Word Problems: Direct Variation
Direct variation type word problems encompass one variable, which is directly proportional to another variable:
y = kx,
where k is referred to as the constant of proportionality.
Direct variation word problems often contain verbiage such as:
- "varies directly as"
- " varied directly as"
- "directly proporation to"
1) Recognizing the word problem consists of direct variation as exemplified by presence of verbiage listed above;
2) Writing an equation containing known values for both variables resulting in an expression containing only the unknown representing the constant of proportionality;
3) Solving the equation for the constant of proportionality;
4) Using the calculated constant of proportionality to determine the value of one of the variables given the other.
Example:
Solution:
The amount of money raised at a charity
fundraiser is directly proportional to the number of attendees.
The amount of money raised for five attendees was $100. How
much money will be raised for 60 attendees?
Solution:
Step 1:
The problem may be recognized as relating to direct variation due to the presence of the verbiage "is directly proportional to";
Step 2:
Using:
y = Money Raised at Fundraiser
x = Number of Fundraiser Attendees
k = Constant of Proporationality
y = kx
Knowing $100 was raised by five attendees:
100 = 5k
Step 3:
5k = 100
k = 20
Step 4:
When x = 60 attendees, what is dollar amount raised, y, using k = 20?
y = 20x
y = 20(60)
y = $1,200
The problem may be recognized as relating to direct variation due to the presence of the verbiage "is directly proportional to";
Step 2:
Using:
y = Money Raised at Fundraiser
x = Number of Fundraiser Attendees
k = Constant of Proporationality
y = kx
Knowing $100 was raised by five attendees:
100 = 5k
Step 3:
5k = 100
k = 20
Step 4:
When x = 60 attendees, what is dollar amount raised, y, using k = 20?
y = 20x
y = 20(60)
y = $1,200