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## Equation answer

you own a pizza restaurant the selling price is \$12 for each pizza the cost to make each pizza is \$5 you have a fixed labor cost of \$230 and you sell 85 pizzas what is your profit? How many pizzas would need to be sold to make a profit of \$700? How many pizzas would need to be sold if you wanted to make a profit greater than \$1,000?

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# 6 Answers

Recall:     Profit  =  Revenue  -  Cost

P(x)  =  R(x)  -  C(x)   ,     where x is the number of pizzas sold

Revenue, R(x), is given by the following:

R(x) = (selling price)·(# of pizzas sold)

R(x) = 12x

Cost, C(x), is given by the following:

C(x) = (variable cost)·(# of pizzas sold) + fixed cost

C(x) = 5x + 230

Therefore, since    Profit = Revenue - Cost

P(x) = R(x) - C(x)

P(x) = 12x - (5x + 230)

P(x) = 12x - 5x - 230

P(x) = 7x - 230

Thus, the profit of selling 85 pizzas is determined by solving P(x) when x=85:

P(x) = 7x - 230

P(85) = 7(85) - 230

= 595 - 230

= 365

Since   P(85) = 365 ,  selling 85 pizzas yield a profit of \$365.

To find how many pizzas need to be sold to yield a profit of \$700, solve for x when P(x)=700:

P(x) = 7x - 230

700 = 7x - 230

700 + 230 = 7x - 230 + 230

930 = 7x

930/7 = 7x/7

132.86 = x

x ≈ 133

Thus, to make a profit of \$700, 133 pizzas need to be sold.

To find how many pizzas need to be sold to yield a profit greater than \$1000, solve for x when P(x)>1000:

P(x) = 7x - 230

7x - 230 > 1000

7x - 230 + 230 > 1000 + 230

7x > 1230

7x/7 > 1230/7

x > 175.7

x ≥ 176

Thus, to make a profit greater than \$1000, you would need to sell at least 176 pizzas.

You can set up the problem with the following equation (x means "times"):

85 x (\$12 - \$5) - \$230 = ?

You multiply your profit for each pizza, which is \$12 cost of pizza minus \$5 to make it, with 85 times, the number of pizzas you make. You subtract form all this the fixed labor cost of \$230.

--> 85 x (12 - 5) - 230 = 365

Answer: Your profit from the sale of 85 pizzas is \$365.

To make a profit of \$700, you will have an unknown number of pizzas that you make, let's call this number U (for unknown) and use the same formula and solve it for the unknown:

U x (12 - 5) - 230 = 700     Now add 230 to both sides.

-->  U x (12 - 5) = 930       Now divide both sides by (12 - 5), which is 7.

-->  U = 132.9

Answer: you have to make 133 pizzas to make a profit of \$700.

To make a profit of more than \$1,000, you set up the equation just about the same way as before, only you need to amek sure that you make more than \$1,000, not equal to \$1,000 profit:

U x (12 - 5) -230 > 1,000            Add 230 to both sides

U x (12 - 5) > 1,230                   Divide by 7

U > 175.7

Answer: since you can't make partial pizzas, you will have a profit or greater than \$1,000 by making a minimum of 176 pizzas.

Check back: 176 x (\$12 - \$5) - \$230 = \$1,002 profit.

Cost to make a pizza for you as a owner of a restaurant is \$y = \$5 · x + \$230

1.
You sold 85 pizzas for \$12 and you made \$1020, but you spent \$655 = \$5*85+230, so your profit is \$365 (which is a difference between earning and spending).

2.
\$700 = \$12x - (\$5x + \$230)
700 = 7x - 230 ---> x = 930/7 ≈ 133 pizzas

3. \$1000 < \$12x - (\$5x + \$230)
1000 < 7x - 230
1000 + 230 < 7x ---> x > 1230 / 7 ---> x > 175 pizzas

There's several steps happening here, so writing a few down to keep track may be good. Also, I find converting word problems to math expressions is best started with words. Let's start with the first question asked.

Question #1

(Profit per pizza)·(# pizzas sold) - fixed costs = Profit   But what is the "profit per pizza"?

[1] (Selling price - cost to make)·(#pizzas sold) - fixed costs = Profit   Ok, now we have all the #s to plug in!

(12-585 - 230 = 85 - 230 = \$365   Don't forget the units...the answer is in dollars (\$).

Question 2

Use equation [1] to plug in what you know for the second question. Note that after you plug in numbers, you'll need to manipulate the equation (move things to one side or the other), because this time you're given the profit and need to find the "#pizzas sold".

(12-5)·(#pizzas sold) - 230 = 700    Get #pizzas sold by itself on one side.

(7)·(#pizzas sold) -230 (+230) = 700 (+230)    Use opposite operations and keep both sides balanced!

(1/7)·7·(#pizzas sold) = 930·(1/7)    Opposite of multiplication is division such that (1/7)·7 = 1

#pizzas sold = 930/7 = (you do the math)  This is how many pizzas need to be sold to make \$700 profit.

Question 3

The final question is setup and solved exactly the same way Question 2 except one more little hook. Instead of a profit "equal to" \$1000, they're asking for a profit "greater than" \$1000. So swap out that '=' sign for a '>' sign and watch your sign rules when dealing with inequalities. (Hint: You won't be multiplying or dividing by 'negatives' so that quirky inequality symbol change rule won't come into play on this one.)

Let Selling price be S=\$12

Let Cost price be C=\$5

Let Labor Cost be L=\$230

Let number of pizza,n=85

Equation:

Profit = (no. of pizza x (Selling Cost- Cost Price))- Labor Cost

Profit=85x(\$12-\$5)-\$230

Profit=(\$85x\$7)-\$230

Profit = \$365

You would basically be using the same equation for all the questions.

The base of the problem is to set up the equation needed

Profit = (no. of pizza x (Selling Cost- Cost Price))- Labor Cost

For the third question, note the word "greater" that means instead of an equation you would be setting up an inequalities using ">" greater than sign.

Profit > (no. of pizza x (Selling Cost- Cost Price))- Labor Cost

Hope that helps! Please vote if it did :)

a) profit = revenue - cost = 12*85 - 5*85 - 230 = \$365

b) profit = 700 = n(12-5) - 230

n = 133 pizzas

c) profit = 1000 = n(12-5) - 230

n = 176 pizzas