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Chapter 6: Problem 5

Convert 0.78 to a fraction. Reduce.

Short Answer

Expert verified
0.78 as a fraction is \( \frac{39}{50} \).

Step by step solution

01

Write the Decimal as a Fraction

To convert the decimal 0.78 to a fraction, write it as 78 over 100. So, we have \( \frac{78}{100} \).
02

Simplify the Fraction

To simplify \( \frac{78}{100} \), find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 78 and 100 is 2. Divide both the numerator and the denominator by 2: \( \frac{78 \div 2}{100 \div 2} = \frac{39}{50} \).
03

Verify Simplification

Check if \( \frac{39}{50} \) can be simplified further. The GCD of 39 and 50 is 1, which means \( \frac{39}{50} \) is already in its simplest form.

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Key Concepts

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

Understanding the Greatest Common Divisor (GCD)
The Greatest Common Divisor, or GCD, is a key element when simplifying fractions. A GCD is the largest positive integer that divides two numbers without leaving a remainder. Think of it as the biggest shared factor between two numbers. To find the GCD, you list the factors of each number and identify the largest one that appears in both lists.
For instance, if you want to find the GCD of 78 and 100:
  • Factors of 78: 1, 2, 3, 6, 13, 26, 39, 78
  • Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
The common factors here are 1 and 2. Therefore, the GCD is 2. It's important to find the GCD when simplifying fractions because it ensures that the fraction is reduced most efficiently.
Simplifying Fractions
Simplifying fractions means reducing them to their smallest, simplest form. This makes fractions easier to understand and work with. The process involves dividing both the numerator (top number) and the denominator (bottom number) by their GCD.
In our example, the fraction starts as \(\frac{78}{100}\). We already found their GCD, which is 2. So, divide both numerator and denominator by 2:
  • \(78 \div 2 = 39\)
  • \(100 \div 2 = 50\)
This leaves us with \(\frac{39}{50}\). After verifying, we find \(\frac{39}{50}\) is in its simplest form, with no common divisor other than 1. Simplifying is essential since it gives you the "tightest" form of the fraction.
Converting Decimals to Fractions
Turning a decimal into a fraction involves placing the decimal over a power of ten. The number of decimal places dictates the power of ten you'll use. For example, 0.78 has two decimal places, so you use 100.
Here's the step-by-step:
  • Write 0.78 as \(\frac{78}{100}\) since it has two decimal places.
  • Simplify \(\frac{78}{100}\) by using your GCD finding skills.
Decimal conversion becomes clearer when you systematically convert them into fractions.Finally, always check to make sure the fraction is in its simplest form for your answer to remain precise and easy to interpret.

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Most popular questions from this chapter

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