Problem 80 - Prealgebra

Question

The following problems review addition and subtraction with fractions and mixed numbers. $$6 \frac{7}{8}+1 \frac{5}{8}$$

Step-by-Step Solution

Answer

The sum is \(8 \frac{1}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions. For \(6 \frac{7}{8}\), multiply the whole number \(6\) by the denominator \(8\) and add the numerator \(7\). This results in \(6 \times 8 + 7 = 55\), giving us the improper fraction \(\frac{55}{8}\). Similarly, for \(1 \frac{5}{8}\), multiply the whole number \(1\) by the denominator \(8\) and add the numerator \(5\). This gives \(1 \times 8 + 5 = 13\), resulting in the improper fraction \(\frac{13}{8}\).

Step 2: Add Improper Fractions

Now add the two improper fractions: \(\frac{55}{8} + \frac{13}{8}\). Since they have the same denominator, you can add the numerators directly: \(55 + 13 = 68\). So, the sum of the improper fractions is \(\frac{68}{8}\).

Step 3: Simplify the Result

Simplify \(\frac{68}{8}\). Divide the numerator and the denominator by their greatest common divisor, which is \(4\). This results in \(\frac{68 \div 4}{8 \div 4} = \frac{17}{2}\).

Step 4: Convert Improper Fraction to Mixed Number

Convert \(\frac{17}{2}\) back to a mixed number by dividing the numerator by the denominator. \(17 \div 2 = 8\) with a remainder of \(1\). This converts into the mixed number \(8 \frac{1}{2}\).

Key Concepts

Mixed Numbers

Mixed numbers are a way to express a value that includes both a whole number and a fraction. They are usually written in the form of

• Whole number + a proper fraction

For example, in a problem like \(6 \frac{7}{8}\), "6" is the whole number, and "\( \frac{7}{8} \)" is the fractional part.
Understanding mixed numbers is crucial because they often appear in everyday measurements, such as in recipes or when telling time.
If you need to manipulate mixed numbers, you often have to convert them into improper fractions first, which makes mathematical operations like addition or subtraction much easier.

Improper Fractions

Improper fractions have numerators that are equal to or larger than their denominators, such as \( \frac{17}{5} \).
These fractions often arise when you convert mixed numbers to perform addition, subtraction, or other mathematical operations efficiently.
For instance, to convert the mixed number \(6 \frac{7}{8}\) to an improper fraction:

• Multiply the whole number (6) by the denominator (8): \(6 \times 8 = 48\)
• Add the numerator (7) to the result: \(48 + 7 = 55\)
• Your improper fraction is \(\frac{55}{8}\)

Improper fractions might seem awkward at first, but they are extremely helpful in simplifying complex mathematical problems.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator cannot be divided by the same number anymore, except 1.
Simplifying helps make fractions easier to understand and compare.
For example, take \(\frac{68}{8}\). To simplify:

• Find the greatest common divisor (GCD) of 68 and 8, which is 4.
• Divide both the numerator and denominator by 4: \(\frac{68 \div 4}{8 \div 4} = \frac{17}{2}\).

Simplifying does not change the fraction's value, just its representation.

Converting Fractions

Converting between mixed numbers and improper fractions is a key concept in math.
It helps make adding, subtracting, multiplying, or dividing arithmetic simpler.
To convert an improper fraction to a mixed number, like \(\frac{17}{2}\):

• Divide the numerator by the denominator: \(17 \div 2 = 8\)
• The remainder is 1, so the mixed number is \(8 \frac{1}{2}\).

Converting fractions makes handling them practical, simplifying mathematical expressions, and often making solutions more intuitive.