"The ratio of cats to dogs is 3:5. If there are 20 dogs
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A ratio is a way to compare two quantities. It shows the relative size of one quantity to another. For example, if we have 3 circles and 5 squares, we say the ratio of circles to squares is 3 to 5, written as 3 colon 5.
The ratio 3 to 5 means that for every 3 cats, there are 5 dogs. This doesn't tell us the exact number of cats and dogs, but it tells us their relative quantities. If we have 3 cats, we must have 5 dogs. If we have 6 cats, we must have 10 dogs, maintaining the same 3 to 5 relationship.
Ratios can be written as fractions, which allows us to set up proportions. If the ratio of cats to dogs is 3 to 5, we can write this as cats over dogs equals 3 over 5. When we have actual numbers, we can substitute them into this proportion to solve for unknowns. This is the foundation of solving ratio problems.
Now we know there are 20 dogs, and we need to find how many cats there are. In our ratio 3 to 5, the 5 represents the dog portion. Since 5 units correspond to 20 dogs, we can find the scale factor by dividing 20 by 5, which equals 4. This means each unit in our ratio represents 4 actual animals. We'll use this scale factor to find the number of cats.
Now we can calculate the number of cats. Since our ratio is 3 to 5, and we've determined that each unit represents 4 animals, we multiply 3 units by 4 to get 12 cats. We can verify this is correct by checking if 12 cats to 20 dogs simplifies to our original ratio of 3 to 5. When we divide both 12 and 20 by their greatest common divisor, 4, we get 3 and 5, confirming our answer is correct.
Let's summarize the steps we took to solve this ratio problem. First, we identified the given ratio of cats to dogs as 3 to 5. Second, we found the scale factor by dividing the known quantity of dogs (20) by the dog portion of the ratio (5), giving us a scale factor of 4. Third, we calculated the unknown quantity of cats by multiplying the cat portion of the ratio (3) by the scale factor (4), resulting in 12 cats. We verified our answer by confirming that 12 cats to 20 dogs simplifies to the original ratio of 3 to 5. Therefore, there are 12 cats.