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Chapter 3: Problem 29

Change the following decimals to fractions, and reduce to lowest terms. 1.01 ______

Short Answer

Expert verified
1.01 as a fraction is \( \frac{101}{100} \).

Step by step solution

01

Identify the Decimal Form

We start with the decimal number 1.01.
02

Express the Decimal as a Fraction

Remove the decimal by writing the number over a power of 10. Since 1.01 has two decimal places, it can be expressed as \( \frac{101}{100} \).
03

Simplify the Fraction

Check if the fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 101 and 100 is 1, so \( \frac{101}{100} \) 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.

Decimal Conversion
Converting decimals to fractions is an essential math skill that helps in understanding number relationships more concretely. To convert a decimal like 1.01 into a fraction, the key is recognizing the position of the decimal places. Here are the steps:
  • Firstly, identify how many decimal places are present. In 1.01, there are two decimal places.
  • Next, remove the decimal point and place the original number over a power of 10. The power depends on the number of decimal places. For 1.01, since there are two decimal places, write it as 101 over 100, because 10 squared (or 100) matches the number of decimal places.
  • This gives the fraction \( \frac{101}{100} \).
Understanding decimal conversion helps to effortlessly bridge the gap between fractions and decimals. Take time to practice with various decimals to become proficient in this process.
Simplification of Fractions
Once you have converted a decimal to a fraction, the next step is simplification. Simplifying a fraction means expressing it in its simplest form. It involves reducing the fraction such that the numerator and denominator are as small as possible while maintaining the same value.
  • First, investigate if the numerator and the denominator shares any common factor greater than one.
  • To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
In our example, after converting 1.01 to a fraction, we have \( \frac{101}{100} \). Since 101 and 100 have no common factor besides 1, the fraction is already at its simplest form.
This demonstrates how some fractions might already be simple, immediately after decimal conversion.
Greatest Common Divisor
The greatest common divisor, often abbreviated as GCD, plays a crucial role in simplifying fractions. It is the largest number that divides both the numerator and denominator evenly without leaving a remainder. Finding the GCD allows you to ensure that a fraction is in its simplest form.
  • To find the GCD of two numbers, start by listing out all divisors of each number.
  • Identify the largest number that appears in both lists. This number is the GCD.
  • For instance, when simplifying a fraction like \( \frac{18}{24} \), the divisors of 18 are 1, 2, 3, 6, 9, and 18; for 24, they are 1, 2, 3, 4, 6, 8, 12, and 24. The largest common divisor is 6.
In the provided example, where the fraction is \( \frac{101}{100} \), the GCD is 1. Because of this, the fraction cannot be simplified further and is thus already in its lowest terms.

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