Sure, let's solve the problem step by step.
1. Understanding the ratio: The ratio between dogs and cats is 4:10. This means that for every 4 dogs, there are 10 cats.
2. Setting up the ratio: We can write the ratio as a fraction. Therefore:
[tex]\[
\frac{\text{dogs}}{\text{cats}} = \frac{4}{10}
\][/tex]
3. Assigning the known value: We know from the problem that there are 14 dogs. So we need to find out how many cats correspond to these 14 dogs while maintaining the same ratio.
4. Using a proportion: We set up a proportion where \( x \) is the number of cats. Thus, the proportion is:
[tex]\[
\frac{4}{10} = \frac{14}{x}
\][/tex]
5. Cross-multiplying to solve for \( x \): To solve for \( x \), we cross-multiply:
[tex]\[
4x = 10 \cdot 14
\][/tex]
6. Solving the multiplication:
[tex]\[
4x = 140
\][/tex]
7. Dividing to isolate \( x \): Finally, divide both sides by 4 to isolate \( x \):
[tex]\[
x = \frac{140}{4}
\][/tex]
8. Finding the result:
[tex]\[
x = 35
\][/tex]
Therefore, there are 35 cats at the shelter.
