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If there are 20 dogs and 60 cats at a pet daycare, fill out all the possible ratios of dogs and cats that could be made.

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To find all the possible ratios of dogs to cats when there are 20 dogs and 60 cats, we can express the ratio in the form of \( \frac{d}{c} \), where \( d \) is the number of dogs, and \( c \) is the number of cats.

Step 1: Set Up the Ratios

The total number of dogs is 20, and the total number of cats is 60. Therefore, the ratios we can create include:

  • The ratio of full numbers: \( \frac{20}{60} \)
  • Ratios that reduce these numbers by their greatest common divisor (GCD)

Step 2: Simplifying the Ratios

To simplify \( \frac{20}{60} \), we find the GCD of 20 and 60, which is 20:

\[ \frac{20}{60} = \frac{20 \div 20}{60 \div 20} = \frac{1}{3} \]

Step 3: List Possible Ratios

The ratios can also be represented in terms of other combinations. The possible ratios of dogs to cats can be represented as:

  1. \( \frac{20}{60} \) (20 dogs to 60 cats)
  2. \( \frac{19}{60} \) (19 dogs to 60 cats)
  3. \( \frac{18}{60} \) (18 dogs to 60 cats)
  4. \( \frac{17}{60} \) (17 dogs to 60 cats)
  5. \( \frac{16}{60} \) (16 dogs to 60 cats)
  6. \( \frac{15}{60} \) (15 dogs to 60 cats)
  7. \( \frac{14}{60} \) (14 dogs to 60 cats)
  8. \( \frac{13}{60} \) (13 dogs to 60 cats)
  9. \( \frac{12}{60} \) (12 dogs to 60 cats)
  10. \( \frac{11}{60} \) (11 dogs to 60 cats)
  11. \( \frac{10}{60} \) (10 dogs to 60 cats)
  12. \( \frac{9}{60} \) (9 dogs to 60 cats)
  13. \( \frac{8}{60} \) (8 dogs to 60 cats)
  14. \( \frac{7}{60} \) (7 dogs to 60 cats)
  15. \( \frac{6}{60} \) (6 dogs to 60 cats)
  16. \( \frac{5}{60} \) (5 dogs to 60 cats)
  17. \( \frac{4}{60} \) (4 dogs to 60 cats)
  18. \( \frac{3}{60} \) (3 dogs to 60 cats)
  19. \( \frac{2}{60} \) (2 dogs to 60 cats)
  20. \( \frac{1}{60} \) (1 dog to 60 cats)

Final Simplified Ratio

The overall simplest ratio remains \( \frac{1}{3} \) which represents the basic relationship in simplest form. Thus:

  • All possible ratios as fractions are \( \frac{20}{60}, \frac{19}{60}, \ldots, \frac{1}{60} \)
  • The simplest form of these ratios consistently shows the relationship between dogs and cats.

Summary of Possible Ratios

In conclusion, all the ratios of dogs to cats, while also accounting for the reduction of increased counts of dogs, will maintain a quantitative relationship expressed ultimately through their simplest fraction \( \frac{1}{3} \). Thus, you can express this either as fractions of the whole number of ratios or reduced into the lowest terms of 1:3.