Problem Solving Properties
"For Complex Problems Use Step-by-Step Solutions"
COMPREHENSION
- Read the problem carefully. Take note of all directions.
- Make sure you understand the described situation.
- Make sure you understand the provided information, and what the question is asking.
- If you don't understand, try drawing a clearly labeled picture. Or ask for support from
your professor or tutors in the ASC.
PLANNING
- Focus on the problem. What do you need to know in order to answer the question?
- Look at the given information. How can you use the provided information to answer the
question? - If you do not see a clear logical path leading from the given information to the solution,
do NOT panic. Attempt a solution. Think about what you can gain from the given
information, even if it does not relate to the question.
(HINT: You will often find that one solution can lead to another.)
WRITING EQUATIONS
You need to express mathematically the logical connections between the given information and
the answer you are seeking. This involves:
- Assigning variable names to 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 assigning
variables is that you want 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 2 years older than John is, you might let x stand for John's age
and (x + 2) stand for Jim's age. - Translate English into Math.
Mathematics is a language particularly well suited to describing logical relationships.
English, on the other hand, is much less precise. Below you will see a table of English
phrases and their corresponding mathematical meanings, but don't take it too literally.
The meaning of English words has to be taken in context.
SOLVING
With a plan in place, you now have to simply solve the equation(s) for the unknown factors.
Remember to answer the question that the problem asks.
CHECK
Think about your answer. Does your answer come to the correct units? Is it reasonable? If you
made a mistake along the way, chances are your answer will not just be a little bit off, but will be
completely off.
WORDS FOR OPERATIONS
ENGLISH TO MATH
NOTE: The English language is notoriously imprecise, and these suggested mathematical
translations should be taken only as a guideline. All italicized letters are variables.
| Subtraction |
minus | "a number minus 2" | x - 2 |
| difference between | "the difference between a number and 8" | x - 8 | |
| from | "2 from a number" | n - 2 | |
| less | "a number less 3" | n - 3 | |
| less than | "3 less than a number" | y - 3 | |
| fewer than | "2 fewer than a number" | y - 2 | |
| decreased by | "a number decreased by 2" | x - 2 | |
| take away | "a number take away 2" | x - 2 | |
|
Addition (NOTE: in addition, the equations can be either order) |
plus | "a number plus 2" | x + 2 or 2 + x |
| and | "3 and a number" | n + 3 or 3 + n | |
| added to | "8 added to a number" | n + 8 or 8 + n | |
| greater than | "3 greater than a number" | n + 3 or 3 + n | |
| more than | "3 more than a number" | n + 3 or 3 + n | |
| increased by | "a number increased by 2" | y + 2 or 2 + y | |
| total | "the total length" | l1 + l 2 | |
| sum of | "the sum of the length and width" |
l + w | |
| Multiplication | times | "5 times a number" | 5n |
| product | "the product of 3 and a number" |
3y | |
| at | "3 at 2" | 3 x 2 | |
| double, triple, etc. | "double a number" | 2x | |
| twice | "twice a number" | 2y | |
| of (fraction of) | "three-fourths of a number" | 3/4y | |
| Division | quotient of | "the quotient of 5 and a number" |
5 / n |
| half of | "half of a number" | n / 2 | |
| goes into | "a number goes into 8 twice" |
8 / n = 2 | |
| per | "the price is $2 per 50" | 2/50 | |
| divided by | "6 divided by a number" | 6 / n | |
| Equals | is, is the same as, gives, will be was, is equivalent to |
= | |
GENERAL WORD PROBLEMS
Recall the general strategy for setting up problems. Refer to the top for more detail.
- Read the problem carefully: Determine what is known, what is unknown, and what the
question is asking. - Represent the unknown quantities in terms of variables ( x, y, n, l)
- Use diagrams where appropriate.
- Find formulas or mathematical relationships between the known and unknown.
- Solve the equation for the unknown.
- Check answers to see if they are reasonable.
NUMBER / GEOMETRY 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:
5 + x / 2 = 3x
Solving: (First multiply by 2 to clear the fraction)
5 + x / 2 = 3x
10 + x = 6x
10 = 5x
2 = x
EXAMPLE: If the perimeter of a rectangle is 10 inches, and one side is 1 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:
Bring all x's together x + x + (x + 1) + (x + 1) = 10
4 x + 2 = 10
4 x = 8
x = 2
RATE-TIME PROBLEMS
- Rate = Quantity / Time
EXAMPLE: A fast employee can assemble 7 radios per hour, and another slower 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 per hour, so their combined rate is 12 radios
per hour or 12 radios/hr.
Using time = Quantity / Rate, time = 26 / 12 = 2 1/6 hours OR 2 hours, 10 minutes.
EXAMPLE: You are driving along at 55mph when you are passed by a car doing 85mph. 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 x time for each car we can write...
D = 55 t, and D + 1 = 85 t
Substituting the first equation into the second,
55 t + 1 = 85 t
-30 t = -1
t = 1 / 30hr OR 2 minute
MIXED 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? (HINT: Turn percent into decimal form)
Let x be the unknown amount of 10% solution. Write an equation for the amount of
vinegar in each mixture:
(amount of vinegar in first solution) + (amount of vinegar in second solution) = (amount
of vinegar in total solution)
0.1 x + 0.3 (2) = 0.2 (x + 2)
0.1 x + 0.6 = 0.2 x + 0.4
-0.1 x = -0.2
x = 2 cups
