Finding the Ratio a:b:c Given a:b and b:c

Understanding how to work with ratios and proportions is a fundamental skill in many areas, from mathematics to everyday problem-solving. This article will guide you through the process of finding the ratio a:b:c given the ratios a:b and b:c. We will also provide examples to clarify the concepts and explain the mathematical methods involved.

Understanding the Problem

The question at hand is: If a:b 3:4 and b:c 7:6, what is the value of a:b:c?

Step-by-Step Solution

Let's break down the solution step by step.

Step 1: Representing the Ratios

We start with the given ratios:

a:b 3:4 b:c 7:6

We need to find a common term for b to combine the ratios.

Step 2: Finding a Common Ratio for b

To find a common term for b, we will scale the ratios to have the same value for b.

a:b 3:4 can be scaled to 21:28 by multiplying both terms by 7. b:c 7:6 can be scaled to 28:24 by multiplying both terms by 4.

Therefore, after scaling, the ratios become:

a:b 21:28 b:c 28:24

The ratio a:b:c can now be obtained by combining these scaled ratios:

a:b:c 21:28:24

Multiple Methods to Verify the Ratio

Let's verify this solution using another method. We can use algebraic representation:

Method 1: Using Algebraic Representation

We can represent the ratios as fractions:

a/b 3/4 b/c 7/6

From the second fraction, we can express b in terms of c:

b 7c/6

Substituting this into the first fraction:

a/(7c/6) 3/4

Multiplying both sides by 6c/7, we get:

a (3/4) * (7c/6)

Simplifying this equation:

a 21c/24 7c/8

Thus, the ratio a:b:c becomes:

a:b:c 7c/8 : 7c/6 : c 21:28:24

Method 2: Using a Common Divisible Factor

Another method involves finding a common factor that can be evenly divided by both 4 and 7. Let's choose 28 as this factor.

a:b 3:4 becomes 21:28 (since 3 * 7 21, 4 * 7 28) b:c 7:6 becomes 28:24 (since 7 * 4 28, 6 * 4 24)

Combining these scaled ratios, we obtain:

a:b:c 21:28:24

Conclusion

The ratio a:b:c given the ratios a:b 3:4 and b:c 7:6 is 21:28:24.

This method not only solves the problem but also demonstrates various approaches to tackling similar mathematical problems. Understanding these methods can help improve your problem-solving skills and enhance your proficiency in handling complex mathematical scenarios.

By following these steps and examples, you should now feel confident in solving such ratio and proportion problems. Remember, practice is key to mastering these skills.