Select all the proportional relationships.

Question

A

B

C

D
y=13xy=\frac{1}{3}xy=
3
1
​
x, where xxx and yyy are both positive numbers.

E
Clare borrowed $100\$100$100 from her father. She pays him back $10\$10$10 each week. MMM is the amount of money Clare still owes, and WWW is the number of weeks: 100−10W=M100-10W=M100−10W=M

Answer 1

To determine which relationships are proportional, we need to see if there exists a constant ratio between the two variables involved.

1. A: y = 13x

   • This is a proportional relationship, where for every unit increase in x, y increases by a constant multiple (13). The ratio \( \frac{y}{x} = 13 \) is constant.

2. B: y = \frac{1}{3}x

   • This is also a proportional relationship, as for every unit increase in x, y increases by a constant multiple (\(\frac{1}{3}\)). The ratio \( \frac{y}{x} = \frac{1}{3} \) remains constant.

3. C: y = 3x

   • This is a proportional relationship as well since y increases by a factor of 3 for every increase in x. The ratio \( \frac{y}{x} = 3 \) is constant.

4. D: y = 1x

   • This is also a proportional relationship since it simplifies to \( y = x \). The ratio \( \frac{y}{x} = 1 \) is constant.

5. E: 100 − 10W = M

   • This is a linear relationship, but it is not directly proportional because it includes a constant term (100). The relationship between M and W is not consistent in terms of a constant ratio; it changes as W increases.

Based on the analysis above, the proportional relationships are:

• A: \( y = 13x \)
• B: \( y = \frac{1}{3}x \)
• C: \( y = 3x \)
• D: \( y = 1x \)

Therefore, A, B, C, and D are proportional relationships. E is not proportional.