Natasha is planning a school celebration and wants to have live music and food for everyone who attends. She has found a band that will charge her $750 and a caterer whoWill provide snacked and drinks for $2.25 per person. If her goal is to keep the average cost per person between $2.75 and $3.75, how many people must attend?
(1) 225<p<325
(2) 325<p<750
(3) 500<p<1000
(4) 750<p<1500
750<p< 1500
To determine how many people must attend the school celebration, we can use the given information about the budget and the desired average cost per person.
Let's use the equation:
Total Cost = Cost of the band + Cost of the food
1. Let's solve using the lower limit of the average cost per person ($2.75):
Total Cost = $750 (cost of the band) + $2.75p (cost of the food)
Total Cost = $750 + $2.75p
2. Let's solve using the upper limit of the average cost per person ($3.75):
Total Cost = $750 (cost of the band) + $3.75p (cost of the food)
Total Cost = $750 + $3.75p
Now we can use the inequality statements provided to find the appropriate range for the number of people attending.
(1) 225 < p < 325:
Using the lower limit, Total Cost = $750 + $2.75 * 225 = $1400
Using the upper limit, Total Cost = $750 + $2.75 * 325 = $1625
Since these values do not fall within the desired cost range, this range is not valid.
(2) 325 < p < 750:
Using the lower limit, Total Cost = $750 + $2.75 * 325 = $1625
Using the upper limit, Total Cost = $750 + $2.75 * 750 = $2700
Since these values fall within the desired cost range ($1625 < Total Cost < $2700), this range is valid.
(3) 500 < p < 1000:
Using the lower limit, Total Cost = $750 + $2.75 * 500 = $2125
Using the upper limit, Total Cost = $750 + $2.75 * 1000 = $3250
Since these values fall beyond the desired cost range, this range is not valid.
(4) 750 < p < 1500:
Using the lower limit, Total Cost = $750 + $2.75 * 750 = $2875
Using the upper limit, Total Cost = $750 + $2.75 * 1500 = $4125
Since these values fall beyond the desired cost range, this range is not valid.
Therefore, the correct answer is 325 < p < 750.
To find out how many people must attend the school celebration, we can set up an inequality based on the given information.
Let's assume the number of people who attend is denoted by "p".
The cost of hiring the band is a fixed cost of $750.
The cost of food and drinks is $2.25 per person.
To keep the average cost per person between $2.75 and $3.75, we can write the following inequality:
2.75 ≤ (750 + 2.25p) / p ≤ 3.75
Let's simplify this inequality:
2.75p ≤ 750 + 2.25p ≤ 3.75p
Subtracting 2.25p from each side:
0.5p ≤ 750 ≤ 1.5p
Dividing each side by 0.5:
p ≤ 1500 ≤ 3p
Dividing each side by 3:
0 ≤ p ≤ 500
So, the number of people who must attend the school celebration falls within the range of 0 to 500 (inclusive).
Among the given options, the correct answer is (3) 500 < p < 1000, as it represents the range within which the number of attendees must fall.
2.75 < 2.25 + (750 / p) < 3.75
.50 < 750 / p < 1.50
2 > p / 750 > 2/3
1500 > p > 500
none of the responses seem to be correct