Let's go through the questions one by one.

13. Tell whether the missing values are equivalent. Justify your answer.

• The first part of the problem says, "9 miles for every ½ hour" and "6 miles for every 20 minutes."
  • To determine if they are equivalent, we can convert both values to the same time unit. Let's convert both to miles per minute.
    • 9 miles per ½ hour = 9 miles per 30 minutes = 9/30 = 0.3 miles per minute.
    • 6 miles per 20 minutes = 6/20 = 0.3 miles per minute.
  • Since both values are 0.3 miles per minute, the rates are indeed equivalent.

14. Tell whether the missing values are equivalent. Justify your answer.

• The first part says, "2/5 gram for every 1/10 serving" and "5/2 grams for every ¼ serving."
  • Let's convert both to grams per serving.
    • 2/5 gram per 1/10 serving = (2/5) ÷ (1/10) = (2/5) × (10/1) = 4 grams per serving.
    • 5/2 grams per ¼ serving = (5/2) ÷ (1/4) = (5/2) × (4/1) = 10 grams per serving.
  • Since 4 grams per serving is not equal to 10 grams per serving, these values are not equivalent.

15. You run 3 miles in 24 minutes. What is your average rate of speed in miles per minute?

• The average speed is given by the formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ So: $\text{Speed} = \frac{3 \, \text{miles}}{24 \, \text{minutes}} = 0.125 \, \text{miles per minute}$

16. Dishwasher detergent is sold in individual packets. You can buy a box of 20, 60, or 90 packets.

• (a) The 20-packet box sells for $5.49. What is the unit rate?

  • Unit rate = Price per packet = $\frac{5.49}{20} = 0.2745$ dollars per packet.

• (b) The 60-packet box sells for $10.97. What is the unit rate?

  • Unit rate = Price per packet = $\frac{10.97}{60} = 0.1828$ dollars per packet.

• (c) The 90-packet box sells for $18.95. What is the unit rate?

  • Unit rate = Price per packet = $\frac{18.95}{90} = 0.2106$ dollars per packet.

• (d) Which box is the best value? Explain your answer.

  • The 60-packet box has the lowest unit rate of $0.1828 per packet, so it is the best value.

Would you like any further details or clarification on these answers? Here are some related questions to explore further:

1. How do you convert different units to compare rates (e.g., miles per minute to miles per hour)?
2. What is the formula for calculating the unit rate for price per packet in any given situation?
3. Why is it important to convert all units to the same form when comparing rates?
4. How would you calculate average speed for a different scenario, such as a bike ride?
5. How can you apply unit rate comparisons in everyday shopping situations to find the best deal?

Tip: When comparing rates, always convert them to the same unit of measurement to make accurate comparisons!