Problem 53 - Prealgebra

Question

The following problems all involve the concept of borrowing. Subtract in case. \(4 \frac{1}{4}-2 \frac{1}{3}\)

Step-by-Step Solution

Answer

The difference is \(1 \frac{11}{12}\).

Step 1: Convert Mixed Numbers to Improper Fractions

The first step is to convert the mixed numbers into improper fractions so that subtraction is easier to handle.- For \(4 \frac{1}{4}\), multiply 4 (whole number) by 4 (denominator) and add 1 (numerator):\[4 \times 4 + 1 = 17\]So, \(4 \frac{1}{4}\) is \(\frac{17}{4}\).- For \(2 \frac{1}{3}\), multiply 2 (whole number) by 3 (denominator) and add 1 (numerator):\[2 \times 3 + 1 = 7\]So, \(2 \frac{1}{3}\) is \(\frac{7}{3}\).

Step 2: Find a Common Denominator

To subtract the fractions, they must have the same denominator. The denominators are 4 and 3. - Find the least common multiple (LCM) of 4 and 3, which is 12. This will be the common denominator.

Step 3: Convert Fractions to Equivalent Fractions with the Common Denominator

Convert each fraction so that it has the common denominator of 12.- For \(\frac{17}{4}\): Multiply both the numerator and denominator by 3 to get:\[\frac{17 \times 3}{4 \times 3} = \frac{51}{12}\]- For \(\frac{7}{3}\): Multiply both the numerator and denominator by 4 to get:\[\frac{7 \times 4}{3 \times 4} = \frac{28}{12}\]

Step 4: Subtract the Fractions

Now, subtract the second fraction from the first:\[\frac{51}{12} - \frac{28}{12} = \frac{23}{12}\]The result is the improper fraction \(\frac{23}{12}\).

Step 5: Convert Improper Fraction Back to Mixed Number

Convert the resulting improper fraction back to a mixed number.- Divide 23 by 12, which equals 1 with a remainder of 11:\[23 \div 12 = 1 \, ext{remainder} \, 11\]- Therefore, \(\frac{23}{12}\) is equivalent to \(1 \frac{11}{12}\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

To work with fractions in operations like addition or subtraction, converting mixed numbers to improper fractions is a helpful first step. Mixed numbers are a combination of a whole number and a proper fraction, like \(4 \frac{1}{4}\). To convert:

• Multiply the whole number by the denominator. In the example, 4 (whole number) times 4 (denominator) = 16.
• Add the product to the numerator. Here, 16 + 1 (numerator) = 17.
• Place this sum over the original denominator to complete the conversion: \(\frac{17}{4}\).

This process makes it easier to operate since you can work purely with fractions, bypassing potential confusion with mixed numbers.

Least Common Multiple (LCM)

Finding the least common multiple (LCM) is essential when adding or subtracting fractions with different denominators. The LCM is the smallest number that is a multiple of each of the denominators involved.

• For instance, with denominators like 4 and 3, list the multiples of each: 4 (4, 8, 12, 16...) and 3 (3, 6, 9, 12...).
• The smallest shared value is 12, which becomes the common denominator.

The LCM is crucial because it allows fractions to be rewritten so they share a denominator, facilitating easier arithmetic operations.

Improper Fractions

Improper fractions are fractions where the numerator is equal to or larger than the denominator, such as \(\frac{17}{4}\). These are common outputs when converting mixed numbers and are useful in calculations.

• They simplify subtraction: having both numbers as improper fractions enables direct subtraction once they are aligned with a common denominator.
• Every improper fraction can also be converted back into a mixed number for easier interpretation of the result.

Keeping these formats straight is key to mastering fraction operations.

Fraction Subtraction

Fraction subtraction, particularly with different denominators, requires precision and attention to structure. Once you have common denominators:

• Align the two fractions so that their denominators match. For example, converting to \(\frac{51}{12}\) and \(\frac{28}{12}\).
• Subtract the numerators directly: \(51 - 28 = 23\).
• Your result is the fraction \(\frac{23}{12}\), an improper fraction often requiring conversion back to a mixed number.

This systematic approach ensures errors are minimized, and results are accurate and easy to understand.