Problem 30 - Calculate With Confidence

Question

Change the following decimals to fractions, and reduce to lowest terms. 0.065 __________

Step-by-Step Solution

Answer

0.065 is \(\frac{13}{200}\) in simplified form.

Step 1: Identify the Decimal

Identify the decimal number given in the exercise, which is 0.065 for this problem.

Step 2: Write as a Fraction

Write the decimal as a fraction with the given decimal's place value. Since 0.065 has three decimal places, place the value over 1000:\[0.065 = \frac{65}{1000}\]

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{65}{1000}\) by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 65 and 1000 is 5. Divide both the numerator and the denominator by 5:\[\frac{65 \div 5}{1000 \div 5} = \frac{13}{200}\]

Step 4: Confirm the Fraction is in Lowest Terms

Check if \(\frac{13}{200}\) is in its simplest form by ensuring there are no common factors between 13 and 200 (they are relatively prime). Thus, \(\frac{13}{200}\) is the fraction in the lowest terms.

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial step in making fractions easier to understand and work with. When you have a fraction like \( \frac{65}{1000} \), simplifying it means finding an equivalent fraction that has a smaller numerator and denominator but represents the same value. This makes the fraction easier to use in calculations or equations. To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD). This process reduces the fraction to its simplest form.

For example, when simplifying \( \frac{65}{1000} \), finding the GCD (in this case, 5), and dividing both the numerator and the denominator by this number results in \( \frac{13}{200} \). This is how we find a fraction in its simplest form.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept in mathematics, particularly when simplifying fractions. It refers to the largest positive integer that divides both the numerator and the denominator of a fraction without leaving a remainder. Understanding the GCD helps in reducing fractions effectively.

To find the GCD, you can list all the factors of both numbers and identify the largest factor they share. In the example of \( \frac{65}{1000} \), the GCD is 5 because both 65 and 1000 can be divided evenly by 5. Therefore, dividing the fraction's numerator and denominator by 5 yields its simplest form, \( \frac{13}{200} \). Recognizing the GCD allows you to simplify fractions swiftly and accurately.

Place Value in Decimals

Understanding place value in decimals is key when converting decimals to fractions. The place value of a decimal indicates the position of a digit relative to the decimal point, which determines its fractional equivalent.

For instance, in the decimal 0.065, the digit 6 is in the hundredths place and the 5 is in the thousandths place. This means that the fraction equivalent is \( \frac{65}{1000} \). Each digit's position is a step further in representing those place values: tenths, hundredths, and thousandths.

Recognizing these decimal places makes it easy to convert decimals into fractions by placing the decimal over 10, 100, 1000, or other powers of 10 based on the furthest decimal place. This is why 0.065 becomes \( \frac{65}{1000} \). This foundational understanding is helpful for both converting and simplifying fractions.