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Chapter 1: Problem 43
Write each decimal as a fraction. 0.25
Short Answer
Expert verified
0.25 as a fraction is \( \frac{1}{4} \).
Step by step solution
01
- Understand the Decimal
The given decimal is 0.25. It's important to note that 0.25 is in the hundredths place.
02
- Write as Fraction
Express 0.25 as a fraction with 25 in the numerator and 100 in the denominator, since the decimal goes to the hundredths place. So, we write it as \[ \frac{25}{100} \].
03
- Simplify the Fraction
Simplify the fraction by finding the Greatest Common Divisor (GCD) of 25 and 100. The GCD of 25 and 100 is 25. Now, divide both the numerator and the denominator by the GCD: \[ \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \].
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Decimal Representation
When you see a number like 0.25, you're looking at a decimal. Decimals represent fractions where the denominator is a power of 10. For example, 0.25 means 25 hundredths, or \(\frac{25}{100}\). The position of the digits after the decimal point is crucial.
Each place has a different value:
- The first place to the right of the decimal is the tenths place.
- The second place is the hundredths place.
Understanding this is the first step in converting a decimal into a fraction.
Fraction Simplification
Turning a decimal into a fraction is just the beginning. Simplifying the fraction is equally important. When you have \(\frac{25}{100}\), you simplify it by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example: \(\frac{25}{100}\) can be reduced by dividing both 25 and 100 by 25: \(\frac{25 \div 25}{100 \div 25} = \frac{1}{4}\).
Simplifying makes the fraction easier to understand and work with. It also ensures consistency in mathematical communication.
Greatest Common Divisor (GCD)
To simplify any fraction, you need to find the greatest common divisor (GCD). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
Here’s how you can find it:
- List the factors of each number.
- Identify the largest factor that both numbers share.
For instance, the factors of 25 are 1, 5, and 25. The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
The largest common factor is 25, so \(\frac{25}{100}\) simplifies to \(\frac{1}{4}\). Finding the GCD simplifies complex fractions, making them more manageable.
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