Problem 21 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$1.68$$

Step-by-Step Solution

Answer

The decimal \(1.68\) is \(1 \frac{17}{25}\) as a mixed number.

Step 1: Understand the Decimal

Recognize that the decimal \(1.68\) can be split into a whole number and a decimal part. Here, the whole number is 1 and the decimal part is 0.68.

Step 2: Convert the Decimal Part to a Fraction

Write the decimal part \(0.68\) as a fraction. Since there are two digits after the decimal point, \(0.68\) becomes \(\frac{68}{100}\).

Step 3: Simplify the Fraction

Simplify \(\frac{68}{100}\) by finding the greatest common divisor (GCD) of 68 and 100. The GCD is 4. Divide both the numerator and the denominator by 4 to get \(\frac{17}{25}\).

Step 4: Combine the Whole Number and the Fraction

Combine the whole number 1 with the simplified fraction \(\frac{17}{25}\) to get the mixed number. The result is \(1 \frac{17}{25}\).

Key Concepts

Simplifying Fractions

Simplifying fractions means rewriting a fraction in its simplest form where the numerator and denominator have no common factors other than 1. This process makes fractions easier to understand and use in calculations. To do this, identify the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that can evenly divide both the numerator and the denominator.

An easy method to find the GCD is to list the factors of each number and then identify the largest common factor. Once the GCD is determined, divide both the numerator and denominator by this number. For example, to simplify the fraction \(\frac{68}{100}\), you find that the GCD is 4. Dividing both 68 and 100 by 4 results in the fraction \(\frac{17}{25}\).

Remember:

• The GCD is crucial for simplifying fractions.
• Simplifying fractions does not change their value, only their form.
• Always check if further simplification is possible by verifying common factors.

This technique ensures your fractions are presented in the easiest, cleanest way possible.

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction, representing numbers greater than 1 in an intuitive way. When you have a decimal, converting it involves separating the whole number from the fractional part. This makes reading and working with these numbers much more straightforward.

For instance, take the decimal 1.68. The whole number is 1, and the decimal part 0.68 can be written as the fraction \(\frac{68}{100}\). Simplifying \(\frac{68}{100}\) results in \(\frac{17}{25}\), as described earlier. Combining these gives a mixed number: \(1 \frac{17}{25}\).

Benefits of Mixed Numbers:

• They make it easier to understand the size of the number at a glance.
• Mixed numbers are convenient for adding or subtracting from whole numbers.
• They are often more intuitive in real-world measurements, like cooking or construction.

Using mixed numbers helps in better conceptualizing tasks that require both whole numbers and fractions.

Greatest Common Divisor

The greatest common divisor (GCD) is invaluable when simplifying fractions. It allows you to reduce fractions to their simplest form by dividing the numerator and the denominator by their largest shared factor. This is crucial in making operations with fractions as simple and effective as possible.

To find the GCD of two numbers, such as 68 and 100, you can use techniques such as:

• **Prime Factorization:** Breaking down each number into its prime factors and identifying the highest common factor.
• **Euclidean Algorithm:** A more advanced method where you repeatedly subtract and take remainders until you arrive at the GCD.

In the example

• The factors of 68 are 1, 2, 4, 17, 34, and 68.
• The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.

The common factors are 1, 2, and 4, with 4 being the greatest. Using the GCD ensures fractions are expressed as clearly and compactly as possible, simplifying further mathematical computations.