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.
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To find out how many people Ellen can invite so that making the food herself is cheaper than hiring a caterer, you can set up the following inequality:
25p < 150 + 15p
Where p represents the number of people Ellen can invite.
To find out how many programs must be sold to make a profit of at least $500, you can set up the following inequality:
2p - (150 + 0.5p) ≥ 500
Where p represents the number of programs that need to be sold.
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's say the number of people Ellen can invite is represented by 'x.'
If Ellen hires a caterer, the cost will be $150 plus $15 per person, so the total cost will be 150 + 15x.
If Ellen makes the food herself, the cost will be $25 per person, so the total cost will be 25x.
We want to find when it is cheaper to make the food herself, so we can set up the following inequality:
25x < 150 + 15x
This inequality states that the cost of making the food herself (25x) should be less than or else not greater than the cost of hiring a caterer (150 + 15x).
Now, we can simplify the inequality:
25x - 15x < 150
10x < 150
Finally, divide both sides of the inequality by 10 to solve for 'x':
x < 15
Therefore, Ellen can invite up to 14 people (since x represents the number of people, and we need to consider positive whole numbers only) for making the food herself to be cheaper than hiring a caterer.
To find out how many programs must be sold to make a profit of at least $500, we can set up an inequality.
Let's say the number of programs to be sold is represented by 'x.'
The cost to print the programs is $150 plus $0.50 per program, so the total cost will be 150 + 0.50x.
The selling price for each program is $2, so the total revenue will be 2x.
We want to find when the profit (revenue - cost) is at least $500, so we can set up the following inequality:
2x - (150 + 0.50x) >= 500
This inequality states that the profit (2x - (150 + 0.50x)) should be greater than or equal to $500.
Now, we can simplify the inequality:
2x - 150 - 0.50x >= 500
1.50x - 150 >= 500
Finally, add 150 to both sides of the inequality:
1.50x >= 650
Now, divide both sides of the inequality by 1.50 to solve for 'x':
x >= 433.33
Since the number of programs must be a whole number, we round up to the nearest whole number. Therefore, the band boosters must sell at least 434 programs to make a profit of at least $500.