how many kilograms of beans should she buy?

Question

Susan is making a special blend of coffee. She mixes ground Columbian beans worth $11.25 per kilogram with ground Kona beans, worth $20.00 per kilogram. Overall, the blend costs $16.50 per kilogram. If Susan has 2 kilograms of Columbian beans already on hand, how many kilograms of the Kona beans should she buy to make her special blend?

Answer 1

Word problems are always the hardest.  Let's identify what we know, and then what we don't.

 

Cost of blend:  $16.50 per kg (kilogram), or $16.50/kg.

Cost of Columbian beans:  $11.25/kg

Cost of Kona beans:  $20.00/kg

 

Amount of Columbian beans:  2 kg

 

Even though we know the amount of Columbian beans, we can start with

x = amount of Columbian beans (in kg)

y = amount of Kona beans (in kg)

 

The total cost for a blend of the two beans is

(kg of Columbian beans) times ($11.25/kg) + (kg of Kona beans) times ($20.00/kg)

= 11.25x + 20y.

 

The total kilograms = x + y.

 

Therefore, the cost per kilogram of the blend is the ratio of the total cost to the total kilograms:

 

Cost per kg = (11.25x + 20y) / ( x + y)

 

We know the cost per kg of the blend (16.50), and we also know how many kg of Columbian beans there are (x=2).  Substituting these values into the above equation yields

 

16.50 = [11.25(2) + 20y] / (2 + y)

 

Solve for y, and you have your answer.

 

 

Answer 2

Hi Madison,

 

This problem requires you to set-up two equations with two variables.  Let's call Columbian beans "c" and Kona beans "k".  I've listed the first equation below:

 

$11.25c + $20.00k = $16.50

 

This means that an unknown amount of Columbian beans ($11.25/kg) and an unknown amount of Kona beans ($20.00/kg) will get us a final price of $16.50/kg.

 

Because "c" and "k" represent percentages, our second equation must show that when added together, they equal 100%:

 

c + k = 1

 

By solving for either "c" or "k" in the second equation, I can then substitute this value into our first equations:

 

c = 1 - k

 

$11.25(1 - k) + $20.00k = $16.50

 

$11.25 - $11.25k + $20.00k = $16.50

 

$11.25 + $8.75k = $16.50

 

$8.75k = $5.25

 

k = 0.60 = 60%

 

This means that the Kona blend accounts for 60% of the beans and that the Columbian beans are 40%.

 

To determine how much of the Kona blend she needs to buy, we take our 2kg of Columbian beans and enter it into the following equation, where "B" represents the total amount of beans we need to buy:

 

2kg ÷ B = 0.4

 

B = 2kg ÷ 0.4 = 5kg of beans

 

Since we need 5kg of total coffee beans, that means that we need 3kg of the Kona blend to make the special blend.

 

Hope this helps.