Using Algebra to Solve Everyday Problems

Frequently, the best way to solve an abstract problem is to substitute actual numbers for variables and see how the numbers relate and how they can be moved around. You can also use your intuition about real problems to solve problems using variables.

Here are some related problems:

1a) I bought 6 sandwiches at 6 dollars each plus 4 fancy coffees that were discounted a dollar from their normal price of $4. Write the equation and say how much money was spent.

1b) I bought “s” number of sandwiches at a cost of “s” dollars plus 4 coffees at a dollar off their normal price “p”. If the original cost of a coffee is 3 dollars less than the cost of a sandwich, write two equations that help you determine the amount spent.

Find the cost if the original price of coffee is $3. Find the amount of money spent if the original price of coffee is $4.

2a) A train delivery of goods takes an average of 8 hours. A comparable truck delivery can be more conveniently scheduled, but takes approximately 4 times more fuel. What is the difference in fuel cost, if the truck drives 500 miles at an average of 6.5 miles per gallon fuel efficiency and each gallon of fuel costs $2.90 a gallon.

2b) Use the same distance and fuel efficiency to calculate the difference in fuel cost if the cost of gasoline rises to $3.10 a gallon. What if the price drops to $2.80 a gallon?

3a) In the United States, Ford Mustang deliveries totaled 15,717 cars in Q2 (the second quarter of the year) 2020, a decrease of about 27 percent compared to 21,625 cars sold in Q2 2019 (info from fordauthority.com). If the sales were distributed equally over the 3,000 sales locations (dealerships), how many car sales would each dealership have this quarter. If the average sale price for a Mustang is $30,000 and the car dealership makes a 5% profit from each sale, how much money should the dealership have made  this quarter?

3b) How much money would the dealership make if they offered a sale and made only 4% profit for the same number of cars?

4a) You want to put together a budget for your pet spending, allowing $600 a year to spend on your pet. If you spend $20 a week on food and budget $200 a year for visits to the vet, how much money can you spend on boarding your dog when you go on vacation and how much can you spend for other supplies and pet toys. Is there enough money in your budget? Compare to the average yearly spending of all pet owners put together by the American Pet Products Association below.

4b) In 2019, about $95.7 billion was spent on pet purchases and care. Figure out the percentages spent on these four categories, given the total amount spent:

Food and Treats: $36.9 billion

Veterinary care: $29.3 billion

Pet boarding and grooming & pet insurance: $10.3 billion

Pet supplies and purchase of small pets: $19.2 billion

If pet ownership and expenses increase by 4 percent in 2020, predict how much will be spent in each category if the percentages remain the same.

A note about percent increase and percent decrease:

The percent that something goes up or down is relative to the starting amount.

Calculate the percent change from a to b:  

Always divide by the starting amount when calculating percent increase or decrease.

5a) A resident on Lake Audubon bought a pontoon boat for $10,000. If the owner plans to sell the boat in 5 years for $2,000, what percent of the purchase price did the owner lose?

5b) Over the boat owner’s 5 years of ownership, the boat was taken out an average of 30 times per year for an average of 2 hours per trip. What is the net cost of the boat per hour of use. If you could rent a similar boat for $ 85 per hour, which option is more cost effective, ownership or rental?

Bonus question: If you don’t consider the cost of gasoline or maintenance, for how many hours per year could you rent a boat before you spent $8,000? How many hours per year would you have to take the boat out in order for the cost of ownership per year to be the same as boat rental or more?

6a) Six members of the high school girl’s basketball team each played more than 20 percent of the game. For each player, calculate their field goal (FG) percentage, their free throw (FT) percentage (percent of made shots vs. shots attempted), their outside shot percentage and determine the player with the best weighted average scoring percentage overall (free throws are worth 1 point, field goals 2 points, and outside shots 3 points each).

Name                FG attempts   FG made     FT attempts     FT made    OS attempts    OS made

6b) Come up with a general formula for calculating the field goal percentage and the weighted average for a player.

Make up some more real problems and solve them!