Problem 29 - Prealgebra

Question

Perform each of the following divisions. $$1 . 1 \longdiv { 2 . 4 2 }$$

Step-by-Step Solution

Answer

The result of \( 1.1 \div 2.42 \) is \( \frac{5}{11} \).

Step 1: Convert Mixed Number to Improper Fraction

The number 1.1 can be expressed as an improper fraction. 1.1 is equivalent to \( \frac{11}{10} \) because it equals 1 plus \( \frac{1}{10} \).

Step 2: Convert Decimal to Fraction

Now, convert 2.42 to a fraction. Since there are two digits after the decimal point, 2.42 can be written as \( \frac{242}{100} \).

Step 3: Division as Multiplication by the Reciprocal

Instead of division, we'll multiply by the reciprocal. Therefore, the problem \( \frac{11}{10} \div \frac{242}{100} \) becomes \( \frac{11}{10} \times \frac{100}{242} \).

Step 4: Simplify the Fraction

Before multiplying the fractions, simplify them. Check for any common factors between the numerators and denominators. 11 and 242 have a common factor of 11, which simplifies to \( 1/22 \). Multiply \( \frac{1}{10} \) and \( \frac{100}{22} \).

Step 5: Perform Multiplication

Multiply the simplified fractions: \( \frac{1}{10} \times \frac{100}{22} = \frac{100}{220} \).

Step 6: Simplify the Result

The fraction \( \frac{100}{220} \) can be simplified further, both numbers are divisible by 20, simplifying to \( \frac{5}{11} \).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator. These fractions can represent numbers greater than one. Let's clarify how to convert a decimal or a mixed number into an improper fraction.

- A mixed number like 1.1 is made up of a whole part and a fractional part. So, 1.1 is the sum of 1 and 0.1.- To convert this into an improper fraction, express the whole number as a fraction with a common denominator. Here, 1 is equivalent to \( \frac{10}{10} \). Add it to 0.1, which is \( \frac{1}{10} \), to get \( \frac{11}{10} \).

This shows how an improper fraction is obtained, which makes it easier to manage calculations, especially when performing operations like division.

Decimal to Fraction Conversion

Transforming a decimal into a fraction is a straightforward process and can help simplify calculations. When you have a decimal such as 2.42, you follow these steps:

- Count the number of digits after the decimal point. For 2.42, there are two digits.- Use that count to place the decimal in a fraction over 100 (because there are two digits).- Hence, 2.42 can be expressed as \( \frac{242}{100} \).

Understanding this conversion is fundamental when you are looking to perform operations like division or multiplication with decimals in a fractional form.

Simplifying Fractions

Simplifying fractions makes them easier to work with and understand. It's about reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than one.

- Look for common factors. In the problem, the fraction \( \frac{11}{242} \) was simplified with the common factor of 11, reducing it to \( \frac{1}{22} \).- Once common factors are eliminated, fractions become easier to multiply or add.

This practice is essential when you're dealing with complex operations, ensuring you handle the simplest terms and thereby minimizing errors.

Multiplying by Reciprocals

Division of fractions involves a clever trick—multiplying by the reciprocal. Instead of directly dividing, by finding the reciprocal, we simplify the operation to multiplication.

- The reciprocal of a fraction is found by swapping the numerator and denominator. For instance, the reciprocal of \( \frac{242}{100} \) is \( \frac{100}{242} \).- By multiplying \( \frac{11}{10} \) by \( \frac{100}{242} \), we sidestep division and simplify the calculation process.

Employing reciprocals is an efficient method in fraction division, ensuring precision with calculations while maintaining the integrity of the operations.