Writing and Simplifying Ratios

Ratios are a way of comparing two or more quantities. They are an important concept in math, especially when dealing with ratios, rates, and real-world problems. In this explanation, we will explore what ratios are, how they are written, and how to simplify them. Understanding these concepts will help improve your math skills and make it easier to solve problems involving comparisons and proportional reasoning.

What is the ratio?

A ratio is simply a comparison between two quantities that shows how much of one thing is present relative to another. For example, if there are 2 apples and 3 oranges in a basket, the ratio of apples to oranges can be written in several ways:

2 to 3

2:3

2/3

All of these expressions mean the same thing: for every 2 apples, there are 3 oranges. This ratio helps to illustrate the relationship between the quantity of apples and oranges.

Writing ratio

There are three common ways to write ratios:

1. Use of the word "to": 2 to 3
2. Use of colon symbol: 2:3
3. Use of fraction: 2/3

Each form is used depending on the context and to make it easier for you to understand. Below are some examples of writing ratios in all three forms:

Example 1: Comparing boys and girls in the class

Imagine a class consisting of 20 boys and 15 girls. The ratio of boys and girls can be written as:

• 20 to 15
• 20:15
• 20/15

Example 2: Using day and night

If a month has 30 sunny days and 10 rainy days, what will be the ratio of sunny days to rainy days?

• 30 to 10
• 30:10
• 30/10

Simplifying ratios

Simplifying ratios means reducing them to their simplest form. This is done in the same way as simplifying fractions. You divide both sides of the ratio by their greatest common divisor (GCD), which is the largest number that can divide both numbers without leaving a remainder.

Steps to simplify ratios

1. Find the GCD of the numbers in the ratio.
2. Divide each number by the GCD.
3. Write down the simplified ratio.

Example 1: Simplifying 20 into 15

To simplify the ratio 20 to 15, follow these steps:

1. Find the GCD of 20 and 15. Since 5 is the largest number that exactly divides both 20 and 15, GCD = 5.
2. Divide both numbers by the GCD.

20 ÷ 5 = 4  15 ÷ 5 = 3

The simplified ratio of boys to girls is 4 to 3, 4:3, or 4/3.

Example 2: Simplifying 30 into 10

Consider the ratio 30 to 10. To simplify this ratio:

1. Find the GCD of 30 and 10. The number 10 is the largest number that divides both of them exactly, so GCD = 10.
2. Divide each quantity by the GCD.

30 ÷ 10 = 3  10 ÷ 10 = 1

The simplified ratio of sunny days to rainy days is 3 to 1, 3:1, or 3/1.

Why simplify ratios?

Simplifying ratios makes it easier to understand the relationship between different quantities. This provides clarity and helps compare data more effectively. In many cases, a simplified ratio communicates the same relationship in a clearer and more concise way.

For example, a ratio of 20:15 may not be as immediately understandable as a simplified 4:3. Simplified ratios can also be important when solving problems, making calculations more manageable, and verifying equality between two ratios (also called checking whether two ratios are equivalent).

Practice problems

Let's try to simplify some ratios. Use the same steps as above. Remember to first find the greatest common divisor for each pair of numbers.

Problem 1

Write and simplify the following ratio: 42 to 56.

Solution steps:

1. Find the GCD of 42 and 56.
2. Divide both numbers by the GCD.

Solution:

GCD of 42 and 56 is 14.  42 ÷ 14 = 3  56 ÷ 14 = 4  The simplified ratio is 3 to 4 or 3:4 or 3/4.

Problem 2

Write the ratio of 48 to 18 and simplify.

Solution steps:

1. Determine the GCD of 48 and 18.
2. Divide both numbers by their GCD.

Solution:

GCD of 48 and 18 is 6.  48 ÷ 6 = 8  18 ÷ 6 = 3  The simplified ratio is 8 to 3 or 8:3 or 8/3.

Problem 3

Simplify the ratio 36:90.

Solution steps:

1. Find the largest number which divides both 36 and 90.
2. Divide the numbers by this greatest common divisor.

Solution:

GCD of 36 and 90 is 18.  36 ÷ 18 = 2  90 ÷ 18 = 5  The simplified ratio is 2 to 5 or 2:5 or 2/5.

Conclusion

Learning how to write and simplify ratios is an essential math skill used in many real-life situations, such as recipes, maps, and finance. Becoming familiar with the process of simplifying ratios will help simplify complex problems and make better decisions when faced with problems involving proportional relationships.

Keep practicing writing and simplifying ratios by solving different types of problems. The more you practice, the easier it will be to recognize patterns and apply your knowledge effectively. When simplifying, always make sure to take out the greatest common denominator first, as this is the key to reducing ratios to their smallest forms. This practice helps not only in math but in any field where comparisons and quantitative analysis are necessary.

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