One Neighborhood Has 18 Large Dogs And 21 Small Dogs. Which Is The Simplified Ratio Of Small Dogs To Large Dogs?A. 7:6 B. 6:7
Ratios are a fundamental concept in mathematics that help us compare the relative sizes of two or more quantities. In this article, we will explore the concept of ratios and how to simplify them using a real-world example involving dogs.
What is a Ratio?
A ratio is a comparison of two or more numbers that shows the relative size of each quantity. It is usually expressed as a fraction or a set of numbers separated by a colon. For example, if we have 12 apples and 8 oranges, the ratio of apples to oranges is 12:8 or 3:2.
The Dog Example
Let's consider a neighborhood with 18 large dogs and 21 small dogs. We want to find the simplified ratio of small dogs to large dogs. To do this, we need to first find the greatest common divisor (GCD) of the two numbers.
Finding the Greatest Common Divisor (GCD)
The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. To find the GCD, we can use the Euclidean algorithm or simply list the factors of each number.
Factors of 18
The factors of 18 are: 1, 2, 3, 6, 9, 18
Factors of 21
The factors of 21 are: 1, 3, 7, 21
Finding the GCD
The greatest common divisor of 18 and 21 is 3.
Simplifying the Ratio
Now that we have found the GCD, we can simplify the ratio by dividing both numbers by the GCD.
Simplified Ratio
The simplified ratio of small dogs to large dogs is 21 ÷ 3 : 18 ÷ 3 = 7:6.
Conclusion
In this article, we have learned how to find the simplified ratio of two quantities using a real-world example involving dogs. We have also learned how to find the greatest common divisor (GCD) of two numbers and how to use it to simplify a ratio. The simplified ratio of small dogs to large dogs is 7:6.
Additional Examples
Here are a few more examples of how to simplify ratios:
- If we have 24 apples and 36 oranges, the ratio of apples to oranges is 24:36 or 2:3.
- If we have 15 cats and 20 dogs, the ratio of cats to dogs is 15:20 or 3:4.
Practice Problems
Try these practice problems to test your understanding of ratios and simplifying them:
- If we have 12 pencils and 18 pens, what is the simplified ratio of pencils to pens?
- If we have 25 books and 30 magazines, what is the simplified ratio of books to magazines?
Answer Key
- The simplified ratio of pencils to pens is 2:3.
- The simplified ratio of books to magazines is 5:6.
Final Thoughts
In our previous article, we explored the concept of ratios and how to simplify them using a real-world example involving dogs. In this article, we will answer some frequently asked questions (FAQs) about ratios to help you better understand this important mathematical concept.
Q: What is a ratio?
A: A ratio is a comparison of two or more numbers that shows the relative size of each quantity. It is usually expressed as a fraction or a set of numbers separated by a colon.
Q: How do I find the greatest common divisor (GCD) of two numbers?
A: To find the GCD of two numbers, you can use the Euclidean algorithm or simply list the factors of each number. The GCD is the largest number that divides both numbers without leaving a remainder.
Q: Why do I need to find the GCD to simplify a ratio?
A: Finding the GCD is necessary to simplify a ratio because it allows you to divide both numbers by the same value, resulting in a simpler ratio.
Q: Can I simplify a ratio if the numbers are not whole numbers?
A: Yes, you can simplify a ratio even if the numbers are not whole numbers. For example, if you have 3.5 apples and 2.5 oranges, the ratio of apples to oranges is 3.5:2.5 or 7:5.
Q: How do I know if a ratio is simplified?
A: A ratio is simplified if the numbers cannot be divided by any common factor other than 1. In other words, if you cannot find any number that divides both numbers without leaving a remainder, then the ratio is simplified.
Q: Can I simplify a ratio with fractions?
A: Yes, you can simplify a ratio with fractions. For example, if you have 1/2 apples and 1/3 oranges, the ratio of apples to oranges is 1/2:1/3 or 3:2.
Q: How do I compare ratios?
A: To compare ratios, you can compare the relative sizes of the numbers. For example, if you have a ratio of 3:2 and another ratio of 6:4, the second ratio is larger because the numbers are greater.
Q: Can I use ratios to solve real-world problems?
A: Yes, you can use ratios to solve real-world problems. For example, if you have a recipe that calls for a ratio of 2:3 of flour to sugar, you can use this ratio to determine the amount of each ingredient needed.
Q: What are some common applications of ratios?
A: Ratios have many common applications in real life, including:
- Cooking and baking: Ratios are used to determine the amount of ingredients needed for a recipe.
- Building and construction: Ratios are used to determine the amount of materials needed for a project.
- Science and engineering: Ratios are used to determine the amount of substances needed for an experiment.
- Finance: Ratios are used to determine the amount of money needed for a loan or investment.
Conclusion
In this article, we have answered some frequently asked questions (FAQs) about ratios to help you better understand this important mathematical concept. We have also explored some common applications of ratios in real life. By understanding ratios, you can solve a wide range of problems and make informed decisions in your personal and professional life.
Additional Resources
If you want to learn more about ratios, here are some additional resources:
- Khan Academy: Ratios and Proportional Relationships
- Mathway: Ratios and Proportions
- Wolfram Alpha: Ratios and Proportions
Practice Problems
Try these practice problems to test your understanding of ratios:
- If you have 12 pencils and 18 pens, what is the simplified ratio of pencils to pens?
- If you have 25 books and 30 magazines, what is the simplified ratio of books to magazines?
- If you have a recipe that calls for a ratio of 2:3 of flour to sugar, how much flour and sugar do you need to make 12 servings?
Answer Key
- The simplified ratio of pencils to pens is 2:3.
- The simplified ratio of books to magazines is 5:6.
- To make 12 servings, you need 24 cups of flour and 36 cups of sugar.