Question

A grocer buys two kinds of tea, one at 32 cents per pound and the other at 40 cents per pound. He mixes together some of each and proposes to sell the blend at 43 cents per pound, making a profit of 25% on the cost. How many pounds of each kind must he use to make a mixture that weighs 100 pounds?

Answer 1

To solve this problem, we'll use the concept of weighted averages. Let's denote the amount of the 32-cent tea as 'x' pounds and the amount of the 40-cent tea as 'y' pounds.

Given that the grocer mixes the teas to make a 100-pound blend, we can establish our first equation based on the total weight of the mixture:

x + y = 100 (Equation 1)

The second set of information tells us that the grocer wants to sell the blend at 43 cents per pound with a 25% profit. To find the cost price per pound, we need to account for the desired profit. Hence, we divide the selling price by 1 plus the profit margin:

Cost price per pound = Selling price per pound / (1 + profit percentage)
Cost price per pound = 43 cents / (1 + 0.25)
Cost price per pound = 43 / 1.25
Cost price per pound = 34.4 cents

Now, let's calculate the total cost of x pounds of the 32-cent tea and y pounds of the 40-cent tea:

Total cost = (32 cents * x) + (40 cents * y)

The average cost per pound of the mixture, which is the weighted average of the costs, should equal the determined cost price per pound, which is 34.4 cents. So the second equation is based on the cost of the tea:

(32x + 40y) / 100 = 34.4 (Equation 2)

We can multiply this equation by 100 to get rid of the denominator:

32x + 40y = 3440 (Equation 3)

Now we have a system of two equations with two variables:

x + y = 100 (Equation 1)
32x + 40y = 3440 (Equation 3)

We can solve this system using the substitution or elimination method. Let's use substitution in this case. We'll solve Equation 1 for x:

x = 100 - y (Equation 4)

Now, substitute Equation 4 into Equation 3:

32(100 - y) + 40y = 3440
3200 - 32y + 40y = 3440
3200 + 8y = 3440
8y = 3440 - 3200
8y = 240
y = 240 / 8
y = 30

Now that we have the value of y, we can substitute it back into Equation 1 to find x:

x + 30 = 100
x = 100 - 30
x = 70

Therefore, the grocer must use 70 pounds of the 32-cent tea and 30 pounds of the 40-cent tea to make a 100-pound mixture that can be sold at 43 cents per pound, allowing for a 25% profit margin on the cost.

Answer 2

He must use 75 pound of first kind and 25 pound of second kind tea.

Solution-

Price of first kind tea per pound = $0.32

Price of second kind tea per pound = $0.40

By selling it at $0.43 per pound he made a profit of 25%

Let the cost price is x, then

`\Rightarrow x+(25)/(100)x=0.43\\\\\Rightarrow (125)/(100)x=0.43\\\\\Rightarrow x=0.43* (100)/(125)\\\\\Rightarrow x=34`

The cost price of the mixed kind is $0.34

Then, let the amount of first kind tea in the 100 pound mixture is y pound, so the amount of second kind tea is (100-y) pound

So, the price of mixture will be equal to the sum of price of each kind,

`\Rightarrow 0.32y+0.40(100-y)=0.34* 100`

`\Rightarrow 0.32y+40-0.40y=34`

`\Rightarrow 0.08y=6`

`\Rightarrow y=(6)/(0.08)=75`

`\Rightarrow y=75`

`\Rightarrow (100-y)=100-75=25`

Therefore, he must use 75 pound of first kind and 25 pound of second kind tea.