Problem 68 - Prealgebra

Question

Simplify the given expression. \(\frac{5}{3}-1.1\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{17}{30} \).

Step 1: Convert Decimal to Fraction

First, let's convert the decimal number in the expression to a fraction. The number 1.1 can be written as \( \frac{11}{10} \). So the expression becomes \( \frac{5}{3} - \frac{11}{10} \).

Step 2: Find a Common Denominator

To subtract these fractions, they need to have a common denominator. The denominators are 3 and 10. The least common denominator is 30.

Step 3: Convert Fractions to Common Denominator

Convert each fraction to have a denominator of 30. Multiply \( \frac{5}{3} \) by \( \frac{10}{10} \) to get \( \frac{50}{30} \), and \( \frac{11}{10} \) by \( \frac{3}{3} \) to get \( \frac{33}{30} \).

Step 4: Subtract Fractions

Now, subtract the two fractions: \( \frac{50}{30} - \frac{33}{30} = \frac{50-33}{30} = \frac{17}{30} \).

Step 5: Check for Simplification

The fraction \( \frac{17}{30} \) is already in its simplest form since 17 is a prime number and does not share any common factors with 30 other than 1.

Key Concepts

Fractions

Fractions are an essential part of mathematics that involve parts of a whole. A fraction consists of two components:

• The numerator, which indicates how many parts we are considering.
• The denominator, which shows how many equal parts make up a whole.

In the expression \( \frac{5}{3} \), 5 is the numerator, and 3 is the denominator. The fraction represents 5 parts of something that is divided into 3 equal parts.
Fractions can be added, subtracted, multiplied, or divided, which allows for extensive flexibility in calculations. Simplifying fractions involves reducing them to their lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Decimal to Fraction Conversion

Decimal numbers are another way of representing non-whole numbers. Converting a decimal to a fraction involves a few simple steps:

• Count the number of digits after the decimal point.
• Use this count to determine the power of 10 as the denominator.
• Write the number without the decimal point as the numerator.
• Reduce if possible.

For example, the decimal 1.1 has one digit after the decimal. Therefore, it translates into \( \frac{11}{10} \), as 1.1 equals 11/10. The conversion helps in unifying different parts of an equation or an expression, like in our problem where having both values as fractions helps with direct subtraction.

Least Common Denominator

Finding a common denominator is vital when adding or subtracting fractions. The least common denominator (LCD) is the smallest multiple that two or more denominators share. It allows fractions to have the same denominator, making them easy to add or subtract. To find the LCD:

• Identify the denominators.
• Find the least common multiple (LCM) of these denominators.

For example, the denominators 3 and 10 have an LCD of 30 because 30 is the smallest number both can divide equally. This ensures all parts of fractions are directly comparable when performing arithmetic operations.

Prime Numbers

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They play a crucial role in mathematics, especially in the simplification of fractions. When reducing fractions, if the numerator is a prime number and does not share any factors with the denominator, the fraction is in its simplest form.In our solution, the final result was \( \frac{17}{30} \). Here, 17 is a prime number, and it does not have common factors with 30 other than 1. This means \( \frac{17}{30} \) is already simplified. Prime numbers serve as building blocks, and knowing them can greatly aid in recognizing when a fraction is simplified.