1. An Introduction to Problem Solving
§ 2.5
An Introduction to Problem Solving

2. Strategy for Problem Solving
General Strategy for Problem Solving
UNDERSTAND the problem
Read and reread the problem
Choose a variable to represent the unknown
Construct a drawing, whenever possible
Propose a solution and check
TRANSLATE the problem into an equation
SOLVE the equation
INTERPRET the result
Check the proposed solution in the problem
State your conclusion

3. Finding an Unknown Number
Example:
The product of twice a number and three is the same as the sum of five times the number and 12. Find the number.
1.) UNDERSTAND
Read and reread the problem. If we let
x = the unknown number, then “twice a number” translates to 2x,
“the product of twice a number and three” translates to 2x · 3,
“five times the number” translates to 5x, and
“the sum of five times the number and 12” translates to 5x + 12.
Continued

4. Finding an Unknown Number
Example continued:
2.) TRANSLATE
The product of
the sum of +
is the same as =
twice a number 2x
5 times the number 5x
and 3 3
and 12 12
Continued

5. Finding an Unknown Number
Example continued:
3.) SOLVE
2x · 3 = 5x + 12
6x = 5x Simplify left side.
6x + (– 5x) = 5x + (– 5x) Add –5x to both sides.
x = Simplify both sides.
4.) INTERPRET
Check: Replace “number” in the original statement of the problem with 12. The product of twice 12 and 3 is 2(12)(3) = 72. The sum of five times 12 and 12 is 5(12) + 12 = 72. We get the same results for both portions.
State: The number is 12.

6. Solving a Problem Example:
A car rental agency advertised renting a Buick Century for $24.95 per day and $0.29 per mile. If you rent this car for 2 days, how many whole miles can you drive on a $100 budget?
1.) UNDERSTAND
Read and reread the problem. Let’s propose that we drive a total of 100 miles over the 2 days. Then we need to take twice the daily rate and add the fee for mileage to get 2(24.95) (100) = = This gives us an idea of how the cost is calculated, and also know that the number of miles will be greater than If we let
x = the number of whole miles driven, then
0.29x = the cost for mileage driven
Continued

7. Solving a Problem Example continued: 2.) TRANSLATE Daily costs
2(24.95)
mileage costs
0.29x
100
maximum budget
plus +
is equal to =
Continued

8. Solving a Problem Example continued: 3.) SOLVE 2(24.95) + 0.29x = 100
x = Simplify left side.
Subtract from both sides.
49.90 – x = 100 – 49.90
0.29x = Simplify both sides.
Divide both sides by 0.29.
x  Simplify both sides.
Continued

9. Solving a Problem Example continued: 4.) INTERPRET
Check: Recall that the original statement of the problem asked for a “whole number” of miles. If we replace “number of miles” in the problem with 173, then (173) = , which is over our budget. However, (172) = 99.78, which is within the budget.
State: The maximum number of whole number miles is 172.