Proportion Problem-Solving: Exercises

Welcome to this tutorial on proportion problem-solving. Here, we'll provide you with a range of exercises designed to help you master the concept of proportions, a fundamental aspect of ratio understanding.

Historically, proportions have been a crucial mathematical tool. Eminent mathematicians like Euclid and Pythagoras relied on the concept of proportions to solve complex geometrical problems. Euclid, in his work 'Elements', gave a precise definition of proportions and introduced the method of 'ex aequali', a significant milestone in the journey of proportion understanding.

Understanding Proportions

Before we dive into the exercises, it's essential to recap what we mean by proportions. In mathematics, two ratios are said to be proportional if the quotient of the first pair is equal to the quotient of the second pair. For example, the ratios 2:4 and 3:6 are proportional because 2/4 is equivalent to 3/6.

Exercise 1

Let's start with a straightforward exercise. Given the proportion 3:4 = 15:?, what number should replace the question mark to make the ratios proportional?

3/4 = 15/?
3 × ? = 4 × 15
? = (4 × 15) / 3
? = 20

The missing number is 20.

Exercise 2

Now a slightly complex problem. If 7/9 = 14/?, what is the value of ?

7/9 = 14/?
7 × ? = 9 × 14
? = (9 × 14) / 7
? = 18

The missing number is 18.

Exercise 3

For a challenge, let's have two unknowns. If 5/? = ?/20, find the value of ?

5/? = ?/20
5 × 20 = ? × ?
100 = ?²
? = √100
? = 10

The missing number is 10.

As you work through these problems, you're treading the same path as Euclid and other great mathematicians. They understood that proportions aren't just a mathematical concept; they're a way of seeing relationships and patterns. The ability to solve proportions is a valuable tool for problem-solving in many areas of life.

Take Away

These exercises should give you a better understanding of how to solve proportions and the role they play in mathematics. As you get more comfortable, try creating your own problems to solve. After all, creating the question is often just as instructive as finding the answer.

Next Steps

Continue exploring the fascinating world of ratios, proportions, and fractions. Understanding these concepts is not just about solving mathematical problems; it's about understanding the relationships that govern our world. Happy exploring!

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