Direct variation means y = kx. Doubling x doubles y, and the graph is a line through the origin.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Inverse variation
Inverse variation means y = k ÷ x. Doubling x halves y, and the graph is a hyperbola that never reaches either axis.
Finding k
One matching pair of values is enough. Substitute them, solve for k, then use the finished equation for every other question.
Joint variation
Joint variation depends on more than one quantity at once, as in y = kxz. Area varies jointly with length and width.
Combined variation
Some quantities vary directly with one thing and inversely with another, such as y = kx ÷ z.
A quick test
If y ÷ x stays constant, the variation is direct. If x × y stays constant, it is inverse.
Direct variation
y varies directly as x means y = kx: doubling x doubles y, and the graph is a line through the origin. It is the proportional relationship from Grade 7, now named and generalised.
Inverse variation
y varies inversely as x means y = k/x: doubling x halves y, and the product xy stays constant. The graph is a hyperbola with both axes as asymptotes. Speed and time for a fixed distance behave this way.
Joint and combined variation
y = kxz is joint variation — y varies directly with both. Combined variation mixes direct and inverse, as in y = kx/z. Reading which variables sit on top and which underneath is the whole of the translation step.
Find k from one data point
Substitute a known pair into the relationship to compute k, then use the completed formula for every other question. Every variation problem has this two-stage structure, and skipping the first stage is the usual mistake.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Omar Model a Journey
At a fixed distance, travel time varies inversely with speed. At 60 km/h the trip takes 3 hours.
- Step 1
Inverse variation means t = k ÷ s.
Step 4: Your Turn
Practice makes it stick.
The Wages
Problem 1 of 2
Pay varies directly with hours. 6 hours pays $90. What does 10 hours pay, in dollars?
dollars
The Workers
Problem 2 of 2
Time varies inversely with the number of workers. 4 workers take 12 days. How many days do 6 workers take?
days
Find the Constant
1 of 8
y varies directly with x, and y = 20 when x = 4. What is k?
2 of 8
Same relationship. What is y when x = 7?
3 of 8
y varies inversely with x, and y = 6 when x = 4. What is k?
4 of 8
Same relationship. What is y when x = 8?
5 of 8
y varies jointly with x and z, with k = 2. What is y when x = 3 and z = 5?
6 of 8
Pressure varies inversely with volume, and k = 100. What is the pressure when the volume is 25?
7 of 8
Sort each pairing by the kind of variation it shows.
Tap something to move it.
- Empty
- Empty
8 of 8
y varies directly with x. When x = 9, y = 27. What is x when y = 12?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y varies inversely with x, and y = 10 when x = 3. What is k?
Question 2 of 2
Which graph belongs to direct variation?
What You Learned
- Direct variation is y = kx; inverse variation is y = k ÷ x.
- One matching pair of values is enough to find k.
- Joint variation depends on two quantities at once, as in y = kxz.