How to Tell Whether the Ratios Form a Proportion

Oct 13, 2025 by miodeya.com 120 views

What is a Proportion?

A proportion is a statement that two ratios are equal. It is often written in the form of a:b = c:d, where a, b, c, and d are the terms of the ratios. To determine if two ratios form a proportion, we need to check if the cross products of the ratios are equal.

How to Tell Whether the Ratios Form a Proportion

To tell whether the ratios form a proportion, we need to follow these steps:

Write the ratios in the form of a:b and c:d.
Multiply the first term of the first ratio (a) by the last term of the second ratio (d) to get the cross product.
Multiply the first term of the second ratio (c) by the last term of the first ratio (b) to get the cross product.
Check if the two cross products are equal.

Example 1: 1 to 4 and 3 to 12

Let's use the first example to illustrate the process. We have the ratios 1:4 and 3:12.

Step 1: Write the Ratios

The ratios are already written in the form of a:b and c:d.

Step 2: Find the Cross Products

To find the cross products, we multiply the first term of the first ratio (1) by the last term of the second ratio (12) to get 1 × 12 = 12. We also multiply the first term of the second ratio (3) by the last term of the first ratio (4) to get 3 × 4 = 12.

Step 3: Check if the Cross Products are Equal

Since the two cross products are equal (12 = 12), we can conclude that the ratios 1:4 and 3:12 form a proportion.

Example 2: $1:7$ and $4:28$

Let's use the second example to illustrate the process. We have the ratios $1:7$ and $4:28$ .

Step 1: Write the Ratios

The ratios are already written in the form of a:b and c:d.

Step 2: Find the Cross Products

To find the cross products, we multiply the first term of the first ratio (1) by the last term of the second ratio (28) to get 1 × 28 = 28. We also multiply the first term of the second ratio (4) by the last term of the first ratio (7) to get 4 × 7 = 28.

Step 3: Check if the Cross Products are Equal

Since the two cross products are equal (28 = 28), we can conclude that the ratios $1:7$ and $4:28$ form a proportion.

Example 3: $2:5$ and $30:80$

Let's use the third example to illustrate the process. We have the ratios $2:5$ and $30:80$ .

Step 1: Write the Ratios

The ratios are already written in the form of a:b and c:d.

Step 2: Find the Cross Products

To find the cross products, we multiply the first term of the first ratio (2) by the last term of the second ratio (80) to get 2 × 80 = 160. We also multiply the first term of the second ratio (30) by the last term of the first ratio (5) to get 30 × 5 = 150.

Step 3: Check if the Cross Products are Equal

Since the two cross products are not equal (160 ≠ 150), we can conclude that the ratios $2:5$ and $30:80$ do not form a proportion.

Example 4: $18:24$ and $15:20$

Let's use the fourth example to illustrate the process. We have the ratios $18:24$ and $15:20$ .

Step 1: Write the Ratios

The ratios are already written in the form of a:b and c:d.

Step 2: Find the Cross Products

To find the cross products, we multiply the first term of the first ratio (18) by the last term of the second ratio (20) to get 18 × 20 = 360. We also multiply the first term of the second ratio (15) by the last term of the first ratio (24) to get 15 × 24 = 360.

Step 3: Check if the Cross Products are Equal

Since the two cross products are equal (360 = 360), we can conclude that the ratios $18:24$ and $15:20$ form a proportion.

Example 5: $35:16$ and $5:2$

Let's use the fifth example to illustrate the process. We have the ratios $35:16$ and $5:2$ .

Step 1: Write the Ratios

The ratios are already written in the form of a:b and c:d.

Step 2: Find the Cross Products

To find the cross products, we multiply the first term of the first ratio (35) by the last term of the second ratio (2) to get 35 × 2 = 70. We also multiply the first term of the second ratio (5) by the last term of the first ratio (16) to get 5 × 16 = 80.

Step 3: Check if the Cross Products are Equal

Since the two cross products are not equal (70 ≠ 80), we can conclude that the ratios $35:16$ and $5:2$ do not form a proportion.

Example 6: 5 to 7

Let's use the sixth example to illustrate the process. We have the ratios 5:7.

Step 1: Write the Ratios

The ratios are already written in the form of a:b and c:d.

Step 2: Find the Cross Products

To find the cross products, we multiply the first term of the first ratio (5) by the last term of the second ratio (7) to get 5 × 7 = 35. We also multiply the first term of the second ratio (7) by the last term of the first ratio (5) to get 7 × 5 = 35.

Step 3: Check if the Cross Products are Equal

Since the two cross products are equal (35 = 35), we can conclude that the ratios 5:7 form a proportion.

Conclusion

In conclusion, to tell whether the ratios form a proportion, we need to follow the steps outlined above. We need to write the ratios in the form of a:b and c:d, find the cross products, and check if the cross products are equal. If the cross products are equal, then the ratios form a proportion. If the cross products are not equal, then the ratios do not form a proportion.

Frequently Asked Questions

Q: What is a proportion?

A: A proportion is a statement that two ratios are equal. It is often written in the form of a:b = c:d, where a, b, c, and d are the terms of the ratios.

Q: How do I tell whether the ratios form a proportion?

A: To tell whether the ratios form a proportion, you need to follow these steps:

Write the ratios in the form of a:b and c:d.
Multiply the first term of the first ratio (a) by the last term of the second ratio (d) to get the cross product.
Multiply the first term of the second ratio (c) by the last term of the first ratio (b) to get the cross product.
Check if the two cross products are equal.

Q: What is the cross product?

A: The cross product is the result of multiplying the first term of the first ratio by the last term of the second ratio, or the first term of the second ratio by the last term of the first ratio.

Q: Why do we need to find the cross products?

A: We need to find the cross products to check if the ratios form a proportion. If the cross products are equal, then the ratios form a proportion.

Q: What if the cross products are not equal?

A: If the cross products are not equal, then the ratios do not form a proportion.

Q: Can I use a calculator to find the cross products?

A: Yes, you can use a calculator to find the cross products. However, make sure to follow the steps outlined above to ensure that you are finding the correct cross products.

Q: Can I use a calculator to check if the ratios form a proportion?

A: Yes, you can use a calculator to check if the ratios form a proportion. Simply enter the cross products into the calculator and check if they are equal.

Q: What if I get a different answer on the calculator?

A: If you get a different answer on the calculator, double-check your work to make sure that you are entering the correct numbers and following the correct steps.

Q: Can I use a calculator to find the cross products and check if the ratios form a proportion?

A: Yes, you can use a calculator to find the cross products and check if the ratios form a proportion. Simply follow the steps outlined above and enter the cross products into the calculator.

Q: What if I am still unsure about how to tell whether the ratios form a proportion?

A: If you are still unsure about how to tell whether the ratios form a proportion, try using a different method, such as using a ratio table or a proportion chart. You can also ask a teacher or tutor for help.

Q: Can I use a ratio table or a proportion chart to tell whether the ratios form a proportion?

A: Yes, you can use a ratio table or a proportion chart to tell whether the ratios form a proportion. A ratio table is a table that lists the ratios and their corresponding cross products, while a proportion chart is a chart that shows the ratios and their corresponding cross products.

Q: What are the benefits of using a ratio table or a proportion chart?

A: The benefits of using a ratio table or a proportion chart include:

It can help you to visualize the ratios and their corresponding cross products.
It can help you to identify patterns and relationships between the ratios.
It can help you to check if the ratios form a proportion.

Q: Can I use a ratio table or a proportion chart to find the cross products?

A: Yes, you can use a ratio table or a proportion chart to find the cross products. Simply list the ratios and their corresponding cross products in the table or chart.

Q: What if I am still unsure about how to use a ratio table or a proportion chart?

A: If you are still unsure about how to use a ratio table or a proportion chart, try using a different method, such as using a calculator or asking a teacher or tutor for help.

Conclusion

In conclusion, telling whether the ratios form a proportion is an important skill that can be used in a variety of real-world applications. By following the steps outlined above and using the tools and resources available, you can become proficient in telling whether the ratios form a proportion.