# The Joy of Algebra Word Problems

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```					Understanding Algebra
Word Problems
How Do I Teach?
   Project a positive attitude
   Explain types
   Emphasize systematic approach
•   Identify type
•   Identify key unknown
•   Express all quantities in terms of key unknown
•   Write equation
•   Check answers in problem (not equation)
Word Problem Types
 Number Problems
 Distance Problems

 Mixture Problems

 Work Problems
Number Problems
Types
 Age Problems
 Integer Problems

 Consecutive Integer Problems

• Always use 2n for consecutive integer
problems
Solving Number Problems
 All unknowns expressed in terms of
one number
 Identify the number

 Write algebraic expressions in terms

of single number
Integer Problem
   The sum of three numbers is 40. The first
number is 4 more than the product of 6
and the second number. The third number
is 9 less than twice the second number.
Find the three numbers.
   Let n represent the second number
• First number is 6n + 4
• Third number is 2n – 9
   (6n + 4) + n + (2n – 9) = 40
Benefits of Number Problems
 Identifying root
 Writing algebraic expressions

 Writing equations
Distance Problems

Distance = (rate)(time)
Types
   Traveling in the Same Direction
• Overtaking
• Separating
   Traveling in Opposite Directions
• Traveling Toward Each Other
• Traveling Away From Each Other
   Going and Returning
Overtaking in the Same
Direction
   Keys
• Slower vehicle travels longer
   ts > tf
• Slower vehicle has head start
   ts = tf
   Slower/Earlier Vehicle = rsts
   Faster/Later Vehicle = rftf
   Simplest Solution: rftf = rsts
• When travel times can be determined
   Alternate Solution: rftf – rstf = Constant
• When given head start distance
Overtaking Problem
   A fishing boat leaves Tampa Bay at 4:00
a.m. and travels at 12 knots. At 5:00 a.m.
a second boat leaves the same dock for
the same destination and travels at 14
knots. How long will it take the second
boat to catch the first?
   Let t be travel time of first boat
   Then t – 1 is the travel time of second
boat
   14(t – 1) = 12t
Separating in Same Direction
 Key: Travel time same for both
 Faster Vehicle = rft

 Slower Vehicle = rst

 Solution: rft - rst = Constant

• Constant is desired distance between
Separating Problem
   At the auto race one car travels 190
mph while another travels 195 mph.
How long will it take the faster car to
gain two laps on the slower car if the
speedway track is 2.5 miles long?

 Let t be the racing time
 195t – 190t = (2)(2.5)
Traveling in Opposite Directions
   Common variants
• Traveling toward each other
• Traveling away from each other
   Key: Net effect is sum of individual
distances
• Distance covered by vehicle1 = r1t1
• Distance covered by vehicle2 = r2t2
   t1 may or may not equal t2
   Solution: r1t1 + r2t2 = Constant
Coming Together Problem
   A freight train leaves Centralia for Chicago at the
same time a passenger train leaves Chicago for
Centralia. The freight train moves at a speed of
45 mph, and the passenger train travels at a
speed of 64 mph. If Chicago and Centralia are
218 miles apart, how long will it take the two
trains to meet?

   Let t be time to meet
   45t + 64t = 218
Going and Returning
   Key: Distance between points always
same
• Speeds usually vary
• Times usually vary
   Common variants
• To and from a location
• Up and down a waterway
   Distance going = rgtg
   Distance returning = rrtr
   Solution: rgtg = rrtr
Going and Returning Problem
   Brandon and Shanda walk to Grandma’s house at
a rate of 4 mph. They ride their bicycles back
home at a rate of 8 mph over the same route
that they walked. It takes one hour longer to
walk than to ride. How long did it take them to
walk to Grandma’s?

   Let h be time to walk
   Then h – 1 is time to ride
   4h = 8(h – 1)
Mixture Problems
Types
   Differently Valued Items
• Money (coins, bills)
• Items priced by pound (candy, nuts,
etc.)
• Tickets
 Different Interest Rates
 Different Liquid Solutions
Differently Valued Items
   Key: Total value equals sum of value of different
items
• All values must be expressed in same units
   Total ValueItem 1=(QuantityItem 1)(ValueItem 1)
   Total ValueItem 2=(QuantityItem 2)(ValueItem 2)
   Total ValueItem 3=(QuantityItem 3)(ValueItem 3)
   Solution: (Q1)(V1) + (Q2)(V2) + (Q3)(V3) + … =
QtVt
Coin Problem
   A coin bank contains              Let n be number of
four more quarters                 nickels
than nickels, twice as
many dimes as                     n + 4 = quarters
nickels, and five more            2n = dimes
than three times as
many pennies as
   3n + 5 = pennies
nickels. If the bank
contains \$22.25, how
many of each coin are
in it?
(3n + 5)(0.01) + n(0.05) + 2n(0.10) + (n + 4)(0.25) = 22.25
Different Interest Rates
   Key: Total interest = sum of interest from
individual investments
• Rate and time must be expressed consistently
   InterestInvestment 1=(Principal1)(Rate1)(Time1)
   InterestInvestment 2=(Principal2)(Rate2)(Time2)
   InterestInvestment 3=(Principal3)(Rate3)(Time3)

(p1)(r1)(t1) + (p2)(r2)(t2) + (p3)(r3)(t3) + . . . = Interest Income
Interest Problem
   An investor has \$500 more invested at 7%
than he does at 5%. If his annual interest
is \$515, how much does he have invested
at each rate?

   Let p be amount at 5%
   Then p + 500 is amount at 7%
   p(0.05)(1) + (p + 500)(0.07)(1) = 515
Different Solutions
   Key: Solute in parts equals solute in whole
   Common variants
   SoluteSolution 1 = QuantitySolution 1ConcentrationSolution 1
   SoluteSolution 2 = QuantitySolution 2ConcentrationSolution 2
   SoluteSolution 3 = QuantitySolution 3ConcentrationSolution 3

   Equation: q1c1 + q2c2 + q3c3 + . . . = qmixturecmixture
Solutions Problem
   How many gallons of cream that is 30%
butterfat must be mixed with milk that
is 3% butterfat to make 45 gallons that
are 12% butterfat?

   Let c be gallons of cream
   Then 45 – c is gallons of milk
   0.30c + 0.03(45 – c) = 0.12(45)
Work Problems

work done = (rate of work)(time spent)
Types
   Working Together
• Sum of contributions = Complete Task
   Working Against Each Other
• Difference between contributions =
Work Problem
   Ron, Mike, and Tim are going to paint a
house together. Ron can paint one side of
the house in 4 hours. To paint an equal
area, Mike takes only 3 hours and Tim 2
hours. If the men work together, how long
will it take them to paint one side of the
house?

   Let t be time needed to paint the side.
   (1/4)t + (1/3)t + (1/2)t = 1
How Do I Test?
   Select typical problems (10 - 15 per test)
• Avoid confusing problems
• Avoid “trick” problems
   Use as bonus questions if used at all
   Focus on developing equations