Name:____________________________________________Class:_____Date:___________
Algebra 1 Linear Word Problems
Direct Variation problems:
example: The number s of tablespoons of sea salt needed in a saltwater fish tank
varies directly with (depends on) the number w of gallons of water in the tank. A pet shop
owner recommends adding 100 tablespoons of sea salt to a 20 gallon tank.
Write a direct variation equation that relates w and s.
How many tablespoons of salt should be added to a 30 gallon saltwater fish tank?
1. The distance d (in meters) you travel on a bicycle varies directly with the number r of
revolutions that the rear tire completes. You travel about 2 meters on a mountain bike for
every revolution of the tire.
a) Write a direct variation equation that relates r and d.
b) How many meters do you travel in 1500 tire revolutions?
2. At one company, the amount of vacation v (in hours) an employee earns varies directly with
the amount of time t (in weeks) he or she works. An employee who works 2 weeks earns 3
hours of vacation.
a) Write a direct variation equation that relates t and v.
3. How many hours of vacation time does an employee earn in 8 weeks?Landscapers plan to
spread a layer of stone on a path. The number s of bags of stone needed depends on the
depth d (in inches) of the layer. They need 10 bags to spread a layer of stone that is 2
inches deep. Write a direct variation equation that relates d a s. Then find the number of
bags needed to spread a layer that is 3 inches deep.
Proportion Problems:
example: Each day, the seals at an aquarium are each fed 8 pounds of food for every 100
pounds of their body weight. A seal at the aquarium weighs 280 pounds. How much food should
the seal be fed per day?
Write a proportion using the words that describe what the numbers represent.
Write a proportion with the correct numbers, and solve,
4. A recipe that yields (that means that how many it makes) 12 buttermilk biscuits calls for 2
cups of flour. How much flour is needed to make 30 biscuits?
5. It took 7.2 minutes to upload 8 digital photographs taken from your computer to a website.
At this rate, how long will it take to upload 20 photographs?
6. An exhibit at Tobu World Square in Japan includes a scale model of the Empire State
Building. The model was built using a scale of 1 m to 25 m (1 m: 25 m). The actual height of
the Empire State Building is 443.2 meters. What is the height of the model?
Slope and Intercept Problems:
Example: To get from one floor to another at a library, you can take either the stairs or the
escalator. You can climb stairs at a rate of 1.75 feet per second, and the escalator rises at
a rate of 2 feet per second. You have to travel a vertical distance of 28 feet. The
equations model the vertical distance d (in feet) you have left to travel after t seconds.
Stairs: d = -1.75t + 28 Escalator: d = -2t + 28
Graph the equations in the same coordinate plane.
How much time do you save by taking the escalator?
Example: A company produced two 30 second commercials, one for $300,000 and the
second for $400,000. Each airing of either commercial on a particular station costs
$150,000. The cost C (in thousands of dollars) to produce the first commercial and air it n
times is given by C = 150n + 300. The cost to produce the second and air it n times is given
by C = 150n + 400.
Graph the equations in the same coordinate plane.
Based on the graphs, what is the difference of the costs to produce each commercial and air
it 2 times? 4 times? What do you notice about the differences of the costs?
7. Your family spends $60 on tickets to a hockey game and $4 per hour for parking. The
total cost C (in dollars) is given by C = 60 + 4t where t is the time (in hours) your family's
car is parked.
a) Graph the equation.
b) Suppose the parking fee is raised to $5.50 per hour so that the total cost of tickets and
parking for t hours is C = 60 + 5.5t. Graph the equation in the same coordinate plane as the
equation in part a).
c) How much more does it cost to go to a game for 4 hours after the parking fee is raised?
8. A service station charges $40 per hour for labor plus the cost of parts to repair a car.
Parts can either be ordered from the car dealership for $250 or from a warehouse for
$200. The equations below give the total repair cost C (in dollars) for a repair that takes t
hours using parts from the dealership or from the warehouse.
Dealership: C = 40t + 250 Warehouse: C = 40t + 200
a) Graph both equations in the same coordinate plane.
b) Use the graphs to find the difference of the costs if the repair takes 3 hours. What if
the repair takes 4 hours? What do you notice about the differences of the costs?
LinearWord Problems:
Example: A recording studio charges musicians an initial fee of $50 to record an album.
Studio time costs an additional $35 per hour.
Write an equation that gives the total cost of an album as a function of studio time (in
hours).
Find the total cost of recording an album that takes 10 hours of studio time.
9. The initial fee to have a website set up using a server is $48. It costs $44 per month to
maintain the website. Write an equation that gives the total cost of setting up and
maintaing a website as a function of the number of months it is maintained. Find the total
cost of setting up and maintaining the website for 6 months.
10. A cameral shop charges $3.99 for an enlargement of a photograph. Enlargements can be
delivered for a charge of $1.49 per order. Write an equation that gives the total cost of an
order with delivery as a function of the number of enlargements. Find the total cost of
ordering 8 photograph enlargements with delivery.
11. Your family spends $30 for tickets to an aquarium an d$3 per hour for parking. Write an
equation that gives the total cost of your family's visit to the aquarium as a function of the
number of hours that you are there Find the total cost of 4 hours at the aquarium.
Point/Rate Word Problems:
example: Your gym membership costs $33 per month after an initial membership fee. You
paid a total of $228 after 6 months. Write and equation that gives the total cost as a
function of the length of your gym membership (in months). Find the total cost after 9
months.
12. A carnival charges $2.50 per ride after an entrance fee. You paid a total of $22.50 after
6 rides. Write an equation that gives the total cost as a function of the number of rides.
Find the total cost for 15 rides.
Two Point Word Problems:
example: In Bicycle Moto Cross (BMX) racing, racers purchase a one year membership to a
track. They also pay an entry fee for each race at that track. One racer paid a total of
$125 after 5 races. A second racer paid a total of $170 after 8 races. How much does the
track membership cost? What is the entry fee per race?
13. You have a subscription to an online magazine that allows you to view 25 articles from
the magazine's archives. You are charged an additional fee for each article after the
first 25 articles viewed. After viewing 28 archived articles, you paid a total of
$34.80. After viewing 30 archived articles, you paid a total of $40.70. What is the
cost per archived article after the first 25 articles are viewed? What is the cost of
the magazine subscription?
14. Four years after a hedge maple tree was planted, its height was 9 feet. Eight years
after it was planted, the hedge maple tree's height was 12 feet. What is the growth
rate of the hedge maple? What was its height when it was planted?
