1. Equations and Problem Solving
4/24/2017
Equations and Problem Solving
Using Algebra to solve Word Problems: Counting Problems and Mixture Problems

2. The sum of 3 consecutive integers is 147. Find the integers.
Definition
CONSECUTIVE INTEGERS Integers that differ by one. The integers 50 and 51 are consecutive and so are -10 and -9
The sum of 3 consecutive integers is 147. Find the integers.
n + n+1 + n+2 = 147
n + n n + 2 = 147
3n + 3 = 147
3n = 144
n = 48 n
+ n+1
+ n+2
Let n = the first integer
Then n+1 = the second integer
And n+2 = the third integer
Let n = 48
Then n+1 = 49
And n+2 = 50
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3. Examples The sum of 3 consecutive integers is 72. Find the integers.
Let n = the first integer
Then n+1 = the second integer
And n+2 = the third integer
Let n = 23
Then n+1 = 24
And n+2 = 25
n + n+1 + n+2 = 72
3n + 3 = 72
3n = 69
n = 23
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4. Examples The sum of 3 consecutive integers is 915. Find the integers.
Let n = the first integer
Then n+1 = the second integer
And n+2 = the third integer
Let n = 304
Then n+1 = 305
And n+2 = 306
n + n+1 + n+2 = 915
3n + 3 = 915
3n = 912
n = 304
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5. Mixture Problems
John has $1.70, all in dimes and nickels. He has a total of 22 coins. How many of each kind does he have?
Step 1:
Assigning variables
d = dimes
n = nickels
Step 2:
Write algebraic equation
d + n = 22
Step 3:
Write value equation
0.10d n = 1.70
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6. Mixture Problems
John has $1.70, all in dimes and nickels. He has a total of 22 coins. How many of each kind does he have?
Step 4:
Solve the mixture problem
(100)
0.10d n = 1.70
10d + 5n = 170
( )
(100)
d + n = 22
10d + 10n = 220
(100)
( )
(100)
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7. Change the sign of the 2nd equation
Mixture Problems
John has $1.70, all in dimes and nickels. He has a total of 22 coins. How many of each kind does he have?
Step 4:
Solve the mixture problem
10d + 5n = 170
10d + 10n = -220
-5n = -50
n = 5
-( )
Change the sign of the 2nd equation -5
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8. Mixture Problems
John has $1.70, all in dimes and nickels. He has a total of 22 coins. How many of each kind does he have?
Step 4:
Solve the mixture problem
10d + 5n = 170
10d + 5(10) = 170
10d + 50 = 170
10d = 120
d = 12
Substitute “n” with 10
-50
-50
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9. The 3 consecutive numbers are 29, 30, and 31.
Warm Up
The sum of 3 consecutive integers is 60. What are the values of the 3 integers?
The 3 consecutive numbers are 29, 30, and 31.
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10. Mixture Problem
Tickets to a movie cost $5.00 for adults and $3.00 for children. If tickets were bought for 50 people for a total of $196 how many adult tickets were sold and how many children tickets were sold?
1: Assigning variables
a = adults
c = children
2: Write algebraic equation
a + c = 50
3: Write value equation
5a + 3c = 196
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11. Mixture Problem a + c = 50 -5a – 5c = -250 5a + 3c = 196 -2c = -54
4: Solve the mixture problem
a + c = 50
-5a – 5c = -250
5a + 3c = 196
-2c = -54
c = 27
(-5)( ) (-5) -2 -2
-5a – 5c = -250
-5a – 5(27) = -250
-5a – 135 = -250
-5a = -115
a = 23
+135
+135
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12. Mixture Problem
Tickets to a movie cost $4.00 for adults and $2.00 for children. If tickets were bought for 80 people for a total of $230 how many adult tickets were sold and how many children tickets were sold?
a = 35
c = 45
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13. Homework dimes = 20 quarters = 12
Amy has 32 coins consisting of dimes and quarters. If Amy has a total of $5 in her pocket, how many of each coin are there?
dimes = 20
quarters = 12
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