I need help please.
Ellen is having a dinner party. She is trying to decide whether to hire a
caterer or make the food herself. The caterer charges $150 in addition to
$15 per person. If she makes the food herself it will cost her $25 per
person in groceries. Write an inequality to model the situation so that we
can find how many people Ellen can invite so that making the food herself
is cheaper than hiring a caterer.
Members of the band boosters are planning to sell programs at football
games. The cost to print the programs is $150 plus $0.50 per program.
They plan to sell each program for $2. Write an inequality to model the
situation so that we can find how many programs must be sold to make a
profit of at least $500.
Can someone please explain to me how to do this
To find out how many people Ellen can invite so that making the food herself is cheaper than hiring a caterer, we can set up an inequality.
Let "x" represent the number of people Ellen can invite.
The cost of hiring a caterer is given by: $150 + $15 per person, so the total cost for a caterer is 150 + 15x dollars.
The cost of making the food herself is $25 per person, so the total cost for self-made food is 25x dollars.
To find the point at which making the food herself is cheaper than hiring a caterer, we need to find the x for which the costs are equal or the self-made food is cheaper.
The inequality to model this situation is:
25x ≤ 150 + 15x
Simplifying the inequality:
25x - 15x ≤ 150
10x ≤ 150
Dividing both sides of the inequality by 10:
x ≤ 15
Therefore, Ellen can invite 15 or fewer people so that making the food herself is cheaper than hiring a caterer.
For the second situation:
Let "p" represent the number of programs that must be sold to make a profit of at least $500.
The cost to print the programs is $150 plus $0.50 per program, so the total cost to print "p" programs is 150 + 0.50p dollars.
The revenue from selling each program is $2. Therefore, the total revenue from selling "p" programs is 2p dollars.
To find the number of programs that must be sold to make a profit of at least $500, we need to find the "p" for which the revenue is at least $500 greater than the cost.
The inequality to model this situation is:
2p - (150 + 0.50p) ≥ 500
Simplifying the inequality:
2p - 150 - 0.50p ≥ 500
1.50p - 150 ≥ 500
1.50p ≥ 650
Dividing both sides of the inequality by 1.50:
p ≥ 650 / 1.50
p ≥ 433.33 (rounded up to the nearest whole number)
Therefore, at least 434 programs must be sold to make a profit of at least $500.
To find the number of people Ellen can invite so that making the food herself is cheaper than hiring a caterer, we can set up an inequality.
Let's say the number of people Ellen invites is represented by x.
For hiring a caterer:
Cost = $150 + $15 per person = $150 + $15x
For making the food herself:
Cost = $25 per person = $25x
We want to find the number of people (x) for which making the food herself is cheaper than hiring a caterer. This can be represented by the inequality:
$150 + $15x > $25x
To solve this inequality, we can subtract $15x from both sides:
$150 > $10x
Finally, divide both sides by $10:
15 > x
So Ellen can invite up to 15 people for making the food herself to be cheaper than hiring a caterer.
For the second question, let's find the number of programs that must be sold to make a profit of at least $500.
Let's say the number of programs sold is represented by p.
The cost to print the programs is $150 plus $0.50 per program, so the cost can be represented by the equation:
Cost = $150 + $0.50p
The selling price of each program is $2, and the profit is calculated by subtracting the cost from the revenue:
Profit = ($2 x p) - ($150 + $0.50p)
We want the profit to be at least $500, so the inequality can be set up as:
($2 x p) - ($150 + $0.50p) ≥ $500
To solve this inequality, we can combine like terms:
$2p - $150 - $0.50p ≥ $500
$1.50p - $150 ≥ $500
Now, add $150 to both sides:
$1.50p ≥ $650
Finally, divide both sides by $1.50:
p ≥ $650 / $1.50
p ≥ 433.33
Since we can't sell fractions of a program, the number of programs that must be sold to make a profit of at least $500 is 434.