Understanding and Solving Ratio Problems: A Comprehensive Guide

Mathematics is an essential tool in various fields, from engineering to finance. One of the fundamental concepts in mathematics is the ratio, which is used to express the relationship between two or more quantities. In this article, we will delve into the intricacies of solving ratio problems and enhance our problem-solving skills by breaking down a specific example.

Introduction to Ratios

A ratio is a comparison of two quantities by division. It is expressed as a:b, where a and b are the quantities being compared. Ratios can be simplified or converted to equivalent ratios by multiplying or dividing both terms by the same non-zero number.

Solving Complex Ratio Problems

Let's consider the problem: If x:y 7:9 and y:z 15:7, what is the value of x:z? This problem involves two ratios, and our goal is to find the relationship between x and z.

Step 1: Express Each Ratio in Terms of a Common Term

To solve this problem, we need to find a common term between the two ratios. In the given problem, the term y is common to both ratios. We will express each ratio in terms of y to find the relationship between x and z.

First Ratio: x:y 7:9

We start by expressing x and y in terms of a common factor. Let's multiply both terms of the ratio by 5 to simplify the relationship:

x/y 7/9 35/45 (multiplied by 5)

This gives us:

Second Ratio: y:z 15:7

Similarly, we express y and z in terms of a common factor. We multiply both terms by 3 to achieve this:

y/z 15/7 45/21 (multiplied by 3)

This gives us:

Step 2: Combine the Ratios to Find x:z

Now that both ratios have a common term y, we can combine them to find the relationship between x and z directly. Since x 35 and y 45, and y 45 and z 21, we can write:

x:y:z 35:45:21

To find x:z, we simply take the corresponding terms from the combined ratio:

x:z 35:21

We can simplify this ratio by dividing both terms by their greatest common divisor (GCD), which is 7:

x:z 5:3 (35/7 5, 21/7 3)

Conclusion

Solving ratio problems involves breaking down the given ratios into simpler terms and combining them to achieve the desired relationship. In this article, we discussed how to solve the problem x:y 7:9 and y:z 15:7 to find the value of x:z. The relationship between x and z is found to be 5:3.

Understanding and mastering ratio problems is crucial for various mathematical concepts and real-world applications. By following a systematic approach, you can solve similar problems and improve your problem-solving skills.