Using Cross Products To See If Each Pair Of Ratios

Introduction

Ratios are an important mathematical concept that we encounter in our daily lives. They are used to compare two quantities, and they can be expressed in different ways. One way to express a ratio is as a fraction. For example, the ratio of the number of girls to the number of boys in a classroom can be expressed as a fraction: girls/boys. Ratios can also be expressed as decimals or percentages.

What are Cross Products?

Cross products are a mathematical tool that can be used to compare two ratios. They are used to see if two ratios are equivalent. To understand how cross products work, let’s consider the following example: Suppose we have two ratios: a/b and c/d. We want to see if these ratios are equivalent. To do this, we can use cross products. The cross product of a/b and c/d is ad and bc. If the cross products are equal, then the ratios are equivalent. If the cross products are not equal, then the ratios are not equivalent.

Example:

Suppose we have the following ratios: 2/3 and 4/6. To see if these ratios are equivalent, we can use cross products. The cross product of 2/3 and 4/6 is 2 x 6 and 3 x 4, which is 12 and 12. Since the cross products are equal, we know that the two ratios are equivalent.

Using Cross Products to Solve Problems

Cross products can be used to solve a variety of problems involving ratios. Let’s consider the following examples:

Example 1:

Suppose we have a recipe that calls for 2 cups of flour and 3 cups of sugar. We want to double the recipe. How much flour and sugar do we need? To double the recipe, we need to multiply each ingredient by 2. This gives us 4 cups of flour and 6 cups of sugar. We can use cross products to verify that the ratios are equivalent. The ratio of flour to sugar in the original recipe is 2/3. The ratio of flour to sugar in the doubled recipe is 4/6. To see if these ratios are equivalent, we can use cross products. The cross product of 2/3 and 6/4 is 2 x 4 and 3 x 6, which is 8 and 18. Since the cross products are not equal, we know that the ratios are not equivalent. This means that the recipe is not doubled correctly.

Example 2:

Suppose we have a bag of marbles with 3 red marbles and 5 blue marbles. What is the ratio of red marbles to blue marbles? The ratio of red marbles to blue marbles is 3/5. We can use cross products to verify that this ratio is equivalent to other ratios involving red and blue marbles. For example, the ratio of blue marbles to red marbles is 5/3. To see if these ratios are equivalent, we can use cross products. The cross product of 3/5 and 5/3 is 3 x 3 and 5 x 5, which is 9 and 25. Since the cross products are not equal, we know that the ratios are not equivalent. This means that the ratio of red marbles to blue marbles is not equivalent to the ratio of blue marbles to red marbles.

Conclusion

Cross products are a useful tool for comparing ratios and solving problems involving ratios. By using cross products, we can determine if two ratios are equivalent and solve problems involving ratios. It is important to remember that ratios can be expressed in different ways, such as fractions, decimals, or percentages. By understanding how to use cross products, we can gain a better understanding of ratios and their applications in our daily lives.