of a scarf increases to $\$20$ and the price of a hat increases to $\$30$, we need to update the equations accordingly.

Q: What are some practice problems related to this topic?

A: Here are two practice problems:

1. A bakery sells two types of bread: whole wheat and white bread. The whole wheat bread sells for $\$2$ per loaf, while the white bread sells for $\$3$ per loaf. If the bakery earns $\$120$ from selling 12 loaves, how many of each type of bread were sold?
2. A clothing store sells two types of shirts: short-sleeve and long-sleeve. The short-sleeve shirts sell for $\$15$ each, while the long-sleeve shirts sell for $\$25$ each. If the store earns $\$180$ from selling 8 shirts, how many of each type of shirt were sold?

Q: How do we solve these practice problems?

A: We can use the same method of substitution or elimination to solve these problems. We need to set up the equations based on the given information and then solve for the variables.

Q: What are some real-world applications of this topic?

A: This topic has many real-world applications, such as:

• Inventory management: Businesses need to keep track of their inventory levels to ensure that they have enough stock to meet customer demand.
• Pricing strategy: Companies need to determine the optimal prices for their products to maximize profits.
• Supply chain management: Businesses need to manage their supply chains to ensure that they have a steady supply of raw materials and components.

Q: What are some common mistakes to avoid when solving these types of problems?

A: Some common mistakes to avoid when solving these types of problems include:

• Not reading the problem carefully and understanding the given information.
• Not setting up the equations correctly.
• Not using the correct method to solve the equations (e.g., substitution or elimination).
• Not checking the solutions to ensure that they are reasonable and make sense in the context of the problem.