Proportional Relationships — Challenge
A proportional relationship is y = kx: a straight line through the origin, with one constant ratio running through every pair.
What these problems cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Unit rate — A rate with denominator 1 — how much for exactly one.
- Constant of proportionality — The unit rate, written k, in y = kx.
- Proportional — Two quantities with a constant ratio: yx = k always.
- Origin — The point (0, 0), which every proportional graph passes through.
- Scale — What one grid step is worth — and where the axis starts.
Problem 1
If y=4x, what is y when x=7?
- 11
- 21
- 24
- 28
- 47
Worked answer
28
Substitute x=7 into the rule y=4x.
y = 4· 7 = 28.
The rule multiplies by 4, so doubling x would double y as well.
Problem 2
The quantity y is proportional to x, and y=15 when x=3. What is y when x=8?
- 20
- 24
- 35
- 40
- 45
Worked answer
40
Proportional means y = kx for a fixed k. From 15 = k· 3 we get k = 5.
Then y = 5· 8 = 40.
Adding 5 to y because x rose by 5 would give 20, which treats the relationship as additive rather than proportional.
Problem 3
A proportional relationship passes through (2,9) and (4,18). What is its constant of proportionality?
- 2
- 4.5
- 5
- 7
- 9
Worked answer
4.5
The constant of proportionality is yx, which must be the same at every point.
At (2,9) it is 92 = 4.5, and at (4,18) it is 184 = 4.5 as well, which confirms the relationship is proportional.
So the constant is 4.5.
Problem 4
Three pencils cost $1.50. At the same rate, what do ten pencils cost in dollars?
- 0.50
- 3.00
- 5.00
- 4.50
- 15.00
Worked answer
5.00
One pencil costs 1.503 = 0.50 dollars.
Ten pencils cost 10· 0.50 = 5.00 dollars.
Problem 5
The line y=kx passes through the point (6,9). What is k?
- 0.67
- 1.5
- 2
- 3
- 15
Worked answer
1.5
Substitute the point: 9 = k· 6.
Divide by 6: k = 96 = 1.5.
Dividing the other way round gives 0.67, which is the reciprocal and not the constant here.
Problem 6
Which of these rules is not a proportional relationship?
- y=3x
- y=x2
- y=2x+1
- y=0.5x
- y=10x
Worked answer
y=2x+1
A proportional relationship has the form y = kx, so its graph is a straight line through the origin.
Putting x=0 into y=2x+1 gives y=1, not 0, so that line misses the origin.
Every other option is k times x with no constant added, so y=2x+1 is the exception.
Problem 7
Five kilograms of flour cost $20. At the same rate, what do eight kilograms cost in dollars?
- 25
- 28
- 32
- 36
- 40
Worked answer
32
One kilogram costs 205 = 4 dollars.
Eight kilograms cost 8· 4 = 32 dollars.
Problem 8
On a map, 1 centimetre represents 50 kilometres. How many kilometres does 7 centimetres represent?
- 150
- 250
- 300
- 350
- 400
Worked answer
350
The relationship is proportional with constant 50 kilometres per centimetre.
7· 50 = 350 kilometres.
Problem 9
If y=6x, what is x when y=42?
- 6
- 7
- 8
- 36
- 48
Worked answer
7
Substitute y=42: 42 = 6x.
Divide both sides by 6: x = 7.
Problem 10
A recipe for four servings uses 300 grams of flour. How many grams are needed for ten servings?
- 600
- 700
- 750
- 800
- 1200
Worked answer
750
One serving uses 3004 = 75 grams.
Ten servings use 10· 75 = 750 grams.