Problem 59 - Prealgebra

Question

The following problems all involve the concept of borrowing. Subtract in case. \(10 \frac{3}{10}-4 \frac{4}{5}\)

Step-by-Step Solution

Answer

The result is \(5 \frac{1}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Start by converting each mixed number to an improper fraction. For \(10 \frac{3}{10}\), multiply the whole number \(10\) by the denominator \(10\), and add the numerator \(3\). This gives \(\frac{103}{10}\). For \(4 \frac{4}{5}\), multiply the whole number \(4\) by the denominator \(5\), and add the numerator \(4\). This gives \(\frac{24}{5}\).

Step 2: Find a Common Denominator

Identify the least common denominator (LCD) for the fractions \(\frac{103}{10}\) and \(\frac{24}{5}\), which is \(10\). Convert \(\frac{24}{5}\) to have this denominator by multiplying both the numerator and denominator by \(2\), resulting in \(\frac{48}{10}\).

Step 3: Subtract the Fractions

With a common denominator, you can subtract: \(\frac{103}{10} - \frac{48}{10} = \frac{55}{10}\).

Step 4: Simplify the Result

Simplify \(\frac{55}{10}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is \(5\). This results in \(\frac{11}{2}\).

Step 5: Convert to a Mixed Number

Convert the improper fraction \(\frac{11}{2}\) back to a mixed number by dividing the numerator \(11\) by \(2\). The quotient is \(5\) with a remainder of \(1\), giving the mixed number \(5 \frac{1}{2}\).

Key Concepts

Improper Fractions

Before diving into subtracting mixed numbers, it's essential to understand improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. This makes the fraction equal to or greater than 1, which is useful when converting mixed numbers. Converting mixed numbers involves multiplying the whole part by the denominator and adding the numerator. For example, to convert the mixed number \(10 \frac{3}{10}\) to an improper fraction:

• Multiply the whole number 10 by the denominator 10 to get 100.
• Add the numerator 3 to this product to get 103.
• Therefore, the improper fraction is \(\frac{103}{10}\).

This conversion simplifies arithmetic operations, like subtraction or addition, since we have only one type of fraction to deal with.

Least Common Denominator

When subtracting fractions, having a common denominator ensures the numerators can be directly subtracted. This process involves finding the least common denominator (LCD). The LCD is the smallest number that each denominator divides into without a remainder. In our example:

• The denominators are 10 and 5, with a least common denominator of 10.
• \(\frac{103}{10}\) already has this denominator.
• For \(\frac{24}{5}\), we adjust the fraction to \(\frac{48}{10}\) by multiplying both its numerator and denominator by 2.

With similar denominators, the operation becomes straightforward, focusing only on the numerators.

Greatest Common Divisor

Simplifying fractions is all about reducing them to their simplest form, and understanding the greatest common divisor (GCD) can be pivotal here. The GCD of two numbers is the largest number that divides both without leaving a remainder. When you have a fraction like \(\frac{55}{10}\):

• Identify the GCD of 55 and 10, which is 5.
• Divide both the numerator and the denominator by this GCD.
• This results in the simplified fraction \(\frac{11}{2}\).

Simplifying fractions makes them easier to interpret and work with, especially when converting them into mixed numbers.

Mixed Number Conversion

Finally, converting improper fractions back into mixed numbers provides a more intuitive way of understanding the solution, especially in real-world contexts. After simplifying to \(\frac{11}{2}\):

• Divide 11 by 2, which equals 5 with a remainder of 1.
• The quotient (5) becomes the whole number part of the mixed number.
• The remainder (1) stays as the numerator over the original denominator (2), forming the fraction \(\frac{1}{2}\).
• Thus, \(\frac{11}{2}\) converts to \(5 \frac{1}{2}\).

This conversion complements the fraction making it easier to interpret the result in practical scenarios, like cooking or measuring.