Word Problems
Problem Solving Strategies
Understand
1. Read the problem carefully.
2. Make sure you understand the situation that is
described.
3. Make sure you understand what information is
provided, and what the question is asking.
4. For many problems, drawing a clearly labeled
picture is very helpful.
Plan
1. First focus on the objective. What do you need to
know in order to answer the question?
2. Then look at the given information. How can you
use that information to get what you need to
know to answer the question?
3. If you do not see a clear logical path leading
from the given information to the solution, just
try something. Look at the given information and
think about what you can find from it, even if it
is not what the question is asking for. Often you
will find another piece of information that you
can then use to answer the question.
Write equations
You need to ex press mathematically the logical
connections between the given information and the
answer you are seeking. This involves:
1. Assigning variable names to the unknown
quantities. The letter x is always popular, but it is
a good idea to use something that reminds you
what it represents, such as d for distance or t for
time. The trickiest part of as signing variables is
that you want to use a minimum number of
different variables (just one if possible ). If you
know how two quantities are related, then you
can express them both with just one variable. For
example, if Jim is two years older than John is,
you might let x stand for John’s age and (x + 2)
stand for Jim’s age.
2. Translate English into Math. Mathematics is a
language , one that is particularly well suited to
describing logical relationships. English, on the
other hand, is much less precise.
Solve
Now you just have to solve the equation(s) for the
unknown(s). Remember to answer the question that
the problem asks.
Check!
Think about your answer. Does your answer come
out in the correct units? Is it reasonable? If you made
a mistake somewhere, chances are your answer will
not just be a little bit off, but will be completely
ridiculous
General Word Problems
General Strategy
Recall the general strategy for setting up word
problems. Refer to the Problem Solving Strategies
page for more detail.
1. Read the problem carefully: De termine what is
known , what is unknown, and what question is
being asked.
2. Represent unknown quantities in terms of a
variable.
3. Use diagrams where appropriate.
4. Find formulas or mathematical relationships
between the knowns and the unknowns.
5. Solve the equations for the unknowns .
6. Check answers to see if they are reasonable.
Number Problems
Example: Find a number such that 5 more than one-half
the number is three times the number.
Let x be the unknown number.
Translating into math: 
Solving:
(First multiply by 2 to clear the fraction)
Geometry Problems
Example: If the perimeter of a rectangle is 10
inches, and one side is one inch longer than the other,
how long are the sides?
Let one side be x and the other side be x + 1.
Then the given condition may be expressed as
x + x + (x + 1) + (x + 1) = 10
Solving:
4x + 2 = 10
4x = 8
x = 2
so the sides have length 2 and 3.
Rate-Time Problems
Rate = Quantity/Time
or
Quantity = Rate x Time
Example 1: A fast employee can assemble 7 radios
in an hour, and another s lower employee can only
assemble 5 radios per hour. If both employees work
together, how long will it take to assemble 26 radios?
The two together will build 7 + 5 = 12 radios in an
hour, so their combined rate is 12 radios/hr.
Using Time = Quantity/Rate, Time = 26/12 = 2 1/6 h
or
2 hours 10 minutes
Example 2: you are driving along at 55 mph when
you are passed by a car doing 85 mph. How long will
it take for the car that passed you to be one mile
ahead of you?
We know the two rates, and we know that the
difference between the two distances traveled will be
one mile, but we don’t know the actual distances. Let
D be the distance that you travel in time t, and D + 1
be the distance that the other car traveled in time t.
Using the rate equation in the form
distance = speed ยท time for each car we can write
D = 55 t, and D + 1 = 85 t
Substituting the first equation into the second,
55t + 1 = 85t
-30t = -1
t = 1/30 hr(or 2 minutes)
Mixture Problems
Example: How much of a 10% vinegar solution
should be added to 2 cups of a 30% vinegar solution
to make a 20% solution?
Let x be the unknown volume of 10% solution. Write
an equation for the volume of vinegar in each
mixture:
(amount of vinegar in first solution) + (amount of
vinegar in second solution) = (amount of vinegar in
total solution)
0.1x + 0.3(2) = 0.2(x + 2)
0.1x + 0.6 = 0.2x + 0.4
-0.1x = -0.2
x = 2 cups
