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Chapter 6: Problem 17

Write the ratio in lowest terms. \(5 \mathrm{mi}\) to \(25 \mathrm{mi}\)

Short Answer

Expert verified
The ratio in lowest terms is 1 to 5.

Step by step solution

01

- Understand the Problem

The problem involves writing the ratio of 5 miles to 25 miles in its simplest form.
02

- Write the Ratio

A ratio can be written as a fraction. So, write the ratio of 5 miles to 25 miles as the fraction \(\frac{5 \text{ mi}}{25 \text{ mi}}\).
03

- Simplify the Fraction

To simplify the fraction, find the greatest common divisor (GCD) of 5 and 25. The GCD of 5 and 25 is 5.
04

- Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator of the fraction \(\frac{5}{25}\) by the GCD, which is 5: \(\frac{5 \text{ mi}}{25 \text{ mi}} = \frac{5 \text{ mi} / 5}{25 \text{ mi} / 5} = \frac{1 \text{ mi}}{5 \text{ mi}}\).
05

- Write the Simplified Ratio

The simplified ratio is 1 mile to 5 miles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractions
When we deal with ratios, we often convert them into fractions, which makes it easier to simplify. A fraction represents how many parts of a certain type are in a whole. For example, writing the ratio of 5 miles to 25 miles as a fraction gives us \(\frac{5}{25}\).
This fraction means 5 parts out of a total of 25 parts.
Fractions help us understand and compare different quantities.
To simplify a fraction, we need to reduce it to its simplest form, which involves other concepts like the greatest common divisor (GCD).

Remember, the top number in a fraction is called the numerator and the bottom number is called the denominator.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder.
In other words, it's the biggest number that can evenly divide both parts of a ratio.
Finding the GCD helps in simplifying fractions by reducing them to their smallest possible form.
For example, in our exercise, the GCD of 5 and 25 is 5. This is because 5 is the largest number that can evenly divide both 5 and 25.
Once we know the GCD, we divide both the numerator and the denominator by this number to simplify the fraction.
Ratios
Ratios are a way to compare two quantities.
They show how much of one thing there is compared to another.
In our exercise, we are comparing 5 miles to 25 miles.

We can write ratios in different ways:
  • as fractions \(\frac{5}{25}\)
  • with a colon 5:25
  • in words '5 to 25'
The goal is often to simplify the ratio to make it easier to understand. Simplifying the ratio 5 to 25 involves turning it into the simplest form 1 to 5, by using the GCD to reduce the numbers.

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