Using Ratios to Solve the Problem

Oct 13, 2025 by vinooalluri.com 38 views

Understanding the Problem

In this problem, we are given the ratio of T-shirts to jumpers in Jacob's wardrobe, which is $8:7$ . We are also told that there are 24 T-shirts in his wardrobe. Our goal is to find out how many jumpers are there in his wardrobe.

Using Ratios to Solve the Problem

To solve this problem, we can use the concept of ratios. A ratio is a way of comparing two or more numbers. In this case, the ratio of T-shirts to jumpers is $8:7$ , which means that for every 8 T-shirts, there are 7 jumpers.

Setting Up a Proportion

We can set up a proportion to solve this problem. A proportion is a statement that two ratios are equal. In this case, we can set up the proportion:

\frac{8}{7} = \frac{24}{x}

where $x$ is the number of jumpers.

Solving the Proportion

To solve this proportion, we can cross-multiply:

8x = 7(24)

8x = 168

Now, we can divide both sides by 8:

x = \frac{168}{8}

x = 21

Conclusion

Therefore, there are 21 jumpers in Jacob's wardrobe.

Understanding Ratios in Real-Life Scenarios

Ratios are used in many real-life scenarios, such as cooking, building, and finance. For example, a recipe for making cookies might call for a ratio of 2 cups of flour to 1 cup of sugar. This means that for every 2 cups of flour, you need 1 cup of sugar.

Using Ratios to Compare Similar Quantities

Ratios can also be used to compare similar quantities. For example, if you have a ratio of 3:5 for the number of boys to girls in a class, and you know that there are 15 boys, you can use the ratio to find the number of girls.

Real-World Applications of Ratios

Ratios have many real-world applications, such as:

Cooking: Recipes often use ratios to ensure that the right proportions of ingredients are used.
Building: Architects use ratios to design buildings and ensure that the proportions of different parts are correct.
Finance: Financial analysts use ratios to compare the performance of different companies and make informed investment decisions.
Science: Scientists use ratios to compare the properties of different substances and make new discoveries.

Conclusion

In conclusion, ratios are an important concept in mathematics that can be used to compare similar quantities and solve problems. By understanding ratios, we can apply them to real-life scenarios and make informed decisions.

Additional Examples

Here are a few more examples of how ratios can be used:

Example 1: A recipe for making a cake calls for a ratio of 2 cups of flour to 1 cup of sugar. If you have 4 cups of flour, how many cups of sugar do you need?
Example 2: A company has a ratio of 3:5 for the number of men to women in its workforce. If there are 15 men, how many women are there?
Example 3: A recipe for making a salad calls for a ratio of 2 cups of lettuce to 1 cup of tomatoes. If you have 6 cups of lettuce, how many cups of tomatoes do you need?

Solutions to Additional Examples

Here are the solutions to the additional examples:

Example 1: Since the ratio is 2:1, you need 2 cups of sugar for every 2 cups of flour. Therefore, you need 2 cups of sugar for 4 cups of flour.
Example 2: Since the ratio is 3:5, you need 5 women for every 3 men. Therefore, you need 5 women for 15 men.
Example 3: Since the ratio is 2:1, you need 1 cup of tomatoes for every 2 cups of lettuce. Therefore, you need 3 cups of tomatoes for 6 cups of lettuce.

Conclusion

Q: What is a ratio?

A: A ratio is a way of comparing two or more numbers. It is a statement that two or more quantities are in a certain proportion to each other.

Q: How do I write a ratio?

A: A ratio is written as a fraction, with the first number (the antecedent) on top and the second number (the consequent) on the bottom. For example, the ratio of 3:5 is written as 3/5.

Q: What is the difference between a ratio and a proportion?

A: A ratio is a statement that two or more quantities are in a certain proportion to each other. A proportion is a statement that two ratios are equal.

Q: How do I solve a proportion?

A: To solve a proportion, you can cross-multiply and then divide both sides by the number that you multiplied by. For example, if you have the proportion 2/3 = x/4, you can cross-multiply and get 2(4) = 3x, which simplifies to 8 = 3x. Then, you can divide both sides by 3 to get x = 8/3.

Q: What is the ratio of T-shirts to jumpers in Jacob's wardrobe?

A: The ratio of T-shirts to jumpers in Jacob's wardrobe is 8:7.

Q: How many jumpers are there in Jacob's wardrobe?

A: To find the number of jumpers, you can set up a proportion using the ratio of T-shirts to jumpers. Since there are 24 T-shirts, you can set up the proportion 8/7 = 24/x and solve for x. This gives you x = 21, which means that there are 21 jumpers in Jacob's wardrobe.

Q: What are some real-world applications of ratios?

A: Ratios have many real-world applications, such as:

Cooking: Recipes often use ratios to ensure that the right proportions of ingredients are used.
Building: Architects use ratios to design buildings and ensure that the proportions of different parts are correct.
Finance: Financial analysts use ratios to compare the performance of different companies and make informed investment decisions.
Science: Scientists use ratios to compare the properties of different substances and make new discoveries.

Q: How do I compare similar quantities using ratios?

A: To compare similar quantities using ratios, you can set up a proportion using the ratio of the two quantities. For example, if you have a ratio of 3:5 for the number of boys to girls in a class, and you know that there are 15 boys, you can set up the proportion 3/5 = 15/x and solve for x. This gives you x = 25, which means that there are 25 girls in the class.

Q: What are some common mistakes to avoid when working with ratios?

A: Some common mistakes to avoid when working with ratios include:

Not simplifying the ratio: Make sure to simplify the ratio to its simplest form before using it.
Not checking the units: Make sure that the units of the two quantities are the same before using the ratio.
Not using the correct proportion: Make sure to use the correct proportion when solving a problem involving ratios.