A merchant mixed 10 lb of a cinnamon tea with 2 lb of spice tea. The 12-pound mixture cost $31. A second mixture included 16 lb of the cinnamon tea and 4 lb of the spice tea. The 20-pound mixture cost $52. Find the cost per pound of the cinnamon tea and of the spice tea.


cinnamon    

 dollars per pound

spice    

 dollars per pound

Let's define the variables:

- Let $ x $ be the cost per pound of the cinnamon tea.
- Let $ y $ be the cost per pound of the spice tea.

We are given two mixtures with their total weights and costs. From the first mixture:

\[
10x + 2y = 31
\]
This equation represents the cost of 10 pounds of cinnamon tea and 2 pounds of spice tea totaling $31.

From the second mixture:

\[
16x + 4y = 52
\]
This equation represents the cost of 16 pounds of cinnamon tea and 4 pounds of spice tea totaling $52.

### Step 1: Simplify the second equation
We can simplify the second equation by dividing everything by 2:

\[
8x + 2y = 26
\]

### Step 2: Subtract the first equation from the simplified second equation
We now subtract the first equation from the simplified second equation:

\[
(8x + 2y) - (10x + 2y) = 26 - 31
\]

This simplifies to:

\[
-2x = -5
\]

Thus:

\[
x = \frac{-5}{-2} = 2.5
\]

So, the cost per pound of the cinnamon tea is **$2.50**.

### Step 3: Substitute $ x = 2.5 $ into the first equation
Substitute $ x = 2.5 $ into the first equation:

\[
10(2.5) + 2y = 31
\]

This simplifies to:

\[
25 + 2y = 31
\]

Now solve for $ y $:

\[
2y = 31 - 25
\]

\[
2y = 6
\]

\[
y = \frac{6}{2} = 3
\]

So, the cost per pound of the spice tea is **$3.00**.

### Final Answer:
- Cost per pound of cinnamon tea: **$2.50**
- Cost per pound of spice tea: **$3.00**

Would you like more details or have any questions? 

Here are 5 related questions:
1. How would the solution change if the cost of the second mixture was $60?
2. What is the proportion of cinnamon to spice tea in each mixture?
3. How can we verify that the solution satisfies both equations?
4. How would the equations change if there was a third type of tea?
5. What if the merchant mixed 8 lb of cinnamon and 4 lb of spice tea for $28?

**Tip:** When solving systems of linear equations, simplifying one equation often helps isolate variables and makes the process easier.

A merchant mixed 10 lb of a cinnamon tea with 2 lb of spice tea. The 12-pound mixture cost $31. A second mixture included 16 lb of the cinnamon tea and 4 lb of the spice tea. The 20-pound mixture cost $52. Find the cost per pound of the cinnamon tea and of the spice tea.

Learn how to solve a system of linear equations to find the cost per pound of cinnamon and spice tea. The problem involves two mixtures of tea with different costs and quantities. The detailed solution breaks down each step using algebraic techniques, making it ideal for students in Grades 8-10.

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