Question

Express the following ratios as fractions. Reduce to lowest terms. 3: 7 ______

Short answer

The ratio 3:7 as a fraction is \( \frac{3}{7} \).

Step-by-step solution

Understand Ratio as Fraction

The given ratio is 3:7. A ratio compares two quantities, and it can be expressed as a fraction. In a ratio a:b, it is equivalent to the fraction \( \frac{a}{b} \). Therefore, the ratio 3:7 can be expressed as the fraction \( \frac{3}{7} \).

Simplify the Fraction

To reduce a fraction to its lowest terms, divide the numerator and the denominator by their greatest common divisor (GCD). The fraction \( \frac{3}{7} \) needs to be checked if it can be simplified. The GCD of 3 and 7 is 1 since they are both prime numbers. Therefore, the fraction \( \frac{3}{7} \) is already in its simplest form.

Key concepts

Understanding Ratios

Ratios are everywhere in our daily life, making this concept both interesting and important. A ratio is a way to compare two or more quantities.
For example, a ratio like 3:7 indicates that for every 3 of one thing, there are 7 of another.
Ratios look much like fractions and can be written as such.

• Reading the ratio 3:7 as "3 to 7" implies comparing these two numbers.
• Writing it as a fraction transforms it into \( \frac{3}{7} \), showing the relation between 3 and 7 more mathematically.

This fraction form allows us to use arithmetic operations on them, simplifying comparisons or calculations. Understanding ratios helps not only in math but also in fields like cooking, finance, and science.

Greatest Common Divisor (GCD)

When dealing with fractions, simplifying them often involves finding the greatest common divisor (GCD).
The GCD of two numbers is the largest number that can evenly divide both numbers.
Let's explore this important concept further.

• For instance, consider the numbers 3 and 7 in the fraction \( \frac{3}{7} \).
• Since 3 and 7 are prime numbers, meaning they cannot be divided evenly by any other number except 1 and themselves, their GCD is 1.

To find the GCD between two numbers, you can list out the divisors of each number and identify the greatest one they have in common.
This is a basic and useful skill, especially since simplifying fractions makes calculations easier.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible by reducing the numerator and denominator to their smallest values.
This is done by dividing both the top and bottom of the fraction by their GCD.

• Let's simplify the fraction \( \frac{3}{7} \).
• First, determine the GCD of the numbers 3 and 7, which, as discussed, is 1.
• Because their GCD is 1, dividing both the numerator and denominator by 1 leaves \( \frac{3}{7} \), which is already simplified.

In this way, fractions can often be reduced to cleaner or simpler forms, making them easier to work with in further calculations or comparisons. Knowing how to simplify fractions quickly is a foundational math skill.