A bicycle store costs $ 3150 per month to operate. The store pays an average of $35 per bike. The average selling price of each bicycle is $105. How many bicycles must the store sell each month to break even?
Q
The store must sell
bicycles each month to break even.
To calculate the break-even point, we need to find the number of bicycles the store must sell to cover its operating costs.
Operating costs = $3150
Cost per bike = $35
Selling price per bike = $105
Let's assume the store sells x number of bikes.
Total Cost = Operating costs + (Cost per bike * Number of bikes)
Total Revenue = Selling price per bike * Number of bikes
Break-even point is when Total Cost = Total Revenue.
Therefore, we have the equation:
Operating costs + (Cost per bike * Number of bikes) = Selling price per bike * Number of bikes
Simplifying the equation:
3150 + 35x = 105x
Subtracting 35x from both sides:
3150 = 105x - 35x
Combining like terms:
3150 = 70x
Dividing both sides by 70:
x = 45
Therefore, the store must sell 45 bicycles each month to break even.
A metalworker has a metal alloy that is %15 copper and another alloy that is %75 copper. How many kilograms of each alloy should the metalworker combine to create 90 kg of a %51 copper alloy?
The metalworker should use
kilograms of the metal alloy that is % copper and
of the metal alloy that is % copper.
(Type whole numbers.)
Let's assume the metalworker needs to use x kilograms of the %15 copper alloy and y kilograms of the %75 copper alloy to create 90 kg of a %51 copper alloy.
The total amount of copper in the %15 copper alloy is x * 0.15 kg.
The total amount of copper in the %75 copper alloy is y * 0.75 kg.
Since the total amount of copper in the final alloy is 90 kg * 0.51 kg, we can write the equation for the total amount of copper:
x * 0.15 + y * 0.75 = 90 * 0.51
Simplifying the equation:
0.15x + 0.75y = 45.9
We also have the equation for the total weight of the alloy:
x + y = 90
Now we have a system of equations:
0.15x + 0.75y = 45.9
x + y = 90
To solve this system, we can use the substitution method.
From the second equation, we can express x in terms of y:
x = 90 - y
Substituting the expression for x in the first equation:
0.15(90 - y) + 0.75y = 45.9
Simplifying:
13.5 - 0.15y + 0.75y = 45.9
0.6y = 45.9 - 13.5
0.6y = 32.4
y = 32.4 / 0.6
y = 54
Now we can substitute the value of y back into the second equation to find x:
x + 54 = 90
x = 90 - 54
x = 36
Therefore, the metalworker should use 36 kilograms of the %15 copper alloy and 54 kilograms of the %75 copper alloy.
Gabe Amodeo, a nuclear physicist, needs 60 liters of a %60 acid solution. He currently has a 40% solution and a 70% solution. How many liters of each does he need to make the needed 60 liters of 60% acid solution?
Gabe needs
liters of the %40 solution.
Let's assume Gabe needs x liters of the 40% solution and y liters of the 70% solution to make the desired 60 liters of a 60% acid solution.
The amount of acid in the 40% solution is x * 0.40 liters.
The amount of acid in the 70% solution is y * 0.70 liters.
The desired acid amount is 60 * 0.60 liters.
We can write the equation for the total acid amount as follows:
x * 0.40 + y * 0.70 = 60 * 0.60
Simplifying the equation:
0.40x + 0.70y = 36
We also have the equation for the total volume of the solution:
x + y = 60
Now we have a system of equations:
0.40x + 0.70y = 36
x + y = 60
To solve this system, we can use the substitution method.
From the second equation, we can express x in terms of y:
x = 60 - y
Substituting the expression for x in the first equation:
0.40(60 - y) + 0.70y = 36
Simplifying:
24 - 0.40y + 0.70y = 36
0.30y = 36 - 24
0.30y = 12
y = 12 / 0.30
y = 40
Therefore, Gabe needs 40 liters of the 40% solution.
wrong
Apologies for the mistake. Let me correct it:
Let's assume Gabe needs x liters of the 40% solution and y liters of the 70% solution to make the desired 60 liters of a 60% acid solution.
The amount of acid in the 40% solution is x * 0.40 liters.
The amount of acid in the 70% solution is y * 0.70 liters.
The desired acid amount is 60 * 0.60 liters.
We can write the equation for the total acid amount as follows:
0.40x + 0.70y = 60 * 0.60
Simplifying the equation:
0.40x + 0.70y = 36
We also have the equation for the total volume of the solution:
x + y = 60
Now we have a system of equations:
0.40x + 0.70y = 36
x + y = 60
To solve this system, we can use the substitution method.
From the second equation, we can express x in terms of y:
x = 60 - y
Substituting the expression for x in the first equation:
0.40(60 - y) + 0.70y = 36
Simplifying:
24 - 0.40y + 0.70y = 36
0.30y = 36 - 24
0.30y = 12
y = 12 / 0.30
y = 40
Therefore, Gabe needs 40 liters of the 70% solution.