Problem 39 - Prealgebra

Question

The problems below will allow you to review subtraction of fractions and mixed numbers. $$2-\frac{4}{3}$$

Step-by-Step Solution

Answer

The answer is \( \frac{2}{3} \).

Step 1: Write the whole number as a fraction

The number 2 can be written as a fraction with a denominator of 1. So, 2 becomes \( \frac{2}{1} \).

Step 2: Find a common denominator

To subtract \( \frac{4}{3} \) from \( \frac{2}{1} \), choose a common denominator. Here, the least common multiple of 1 and 3 is 3.

Step 3: Convert the whole number

Convert \( \frac{2}{1} \) to have the common denominator 3. Multiply both the numerator and denominator by 3 to get \( \frac{6}{3} \).

Step 4: Subtract the fractions

Now subtract the fractions: \( \frac{6}{3} - \frac{4}{3} = \frac{6-4}{3} = \frac{2}{3} \).

Key Concepts

Least Common Multiple

When subtracting fractions, it’s essential to have a common denominator. The least common multiple (LCM) helps us find that shared base for the fractions. In essence, the LCM is the smallest number that both denominators can divide into without leaving a remainder.

For example, if you want to subtract \( \frac{4}{3} \) from \( \frac{2}{1} \), you first need to find the LCM of the denominators, 1 and 3. Since every number is a multiple of 1, the LCM in this case is simply 3.

By using the least common multiple, you can transform each fraction into an equivalent one that can be easily subtracted. This step is crucial in ensuring that you’re working with fractions in a compatible form, allowing you to perform subtraction correctly.

Mixed Numbers

Mixed numbers contain both a whole number and a fraction part. For instance, 3\( \frac{1}{2} \) is a mixed number. While the exercise itself doesn’t include mixed numbers, understanding this concept is useful because many operations in fraction arithmetic do involve them.

When dealing with mixed numbers, you often need to convert them into improper fractions before performing subtraction or addition. An improper fraction is one where the numerator (top number) is greater than the denominator (bottom number).

To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Then, place this result over the original denominator. This step simplifies arithmetic operations by dealing directly with numerators and denominators without the complexity of whole numbers mixed in.

Fraction Subtraction Steps

Subtracting fractions follows a specific set of steps to ensure the calculation is accurate. These steps help you manage both simple and complex fractions.

Here’s how you can successfully subtract fractions like in the exercise:

• Step 1: Convert the Whole Number
  First, express any whole number as a fraction. For instance, the number 2 can be expressed as \( \frac{2}{1} \). Doing this ensures both numbers are in fraction form for easy manipulation.


• Step 2: Find a Common Denominator
  Identify the least common multiple of the denominators involved. For \( \frac{2}{1} \) and \( \frac{4}{3} \), the LCM of 1 and 3 is 3. Having a common denominator is necessary for subtraction.


• Step 3: Adjust Fractions
  Adjust the fractions so that they have this common denominator. This typically involves multiplying both the numerator and the denominator of the fractions by the appropriate number. So you would convert \( \frac{2}{1} \) to \( \frac{6}{3} \).


• Step 4: Subtract the Fractions
  Once the fractions have a common denominator, subtract the numerators: \( \frac{6}{3} - \frac{4}{3} = \frac{2}{3} \). The denominators remain the same.

These steps make subtracting fractions a breeze, transforming what might seem complicated into a series of straightforward tasks.