Problem 45 - Prealgebra

Question

The following problems all involve the concept of borrowing. Subtract in case. \(8-1 \frac{3}{4}\)

Step-by-Step Solution

Answer

The result of the subtraction is \(6 \frac{1}{4}\).

Step 1: Convert Mixed Number to Improper Fraction

The mixed number in the problem is \(1 \frac{3}{4}\). We need to convert this into an improper fraction to make subtraction easier. Multiply the whole number (1) by the denominator (4) and add the numerator (3) to it: \(1 \times 4 + 3 = 7\). Thus, \(1 \frac{3}{4}\) becomes \(\frac{7}{4}\).

Step 2: Convert Whole Number to Fraction

The number 8 is a whole number. To subtract a fraction from it, we need to express it as a fraction with the same denominator as \(\frac{7}{4}\). Multiply 8 by 4 to get the numerator: \(8 \times 4 = 32\). So, \(8\) is equivalent to \(\frac{32}{4}\).

Step 3: Perform the Subtraction

Now, subtract the fractions: \(\frac{32}{4} - \frac{7}{4}\). Since they have the same denominator, you simply subtract the numerators: \(32 - 7 = 25\). So, the answer is \(\frac{25}{4}\).

Step 4: Convert Improper Fraction to Mixed Number

Convert \(\frac{25}{4}\) back to a mixed number. Divide 25 by 4, which results in 6 with a remainder of 1. This gives us the mixed number \(6 \frac{1}{4}\).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator is larger than or equal to the denominator. This means the fraction is actually representing a number greater than one. They are useful in mathematical operations because they make it easier to perform addition, subtraction, multiplication, and division with fractions. To change a mixed number into an improper fraction, you do the following:

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• This sum becomes the new numerator, while the denominator remains the same.

This process bridges the gap between whole numbers and fractions, allowing for seamless calculations.

Mixed Numbers

A mixed number combines a whole number and a fraction, like this: 1 3/4. They are typically easier for people to understand because they illustrate exactly how many times a whole number fits into a number plus the extra fraction.

• Mixed numbers are often converted to improper fractions for calculations.
• This makes mathematical operations straightforward and less cumbersome.
• After calculations, converting back to mixed numbers can make the result easier to interpret.

For instance, converting 1 3/4 to an improper fraction gives us 7/4, simplifying subtraction.

Borrowing in Subtraction

Borrowing in subtraction with fractions is similar to borrowing in normal subtraction. However, to "borrow" you first convert numbers to fractions with a common denominator. This ensures that both numbers are compatible for direct subtraction.

• Begin with converting the whole number to a fraction with the same denominator as the other fraction.
• Perform the necessary "borrowing" by adjusting whole numbers to facilitate the subtraction.
• It often involves breaking down the whole number into manageable parts.

This concept helps tackle subtraction problems involving fractions more effectively, ensuring clarity in results and understanding.