Math / Algebra

Questionovement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. A store is selling two mixtures of coffee beans in one-pound bags. The first mixture has 12 ounces of Sumatra combined with ounces of Celebes Kalossi, and costs $\$ 15$ . The second mixture has 4 ounces of Sumatra and 12 ounces of Celebes Kalossi, and costs $\$ 21$ . How much does one ounce of Sumatra and one ounce of Celebes Kalossi cost? $\$ 1.65$ for Sumatra and $\$ 1.20$ for Celebes Kalossi There is no solution. $\$ 0.75$ for Sumatra and $\$ 1.50$ for Celebes Kalossi $\$ 1.50$ for Sumatra and $\$ 0.75$ for Celebes Kalossi

Studdy Solution

STEP 1

What is this asking? We need to find the price per ounce of each type of coffee bean, given the prices of two different mixtures. Watch out! The mixtures are in one-pound bags, which means 16 ounces, not 12!
Don't get tricked!
Also, make sure to keep track of which coffee is which.
Sumatra and Celebes Kalossi... exotic!

STEP 2

1. Setup the Equations
2. Solve for Sumatra
3. Solve for Celebes Kalossi

STEP 3

Let's define $\text{S}$ as the cost per ounce of Sumatra and $\text{C}$ as the cost per ounce of Celebes Kalossi.
We're dealing with two mixtures, so we'll need two equations!
Exciting!

STEP 4

The first mixture has 12 ounces of Sumatra and 4 ounces of Celebes Kalossi (because $16 - 12 = 4$ ), and costs $\$15$ .
This gives us our first equation: $12\text{S} + 4\text{C} = 15$

STEP 5

The second mixture has 4 ounces of Sumatra and 12 ounces of Celebes Kalossi (because $16 - 4 = 12$ ), and costs $\$21$ .
That's equation number two: $4\text{S} + 12\text{C} = 21$

STEP 6

Let's multiply the second equation by $-3$ so that the $\text{S}$ terms will add to zero when we combine it with the first equation.
This is a strategic move! $-3 \cdot (4\text{S} + 12\text{C}) = -3 \cdot 21$ $-12\text{S} - 36\text{C} = -63$

STEP 7

Now, let's add this modified equation to our first equation: $(12\text{S} + 4\text{C}) + (-12\text{S} - 36\text{C}) = 15 + (-63)$ $-32\text{C} = -48$

STEP 8

Divide both sides by $-32$ to find the cost of Celebes Kalossi: $\text{C} = \frac{-48}{-32} = \frac{48}{32} = \frac{3}{2} = 1.5$ So, Celebes Kalossi costs $\$1.50$ per ounce.
Almost there!

STEP 9

We already solved for Celebes Kalossi in the previous step!
Now, we need to solve for Sumatra.
Let's substitute the value of $\text{C}$ (\$1.50) back into our first equation: \[12\text{S} + 4(1.5) = 15\] \[12\text{S} + 6 = 15\]

STEP 10

Subtract $6$ from both sides: $12\text{S} = 15 - 6$ $12\text{S} = 9$

STEP 11

Finally, divide both sides by $12$ to find the cost of Sumatra: $\text{S} = \frac{9}{12} = \frac{3}{4} = 0.75$ Sumatra costs $\$0.75$ per ounce!
We did it!

STEP 12

One ounce of Sumatra costs $\$0.75$ and one ounce of Celebes Kalossi costs $\$1.50$ .

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Studdy Solution

STEP 1

STEP 2

1. Setup the Equations
2. Solve for Sumatra
3. Solve for Celebes Kalossi

STEP 3

Let's define $\text{S}$ as the cost per ounce of Sumatra and $\text{C}$ as the cost per ounce of Celebes Kalossi.
We're dealing with two mixtures, so we'll need two equations!
Exciting!

STEP 4

The first mixture has 12 ounces of Sumatra and 4 ounces of Celebes Kalossi (because $16 - 12 = 4$ ), and costs $\$15$ .
This gives us our first equation: $12\text{S} + 4\text{C} = 15$

STEP 5

The second mixture has 4 ounces of Sumatra and 12 ounces of Celebes Kalossi (because $16 - 4 = 12$ ), and costs $\$21$ .
That's equation number two: $4\text{S} + 12\text{C} = 21$

STEP 6

Let's multiply the second equation by $-3$ so that the $\text{S}$ terms will add to zero when we combine it with the first equation.
This is a strategic move! $-3 \cdot (4\text{S} + 12\text{C}) = -3 \cdot 21$ $-12\text{S} - 36\text{C} = -63$

STEP 7

Now, let's add this modified equation to our first equation: $(12\text{S} + 4\text{C}) + (-12\text{S} - 36\text{C}) = 15 + (-63)$ $-32\text{C} = -48$

STEP 8

Divide both sides by $-32$ to find the cost of Celebes Kalossi: $\text{C} = \frac{-48}{-32} = \frac{48}{32} = \frac{3}{2} = 1.5$ So, Celebes Kalossi costs $\$1.50$ per ounce.
Almost there!

STEP 9

We already solved for Celebes Kalossi in the previous step!
Now, we need to solve for Sumatra.
Let's substitute the value of $\text{C}$ (\$1.50) back into our first equation: \[12\text{S} + 4(1.5) = 15\] \[12\text{S} + 6 = 15\]

STEP 10

Subtract $6$ from both sides: $12\text{S} = 15 - 6$ $12\text{S} = 9$

STEP 11

Finally, divide both sides by $12$ to find the cost of Sumatra: $\text{S} = \frac{9}{12} = \frac{3}{4} = 0.75$ Sumatra costs $\$0.75$ per ounce!
We did it!

STEP 12

One ounce of Sumatra costs $\$0.75$ and one ounce of Celebes Kalossi costs $\$1.50$ .

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Studdy Solution

STEP 1

STEP 2

1. Setup the Equations2. Solve for Sumatra3. Solve for Celebes Kalossi

STEP 3

Let's **define** S\text{S}S as the cost per ounce of Sumatra and C\text{C}C as the cost per ounce of Celebes Kalossi.We're dealing with two mixtures, so we'll need two equations!Exciting!

STEP 4

The first mixture has **12** ounces of Sumatra and **4** ounces of Celebes Kalossi (because 16−12=416 - 12 = 416−12=4), and costs $15\$15$15.This gives us our first equation: 12S+4C=1512\text{S} + 4\text{C} = 1512S+4C=15

STEP 5

The second mixture has **4** ounces of Sumatra and **12** ounces of Celebes Kalossi (because 16−4=1216 - 4 = 1216−4=12), and costs $21\$21$21.That's equation number two: 4S+12C=214\text{S} + 12\text{C} = 214S+12C=21

STEP 6

STEP 7

Now, let's **add** this modified equation to our first equation: (12S+4C)+(−12S−36C)=15+(−63)(12\text{S} + 4\text{C}) + (-12\text{S} - 36\text{C}) = 15 + (-63)(12S+4C)+(−12S−36C)=15+(−63) −32C=−48-32\text{C} = -48−32C=−48

STEP 8

**Divide** both sides by −32-32−32 to find the **cost** of Celebes Kalossi: C=−48−32=4832=32=1.5\text{C} = \frac{-48}{-32} = \frac{48}{32} = \frac{3}{2} = 1.5C=−32−48​=3248​=23​=1.5 So, Celebes Kalossi costs $1.50\$1.50$1.50 per ounce.Almost there!

STEP 9

We already solved for Celebes Kalossi in the previous step!Now, we need to solve for Sumatra.Let's **substitute** the value of C\text{C}C (\$1.50) back into our first equation: \[12\text{S} + 4(1.5) = 15\] \[12\text{S} + 6 = 15\]

STEP 10

**Subtract** 666 from both sides: 12S=15−612\text{S} = 15 - 612S=15−6 12S=912\text{S} = 912S=9

STEP 11

Finally, **divide** both sides by 121212 to find the **cost** of Sumatra: S=912=34=0.75\text{S} = \frac{9}{12} = \frac{3}{4} = 0.75S=129​=43​=0.75 Sumatra costs $0.75\$0.75$0.75 per ounce!We did it!

STEP 12

One ounce of Sumatra costs $0.75\$0.75$0.75 and one ounce of Celebes Kalossi costs $1.50\$1.50$1.50.

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1. Setup the Equations
2. Solve for Sumatra
3. Solve for Celebes Kalossi

Let's define $\text{S}$ as the cost per ounce of Sumatra and $\text{C}$ as the cost per ounce of Celebes Kalossi.
We're dealing with two mixtures, so we'll need two equations!
Exciting!

The first mixture has 12 ounces of Sumatra and 4 ounces of Celebes Kalossi (because $16 - 12 = 4$ ), and costs $\$15$ .
This gives us our first equation: $12\text{S} + 4\text{C} = 15$

The second mixture has 4 ounces of Sumatra and 12 ounces of Celebes Kalossi (because $16 - 4 = 12$ ), and costs $\$21$ .
That's equation number two: $4\text{S} + 12\text{C} = 21$

Now, let's add this modified equation to our first equation: $(12\text{S} + 4\text{C}) + (-12\text{S} - 36\text{C}) = 15 + (-63)$ $-32\text{C} = -48$

Divide both sides by $-32$ to find the cost of Celebes Kalossi: $\text{C} = \frac{-48}{-32} = \frac{48}{32} = \frac{3}{2} = 1.5$ So, Celebes Kalossi costs $\$1.50$ per ounce.
Almost there!

We already solved for Celebes Kalossi in the previous step!
Now, we need to solve for Sumatra.
Let's substitute the value of $\text{C}$ (\$1.50) back into our first equation: \[12\text{S} + 4(1.5) = 15\] \[12\text{S} + 6 = 15\]

Subtract $6$ from both sides: $12\text{S} = 15 - 6$ $12\text{S} = 9$

Finally, divide both sides by $12$ to find the cost of Sumatra: $\text{S} = \frac{9}{12} = \frac{3}{4} = 0.75$ Sumatra costs $\$0.75$ per ounce!
We did it!

One ounce of Sumatra costs $\$0.75$ and one ounce of Celebes Kalossi costs $\$1.50$ .