1. MATHEMATICAL KEY POINTS
Word Problem Solving Strategies
Read through the entire problem.
Highlight the important information and key words that you need to solve the problem.
Identify your variables.
Write the equation or inequality.
Solve.
Write your answer in a complete sentence.
Check or justify your answer.
Solving Work Problems Involving Rational Equations
Work problems often ask us to calculate how long it will take different people working at different
speeds to finish a task. The algebraic models of such situations often involve rational equations derived
from the work formula, W = rt. The amount of work done (W) is the product of the rate of work (r) and
the time spent working (t). The work formula has 3 versions:
Some work problems have multiple machines or people working on a project together for the same
amount of time but at different rates. In that case, we can add their individual work rates together to
get a total work rate. Let’s look at an example:
Example
Problem Myra takes 2 hours to plant 500 flower bulbs. Francis takes 3
hours to plant 450 flower bulbs. Working together, how
long should it take them to plant 1500 bulbs?
Myra: 500 bulbs/2 hours = 250
bulbs/hour
Francis: 450 bulbs/3 hours =
150 bulbs/hour
Think about how many bulbs
each person can plant in one
hour. This is their planting rate.
Myra & Francis together:
250 + 150 bulbs/hour = 400
bulbs/hour
Combine their hourly rates to
determine the rate they work
together.
2. Use one of the work formulas
to write a rational equation, for
example . We know r,
the combined work rate, and
we know W, the amount of
work that must be done. What
we don't know is how much
time it will take to do the
required work at the
designated rate.
t = 3 ¾ hours
Solve the equation by
multiplying both sides by the
common denominator, then
isolating t.
Answer It takes 3 hours 45 minutes for Myra and Francis to plant 1500 bulbs
together.
Inequality Key Words
at least - means greater than or equal to
no more than - means less than or equal to
more than - means greater than
less than - means less than
Example 1: Inequality Word Problems
Keith has $500 in a savings account at the beginning of the summer. He wants to have at least $200 in
the account by the end of the summer. He withdraws $25 each week for food, clothes, and movie
tickets.
Write an inequality that represents Keith's situation.
How many weeks can Keith withdraw money from his account? Justify your answer.
3. Solution:
Step 1: Highlight the important information in this problem.
Note: At least is a key word that notes that this problem must be written as an inequality.
Step 2: Identify your variable. What don't you know? The question verifies that you don't know how
many weeks.
Let w = the number of weeks
Step 3: Write your inequality.
500 - 25w > 200
Step 4: Solve the inequality.
The number of weeks that Keith can withdraw money from his account is 12 weeks or less.
Step 5: Justify (prove your answer mathematically).
I'm going to prove that the largest number of weeks is 12 by substituting 12 into the inequality for w.
You could also substitute any number less than 12.
4. Rational Functions Word Problems - Work, Tank and Pipe
Here are a few examples of work problems that are solved with rational equations.
Examples:
1. Sam can paint a house in 5 hours. Gary can do it in 4 hours. How long will it take the two working
together?
2. Joy can file 100 claims in 5 hours. Stephen can file 100 claims in 8 hours. If they work together,
how long will it take to file 100 claims?
3. A water tank is emptied through two drains in 50 minutes. If only the larger drain is used, the tank
will empty in 85 minutes. How long would it take to empty if only the smaller drain is used?
4. One computer can run a sorting algorithm in 24 minutes. If a second computer is used together
with the first, it takes 13 minutes. How long would it take the second computer alone?
5. Two pipes are filling a tank. One pipe fills three times as fast as the other. With both pipes
working, the tank fills in 84 minutes. How long would each pipe take working alone?
