11. The sum of two numbers is 19, and their difference is 55. What are the two
numbers?
12. For the fundraiser, Will sold 225 candy bars. He earns $1 for each almond candy
bar he sells and $0.75 for each caramel candy bar he sells. If he earned a total of
$187.50, how many of each type of candy bar did he sell for the fundraiser?
13. There were 155 people at the basketball game. Tickets for the game are $2.50 for
students and $4 for adults. If the total money received for admission was
$492.50, how many students and adults attended the game?
14. Jocelyn has $0.75 in her pocket made up of 27 nickels and dimes. How many of
each type of coin does she have?
11. Let's call the two numbers x and y. We can set up a system of equations:
x + y = 19 (the sum of the two numbers is 19)
x - y = 55 (the difference of the two numbers is 55)
We can solve this system of equations by adding the two equations together:
(x + y) + (x - y) = 19 + 55
2x = 74
x = 37
Substituting this value back into one of the original equations, we can find the other number:
37 + y = 19
y = 19 - 37
y = -18
So the two numbers are 37 and -18.
12. Let's call the number of almond candy bars x and the number of caramel candy bars y. We can set up a system of equations:
x + y = 225 (the total number of candy bars sold is 225)
1x + 0.75y = 187.50 (the total amount of money earned is $187.50)
We can solve this system of equations by substitution. Let's solve the first equation for x:
x = 225 - y
Substituting this into the second equation, we have:
1(225 - y) + 0.75y = 187.50
225 - y + 0.75y = 187.50
225 + 0.25y = 187.50
0.25y = -37.50
y = -37.50 / 0.25
y = 150
Substituting this value back into the first equation, we can find the number of almond candy bars:
x + 150 = 225
x = 225 - 150
x = 75
So Will sold 75 almond candy bars and 150 caramel candy bars.
13. Let's call the number of students x and the number of adults y. We can set up a system of equations:
x + y = 155 (the total number of people at the game is 155)
2.50x + 4y = 492.50 (the total money received is $492.50)
We can solve this system of equations by substitution. Let's solve the first equation for x:
x = 155 - y
Substituting this into the second equation, we have:
2.50(155 - y) + 4y = 492.50
387.50 - 2.50y + 4y = 492.50
1.50y = 105
y = 105 / 1.50
y = 70
Substituting this value back into the first equation, we can find the number of students:
x + 70 = 155
x = 155 - 70
x = 85
So 85 students and 70 adults attended the game.
14. Let's call the number of nickels x and the number of dimes y. We can set up a system of equations:
x + y = 27 (the total number of coins is 27)
0.05x + 0.10y = 0.75 (the total amount of money is $0.75)
We can solve this system of equations by substitution. Let's solve the first equation for x:
x = 27 - y
Substituting this into the second equation, we have:
0.05(27 - y) + 0.10y = 0.75
1.35 - 0.05y + 0.10y = 0.75
0.05y = -0.60
y = -0.60 / 0.05
y = -12
Substituting this value back into the first equation, we can find the number of nickels:
x + (-12) = 27
x = 27 + 12
x = 39
Since it doesn't make sense to have a negative number of coins, we can conclude that there was an error in the problem statement or calculation.