Problem 45 - Prealgebra

Question

Find each sum or difference. Write in simplest form. \(12 \frac{7}{8}-7 \frac{3}{8}+2 \frac{5}{8}\)

Step-by-Step Solution

Answer

The result is \(8 \frac{1}{8}\).

Step 1: Convert Mixed Numbers to Improper Fractions

To simplify the calculations, convert each mixed number into an improper fraction. For \(12 \frac{7}{8}\), multiply 12 by 8 and add 7: \((12 \times 8) + 7 = 103\), so it becomes \(\frac{103}{8}\). For \(7 \frac{3}{8}\), multiply 7 by 8 and add 3: \((7 \times 8) + 3 = 59\), so it becomes \(\frac{59}{8}\). For \(2 \frac{5}{8}\), multiply 2 by 8 and add 5: \((2 \times 8) + 5 = 21\), so it becomes \(\frac{21}{8}\).

Step 2: Perform Addition and Subtraction with Fractions

Since all fractions have the same denominator (8), you can directly perform addition and subtraction on the numerators. First, calculate the subtraction: \(\frac{103}{8} - \frac{59}{8} = \frac{103 - 59}{8} = \frac{44}{8}\). Then, add the result to \(\frac{21}{8}\): \(\frac{44}{8} + \frac{21}{8} = \frac{44 + 21}{8} = \frac{65}{8}\).

Step 3: Convert the Improper Fraction to a Mixed Number

To convert \(\frac{65}{8}\) into a mixed number, divide 65 by 8. The quotient is 8 and the remainder is 1. So, \(\frac{65}{8}\) can be written as \(8 \frac{1}{8}\).

Step 4: Simplify the Mixed Number

Check to confirm the fractional part \(\frac{1}{8}\) is in its simplest form. Since 1 and 8 have no common factors other than 1, it is already simplified. Thus, the final answer is \(8 \frac{1}{8}\).

Key Concepts

Mixed Numbers

Mixed numbers are simply a combination of a whole number and a fraction. This way of expressing a number makes it easier to visualize certain quantities. For example, imagine you have 12 full pizzas and 7 pieces, each cut into 8 parts. Instead of saying you have 103 eighth pieces, you'd naturally say you have 12 whole pizzas and 7 out of 8 slices of another. Hence, we say you have \(12 \frac{7}{8}\).To work with these in math problems, it can be helpful to transform them into improper fractions, where the numerator (the top number) is larger than the denominator (the bottom number). This transformation makes it easier to add, subtract, multiply, or divide mixed numbers. The process involves multiplying the whole number by the fraction's denominator and adding the numerator to the result, as shown in the original solution. This converts \(12 \frac{7}{8}\) into \(\frac{103}{8}\). By using an improper fraction, the math problem becomes easier to manage.

Improper Fractions

Improper fractions have numerators larger than their denominators, which can seem confusing at first. Think of it as having more pieces than one full set. So if you had \(\frac{103}{8}\), it means you have 103 pieces of something that's divided into 8 parts per whole unit.Improper fractions are incredibly useful, especially in calculations. They allow for straightforward arithmetic operations without the need to handle the separate portioned whole numbers of mixed numbers. To convert from an improper fraction to a mixed number, use division. Divide the numerator by the denominator. The quotient becomes the whole number, and the remainder is the new numerator over the original denominator. For instance, when \(\frac{65}{8}\) is divided, it results in \(8 \frac{1}{8}\). This conversion is essential for interpreting the result of a calculation in a recognizable or practical form.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their smallest form, where the numerator and the denominator have no common factors other than 1. This process ensures that the fraction is expressed in the simplest and most understandable form.To check if a fraction is simplified, determine if both the numerator and denominator can be divided by the same number greater than one. If they can't, the fraction is already as simple as it gets, like \(\frac{1}{8}\). If possible, divide both by their greatest common divisor (GCD) to simplify the fraction further.Simplified fractions often make the result neater and easier to comprehend. For example, while \(\frac{44}{8}\) isn't its simplest form, dividing both by 4 gives \(\frac{11}{2}\). Simpler fractions are preferred because they offer a clearer picture of the numbers involved and help in avoiding unnecessary complexity in further calculations.