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QuestionA merchant mixes xx kg of tea at \72/kgwith72/kg with ykgat$97/kgforamixturecostof$82/kg.Find kg at \$97/kg for a mixture cost of \$82/kg. Find x:y$ and profit from 15 kg of tea P sold at \$120/kg.

Studdy Solution

STEP 1

Assumptions1. The cost of tea is 72/kg.ThecostofteaQis72/kg. The cost of tea Q is 97/kg3. The cost of the mixture is $82/kg4. The merchant mixes x kg of tea with y kg of tea Q5. The cost of the mixture is calculated as a weighted average of the costs of tea and Q, weighted by the amounts of each tea in the mixture.

STEP 2

First, we need to express the cost of the mixture in terms of x and y. The cost of the mixture is the total cost of the teas divided by the total weight of the teas.
Cost of mixture=Cost of tea ×Weight of tea +Cost of tea Q×Weight of tea QTotal weightCost\, of\, mixture = \frac{Cost\, of\, tea\, \times Weight\, of\, tea\, + Cost\, of\, tea\, Q \times Weight\, of\, tea\, Q}{Total\, weight}

STEP 3

Now, plug in the given values for the costs of tea and Q, and the weights of tea and Q.
Cost of mixture=$72×x+$97×yx+yCost\, of\, mixture = \frac{\$72 \times x + \$97 \times y}{x + y}

STEP 4

We know that the cost of the mixture is 82/kg,sowecansettheaboveequationequalto82/kg, so we can set the above equation equal to 82 and solve for x and y.
$82=$72×x+$97×yx+y\$82 = \frac{\$72 \times x + \$97 \times y}{x + y}

STEP 5

Multiply both sides of the equation by (x+y)(x + y) to get rid of the denominator on the right side.
$82×(x+y)=$72×x+$97×y\$82 \times (x + y) = \$72 \times x + \$97 \times y

STEP 6

Rearrange the equation to isolate the terms with x and y on one side.
$82×x+$82×y=$72×x+$97×y\$82 \times x + \$82 \times y = \$72 \times x + \$97 \times y

STEP 7

Further rearrange the equation to isolate x and y.
$10×x=$15×y\$10 \times x = \$15 \times y

STEP 8

Divide both sides by5 to simplify the equation.
2×x=3×y2 \times x =3 \times y

STEP 9

Therefore, the ratio of x to y is23.

STEP 10

Suppose that15 kg of tea is used. Since the ratio of x to y is23, we can find the amount of tea Q used by multiplying the amount of tea by3/2.
y=15×32=22.5 kgy =15 \times \frac{3}{2} =22.5 \, kg

STEP 11

The total cost of the mixture is the sum of the costs of tea and Q.
Total cost=$72×15+$97×22.5Total\, cost = \$72 \times15 + \$97 \times22.5

STEP 12

Calculate the total cost.
Total cost=$1080+$2182.5=$3262.5Total\, cost = \$1080 + \$2182.5 = \$3262.5

STEP 13

The total revenue from selling the mixture is the selling price per kg times the total weight of the mixture.
Total revenue=$120×(15+22.5)Total\, revenue = \$120 \times (15 +22.5)

STEP 14

Calculate the total revenue.
Total revenue=$120×37.=$4500Total\, revenue = \$120 \times37. = \$4500

STEP 15

The profit is the total revenue minus the total cost.
Profit=Total revenue−Total costProfit = Total\, revenue - Total\, cost

STEP 16

Plug in the values for the total revenue and the total cost to calculate the profit.
Profit=$4500−$3262.5Profit = \$4500 - \$3262.5

STEP 17

Calculate the profit.
Profit=$4500−$3262.5=$1237.5Profit = \$4500 - \$3262.5 = \$1237.5The profit of selling the mixture is $1237.5.

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