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Chapter 1: Problem 2

Change \(0.45\) to a fraction. Reduce to lowest terms.

Short Answer

Expert verified
Decimal 0.45 as a fraction is \(\frac{9}{20}\).

Step by step solution

01

Understand the Decimal

The decimal 0.45 represents 45 hundredths, which means it can be written as \(\frac{45}{100}\).
02

Reduce the Fraction

To reduce the fraction, find the greatest common divisor (GCD) of 45 and 100. The GCD of 45 and 100 is 5. Divide both the numerator and the denominator by 5.
03

Perform the Division

Divide 45 by 5 to get 9, and divide 100 by 5 to get 20. Thus, \(\frac{45}{100} = \frac{9}{20}\).
04

Confirm Simplification

Check if \(\frac{9}{20}\) is in the lowest terms. The numbers 9 and 20 have no common factors other than 1, so \(\frac{9}{20}\) is indeed in its simplest form.

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Key Concepts

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

Simplifying Fractions
When you have a fraction like \( \frac{45}{100} \), simplifying it makes it easier to work with. Simplification involves reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1.
This process always starts by determining the greatest common divisor (GCD), as that will help us divide both parts of the fraction by the largest number they both share.
By simplifying fractions, we make them cleaner and simpler, which enhances understanding and computations.
  • Simplified fractions are more intuitive and usually smaller.
  • They represent the same value as the original fractions.
Greatest Common Divisor
Discovering the greatest common divisor (GCD) is a key step in simplifying fractions.
The GCD of two numbers is the largest positive integer that divides each of the numbers without a remainder.
For instance, when we wanted to simplify \( \frac{45}{100} \), finding the GCD of 45 and 100, which is 5, allowed us to divide effectively.
To find the GCD, you can use methods like:
  • Listing out the factors of each number and choosing the largest common one.
  • Using the Euclidean algorithm, a more efficient and systematic approach.
By identifying the GCD, you ensure that the fraction is reduced effectively and accurately.
Fraction Reduction
Fraction reduction, often simply called simplifying, is the process of transforming a fraction into its simplest form.
After determining the GCD, the next step is to use it to divide both the numerator and the denominator.
In our example with \( \frac{45}{100} \), dividing both 45 and 100 by their GCD, 5, gives us \( \frac{9}{20} \).
This step guarantees that the fraction is expressed in the least possible terms.
  • Reduction helps in accurately representing the size of the parts in a fraction.
  • It is essential for solving mathematical problems efficiently, preventing unnecessary complexity.
By doing this, you ensure clarity and correctness in mathematical computations and comparisons.

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Most popular questions from this chapter

Simplify: \(-\frac{4}{9} \div \frac{27}{81}\).

Simplify: \(-18-(-54)\).

Simplify: \(13+28\).

Use the graphing calculator to evaluate: $$ \frac{10+10}{10+10} $$

Given \(z=-0.4\), evaluate: \(5-\frac{4}{5} z\).
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