1) George Polya developed a four-step process for solving word problems: understand the problem, devise a plan, carry out the plan, and look back. 2) Common representations for numbers in word problems include using variables to represent unknowns, sums and differences, multiples, and consecutive integers. 3) Strategies are provided for different types of word problems involving age, motion, mixtures, collections, work, and formulas in geometry.

1. Mathematics Reviewer
WORD PROBLEMS IN ALGEBRA
Solving Word Problems
George Polya’s Four-step Problem-Solving 2. Check each step of the plan as you proceed. 5. If the first number is x and another
Process This may be intuitive checking or a formal number is k less than the first, then the
As part of his work on problem solving, Polya proof of each step. other number is x  k .
developed a four-step problem-solving process 3. Keep an accurate record of your work. 6. Consecutive integers: If x is the first
similar to the following: D. Looking Back integer, then x  1 is the 2nd, x  2 is the
A. Understanding the Problem 1. Check the results in the original problem. In 3rd, etc.
1. Can you state the problem in your own some cases, this will require a proof. 7. Consecutive even/odd integers: If x is the
words? 2. Interpret the solution in terms of the original first even/odd integer, then x  2 is the
2. What are you trying to find or do? problem. Does your answer make sense? Is 2nd even/odd integer, x  4 is the 2nd
3. What are the unknowns? it reasonable? even/odd integer, etc.
4. What information do you obtain from the 3. Determine whether there is another method 8. Digits. A two digit number can be written
problem? of finding the solution. in the from 10T  U , where T is the ten’s
5. What information, if any, is missing or not 4. If possible, determine other related or more digit and U is the unit’s digit. A three-
needed? general problems for which the techniques digit number can be written in the form
B. Devising a Plan will work.
100H  10T  U , where H is the
The following list of strategies, although not (Source:
hundred’s digit.
exhaustive, is very useful: http://www.drkhamsi.com/classe/polya.html)
N.B. If the digit of a two-digit number
1. Look for a pattern.
10T  U is REVERSED, the new number
2. Examine related problems and determine if Solution Strategies & Tips for Particular Word
the same technique can be applied. becomes 10U  T . Reversing a three-digit
Problems
3. Examine a simpler or special case of the number means reading the number backwards;
problem to gain insight into the solution of A. Representations for Number Problems i.e., 100H  10T  U becomes 100U  10T  H .
the original problem. 1. If the sum of two numbers is s and x is one
4. Make a table. number, then the other number is s  x . B. Age Problems
5. Make a diagram. 2. If the difference between two numbers is d If a is the present age of person A, then a  y
6. Write an equation. and x is the smaller number, then the larger is the age of A y years ago, while a  y is the
7. Use a guess and check. number is d  x . age of A y years from now (or hence).
8. Work backward. 3. If the first number is x and another is k times Age table for age problems:
9. Identify a sub goal. the first, then the other number is kx . Age some
C. Carrying out the Plan 4. If the first number is x and another number Persons Age now years ago or
1. Implement the strategy in Step 2 and is k more than the first, then the other from now
perform any necessary actions or number is x  k .
computations.

2. C. Mixture/Collection Problems meeting point Note: If B undoes what A does, the “+” becomes a
A dA B
For mixtures, the general equation is “” between 1/a and 1/b in #s 2, 3, and 4.
 amount of   % concen-   amount of  dB
 × =  dapart  d A  d B The situations described above can be extended for
 solution   tration   substance  3 or more persons doing a job.
while for those involving prices or money,
4. For bodies traveling in air/current:
 price or cost   totol cost 
 quantity  ×  =   rate against the   rate in still   rate of wind 
F. Some Formulas in Geometry
 
 per unit   or price  1. Perimeter Formulas
 
 wind/current   wind/water   or current  a. square: P  4s, s  length of one side
Types of No. of Amount/price/percent Total  rate with the   rate in still   rate of wind  b. rectangle: P  2l  2w (l = length, w
quantities units per unit amount     = width
 wind/current   wind/water   or current 
c. triangle: P  a  b  c (a, b, and c are
E. Work Problems the sides of the triangle)
TOTAL = sum of all entries at last column If person A can finish a job alone in a time units d. circle (circumference): C  2r (r =
and B can finish the same job alone in b time radius)
D. Motion Problems units, then 2. Area formulas
 distance    rate  time  1 a. square: A  s 2
General Formula: 1. A can finish of the job in 1 time unit,
d  rt a b. rectangle: A  lw
Situations: 1 c. triangle: A  1 bh
2
1. Overtaking = equal distances covered while B can finish of the job in 1 time
b d. circle: A  r 2
Starting point unit. So after k time units, A and B can 3. Volume Formulas
dA A
1 1 a. cube: V  s3
finish k   and k   of the job,
a b b. rectangular solid/prism: V  lwh
d B  d A  rAt0
B respectively. c. sphere: V  3 r 3
4
d A  dB
2. Together, A and B can finish   of
1 1 1 d. right circular cylinder: V  r 2 h
2. Bodies moving in opposite directions = the a b x e. right circular cone: V  1 r 2 h
3
distance apart is the sum of the distances the job in 1 time unit, where x is the no. of
traveled by each body time units that A and B can finish the job
together. Anything you can solve in
Starting point 3. Suppose that A started working the job five minutes should not
A dA B
along in the first p time units then B joined
be considered a problem.
dB for x more time units until they finished the
-- George Polya
dapart  d A  d B 1 1 1
job, then p    x     1 where 1
 a a b
3. Bodies moving toward each other = the sum of represents the whole job.
the distance between the origin of 1st body and 4. If A and B were doing the job together for
2nd body to the meeting point is equal to the sum the first q time units until B left, letting A
of the distances between the origin of each point finish the job alone in x more time units, © gjnabueg 2 August 2003
1 1 1 Revised 19 July 2009
then q     x    1 . All rights reserved 
a b a