Let's solve each problem on the worksheet step by step and explain how to find the
constant of proportionality, denoted as
k.
---
🔹 What is the Constant of Proportionality?
In a proportional relationship, the constant of proportionality (
k) is the ratio of
y to x, or:
$$
k = \frac{y}{x}
$$
This means that for every unit of
x, you get
k units of
y. It’s also known as the
unit rate.
The general equation is:
$$
y = kx
$$
We will use this concept to solve each problem.
---
##
✔ Problem 1: Table (Vinegar and Oil)
| x (vinegar) | 1 | 2 | 3 | 5 |
|-------------|---|---|---|---|
| y (oil) | 3 | 6 | 9 | 15 |
We need to find $ k = \frac{y}{x} $ for any pair.
Try first pair:
$ \frac{3}{1} = 3 $
Check others:
- $ \frac{6}{2} = 3 $
- $ \frac{9}{3} = 3 $
- $ \frac{15}{5} = 3 $
✔ All ratios are 3 → So,
k = 3
✔️ Answer:
k = 3 ✔
---
##
✔ Problem 2: Equation — $ y = 6x $
This is already in the form $ y = kx $
So,
k = 6
✔️ Answer:
k = 6 ✔
---
##
✔ Problem 3: Equation — $ y = 12x $
Again, this is in the form $ y = kx $
So,
k = 12
✔️ Answer:
k = 12 ✔
---
##
✔ Problem 4: Table (Hours Worked vs. Earnings)
| x (hours) | 2 | 4 | 6 | 10 |
|-----------|---|---|---|----|
| y (dollars)| $30 | $60 | $90 | $150 |
Find $ k = \frac{y}{x} $
Try first:
$ \frac{30}{2} = 15 $
Check others:
- $ \frac{60}{4} = 15 $
- $ \frac{90}{6} = 15 $
- $ \frac{150}{10} = 15 $
✔ All ratios are 15 → So,
k = 15
⚠️ But the student wrote
k = $30 — this is incorrect.
💡 The
constant of proportionality is
$15 per hour, not $30.
✔️ Correct answer:
k = 15
>
✘ Mistake: The student likely took $30 (the first y-value) but forgot to divide by x=2.
---
##
✔ Problem 5: Graph (Mugs vs. Clay Used)
The graph shows a straight line through the origin (0,0), which indicates a proportional relationship.
We need to find
k = y/x using any point on the line.
Look at the graph:
- When x = 1 mug → y = 14 ounces
- When x = 2 mugs → y = 28 ounces
- When x = 3 mugs → y = 42 ounces
- When x = 4 mugs → y = 56 ounces
Use one point: $ \frac{14}{1} = 14 $
Check: $ \frac{28}{2} = 14 $, $ \frac{42}{3} = 14 $, $ \frac{56}{4} = 14 $
✔ So,
k = 14
But the student wrote “proportional” — which is not a number.
✘ That’s incorrect. The constant of proportionality is a
number, not just "proportional".
✔️ Correct answer:
k = 14
---
##
✔ Final Answers with Corrections:
| Problem | Given | Your Answer | Correct Answer | Explanation |
|--------|-------|-------------|----------------|------------|
| 1 | Table | k = 3 |
✔ k = 3 | All y/x = 3 |
| 2 | y = 6x | k = 6 |
✔ k = 6 | Coefficient of x |
| 3 | y = 12x | k = 12 |
✔ k = 12 | Coefficient of x |
| 4 | Table | k = $30 |
✘ k = 15 | $30/2 = $15 per hour |
| 5 | Graph | k = proportional |
✘ k = 14 | Use any point: 14 oz / 1 mug = 14 |
---
🟩 Summary:
The
constant of proportionality (k) is found by:
- For tables: $ k = \frac{y}{x} $ (must be same for all pairs)
- For equations: $ y = kx $ → k is the coefficient
- For graphs: pick any point (x, y), then $ k = \frac{y}{x} $
---
✔ Final Correct Answers:
1.
k = 3
2.
k = 6
3.
k = 12
4.
k = 15
5.
k = 14
Let me know if you'd like this printed or formatted for a student!
Parent Tip: Review the logic above to help your child master the concept of tables graphs and equations worksheet.
