Proportional Relationships — Practice
A proportional relationship is y = kx: a straight line through the origin, with one constant ratio running through every pair.
What these questions 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
Which equation represents a proportional relationship?
- y = 4x
- y = 4x + 3
- y = 4x
- y = x2
Worked answer
y = 4x
y = 4x has the form y = kx with k = 4.
The others add a constant, divide, or square — none is proportional.
Problem 2
A table shows x = 2, 4, 6 with y = 10, 20, 30. Find the constant of proportionality k.
Worked answer
5
102 = 5, 204 = 5, 306 = 5
k = 5
Problem 3
A proportional relationship passes through (3, 21). Find k.
Worked answer
7
k = 213 = 7
So y = 7x.
Problem 4
Every proportional graph must pass through which point?
- (1, 1)
- (0, 0)
- (1, 0)
- (0, 1)
Worked answer
(0, 0)
Zero of one means zero of the other, so the line goes through the origin (0, 0).
Problem 5
Five pencils cost $2.00. What do 12 pencils cost?
Worked answer
4.80
2.00 ÷ 5 = $0.40 each
0.40 × 12 = $4.80
Problem 6
A table shows x = 3, 5, 8 with y = 12, 20, 32. Find y when x = 11.
Worked answer
44
123 = 4, and the other rows agree, so k = 4
y = 4x
y = 4 × 11 = 44
Problem 7
Which relationship is NOT proportional?
- 8 apples for $4; 12 apples for $6
- A car travelling 50 miles every hour
- A gym charging $25 to join plus $15 a month
- 3 cups of flour per dozen cookies
Worked answer
A gym charging $25 to join plus $15 a month
With the gym, zero months still costs $25, so the graph misses (0,0).
It is linear (y = 15x + 25) but not proportional.
Problem 8
A car travels 224 miles on 7 gallons. How far on 11 gallons?
Worked answer
352
k = 2247 = 32 mpg
y = 32x
32 × 11 = 352 miles
Problem 9
A proportional graph passes through (4, 10). Find k as a decimal.
Worked answer
2.5
k = 104 = 2.5
Problem 10
Eight pounds of rice cost $20. How many pounds can you buy with $32.50?
Worked answer
13
20 ÷ 8 = $2.50 per pound
32.50 ÷ 2.50 = 13 pounds
Problem 11
A recipe uses 3 cups of flour for 24 cookies. How much flour for 60 cookies?
Worked answer
7.5
324 = 18 cup per cookie
18 × 60 = 7.5 cups
Problem 12
Printer A follows y = 18x pages in x minutes. Printer B prints 130 pages in 10 minutes. How many more pages per minute does the faster one print?
Worked answer
5
A: 18 pages per minute
B: 130 ÷ 10 = 13 pages per minute
18 - 13 = 5
Problem 13
In a proportional relationship, y = 45 when x = 12. Find x when y = 75.
Worked answer
20
k = 4512 = 3.75
75 = 3.75x
x = 75 ÷ 3.75 = 20
Problem 14
Line A passes through (2, 5) and line B through (3, 7). Both are proportional. Which is steeper?
- Line A
- Line B
- They have the same steepness
Worked answer
Line A
A: k = 52 = 2.5
B: k = 73 ≈ 2.33
2.5 > 2.33, so A is steeper.
Problem 15
On a map, 2.5 cm represents 15 km. How many km does 7 cm represent?
Worked answer
42
15 ÷ 2.5 = 6 km per cm
6 × 7 = 42 km
Problem 16
A table shows x = 2, 4, 7 with y = 7, 14, 24. Is it proportional?
- Yes, with k = 3.5
- No — the last pair breaks the pattern
- No — it does not pass through the origin
Worked answer
No — the last pair breaks the pattern
72 = 3.5 and 144 = 3.5 — so far so good
But 247 ≈ 3.43, which does not match
One disagreeing row is enough to rule it out.