Problem 276 - Intermediate Algebra

Question

In the following exercises, write each decimal as a fraction. $$ 1.464 $$

Step-by-Step Solution

Answer

1.464 as a fraction is \(\frac{183}{125}\).

Step 1: Identify the Decimal

Examine the given decimal, which is 1.464.

Step 2: Express Decimal as a Fraction

Write the decimal as a fraction with a denominator that corresponds to its decimal places. Since 1.464 has three decimal places, it can be written as \(\frac{1464}{1000}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{1464}{1000}\) by finding the greatest common divisor (GCD) of 1464 and 1000. The GCD is 8. Divide both the numerator and the denominator by 8 to get the fraction in its simplest form: \(\frac{183}{125}\). This gives \(\frac{1464 \,/ \, 8}{1000 \,/ \, 8} = \frac{183}{125}\).

Key Concepts

simplifying fractions

Simplifying fractions is an essential step in many math problems. When you have a fraction, you want it to be in its simplest form, which means the numerator and the denominator have no common factors other than 1. To simplify a fraction, you need to:

• Identify the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

For instance, in the given exercise, we started with the fraction \(\frac{1464}{1000}\). Finding the GCD (which is 8) allowed us to simplify it to \(\frac{183}{125}\).

greatest common divisor (GCD)

The greatest common divisor (GCD) is the largest positive integer that divides two or more numbers without leaving a remainder. To find the GCD of two numbers:

• List the factors of each number.
• Identify the common factors.
• Select the largest common factor, which will be the GCD.

Alternatively, the Euclidean algorithm provides a quicker way to find the GCD. For example, the GCD of 1464 and 1000 can be found through repetitive division:
\(\text{GCD}(1464, 1000) = \text{GCD}(1000, 464)\).
Continuing, \(\text{GCD}(1000, 464) = \text{GCD}(464, 72)\) and finally, \(\text{GCD}(72, 8) = 8\). Therefore, the GCD is 8. This enables us to simplify the fraction as shown in the exercise.

decimal places

Decimals represent fractions where the denominator is a power of ten. The number of decimal places tells you the power of ten to use in the denominator.

• One decimal place means the denominator is 10. Example: 0.3 is \(\frac{3}{10}\)
• Two decimal places mean the denominator is 100. Example: 0.75 is \(\frac{75}{100}\)
• Three decimal places mean the denominator is 1000. Example: 1.464 is \(\frac{1464}{1000}\)

Understanding and identifying the correct number of decimal places is essential in converting decimals to fractions accurately, as demonstrated in this exercise.