Question

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

Short answer

0.065 as a fraction in lowest terms is \(\frac{13}{200}\).

Step-by-step solution

Write the Decimal as a Fraction

We start by writing the decimal 0.065 as a fraction. The decimal 0.065 has three decimal places, so it can be written as \(\frac{65}{1000}\).

Simplify the Fraction

To simplify \(\frac{65}{1000}\), we need to find the greatest common divisor (GCD) of 65 and 1000. Both 65 and 1000 are divisible by 5. Dividing the numerator and the denominator by their GCD, we get: \(\frac{65 \div 5}{1000 \div 5} = \frac{13}{200}\).

Confirm the Fraction is in Lowest Terms

Check whether \(\frac{13}{200}\) is in its simplest form. The number 13 is a prime number, and it does not divide 200. Hence, \(\frac{13}{200}\) is already in the lowest terms.

Key concepts

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator (top number) and the denominator (bottom number) have no common factors other than 1.
This process makes fractions easier to understand and work with, aiding in accurate calculations.

Consider the fraction \(\frac{65}{1000}\) which derived from the decimal 0.065. Simplifying it involves a few critical steps. First, identify the greatest factor that divides both the numerator and the denominator evenly. In our example, both 65 and 1000 can be divided by the number 5.
Dividing them by 5 results in \(\frac{13}{200}\), an equivalent fraction that cannot be further reduced.

A fraction is fully simplified when no further division can be performed except by one. Once simplified, it represents the same portion of a whole as the original fraction but in the simplest expression possible.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is critical for reducing fractions. It is the biggest number that divides two or more numbers without leaving a remainder.
To simplify a fraction like \(\frac{65}{1000}\), finding the GCD of 65 and 1000 is essential.

To determine the GCD:

• List the factors of each number.
• Identify the highest number common to both lists. In this case, factors of 65 are 1, 5, 13, and 65; for 1000 they are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000.
• The biggest number they share is 5, which is the GCD.

Once found, the GCD is used to divide both numerator and denominator, making the fraction as simple as possible without changing its value.

Prime Numbers

Prime numbers are the basics of understanding fractions and their simplification. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, like 2, 3, 5, 7, 11, and 13.

The role of prime numbers is not only crucial but also practical. When simplifying fractions, checking if the top number is a prime helps determine if the fraction is already in its simplest form.

In the fraction \(\frac{13}{200}\), 13 stands out because it is a prime number. If the numerator or the denominator is prime, and if they don't divide each other, the fraction is already reduced to the lowest terms. Understanding prime numbers simplifies recognizing when fractions no longer need reducing and saves time in mathematical computations.