Equations and Problem Solving 9/14/2018 Equations and Problem Solving Using Algebra to solve Word Problems: Counting Problems and Mixture Problems www.numberbender.com

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 + 1 + 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 9/14/2018 www.numberbender.com

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 9/14/2018 www.numberbender.com

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 9/14/2018 www.numberbender.com

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 + 0.05n = 1.70 9/14/2018 www.numberbender.com

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 + 0.05n = 1.70 10d + 5n = 170 ( ) (100) d + n = 22 10d + 10n = 220 (100) ( ) (100) 9/14/2018 www.numberbender.com

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 9/14/2018 www.numberbender.com

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 9/14/2018 www.numberbender.com

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. 9/14/2018 www.numberbender.com

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 9/14/2018 www.numberbender.com

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 9/14/2018 www.numberbender.com

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 9/14/2018 www.numberbender.com

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 9/14/2018 www.numberbender.com