Problem 38 - Prealgebra

Question

Find the following sums. (Add.) \(\frac{5}{8}+1 \frac{1}{8}+7\)

Step-by-Step Solution

Answer

The sum is \(8 \frac{3}{4}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number, which is \(1 \frac{1}{8}\), into an improper fraction. The whole number is 1 and the fraction is \(\frac{1}{8}\). To convert, multiply the whole number 1 by the denominator 8, and add the numerator 1: \(1 \times 8 + 1 = 9\). Hence, \(1 \frac{1}{8} = \frac{9}{8}\).

Step 2: Write All Terms with the Same Denominator

Now, express all terms as fractions with the same denominator 8. The first term is already \(\frac{5}{8}\). The second term is now \(\frac{9}{8}\). For the whole number 7, rewrite it as a fraction \(\frac{56}{8}\) since \(7 = \frac{56}{8}\) because \(7 \times 8 = 56\).

Step 3: Add the Fractions

Now add the three fractions: \(\frac{5}{8} + \frac{9}{8} + \frac{56}{8}\). Since all fractions have the common denominator 8, add their numerators: \(5 + 9 + 56 = 70\).

Step 4: Simplify the Resulting Fraction

The result of the addition is \(\frac{70}{8}\). To simplify it, divide the numerator and denominator by their greatest common divisor, which is 2. \(\frac{70 \div 2}{8 \div 2} = \frac{35}{4}\). This is the simplified improper fraction.

Step 5: Convert Improper Fraction to Mixed Number

The improper fraction \(\frac{35}{4}\) can be converted to a mixed number by dividing 35 by 4. The quotient is 8, and the remainder is 3. Thus, \(\frac{35}{4} = 8 \frac{3}{4}\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers are unique because they consist of two parts: a whole number and a fraction. This combination is very common in everyday life. For example, if you eat one and a half pizzas, you’ve had 1 whole pizza and half of another, hence you’d express this as a mixed number: “1 1/2”.
When you have a mixed number, it can be slightly tricky to work with them in mathematical calculations. This is where converting them to improper fractions can be helpful. To do so, you take the whole number, multiply it by the denominator of the fraction, and add the numerator to this product.
With our example in the exercise, the mixed number is 1 1/8. Here's how you convert it:

• Multiply the whole number (1) by the denominator (8): \(1 \times 8 = 8\)
• Add the numerator (1): \(8 + 1 = 9\)
• The improper fraction is \(\frac{9}{8}\)

Now, it’s ready to be used in operations like addition, where it behaves similarly to other fractions.

Exploring Improper Fractions

An improper fraction occurs when the numerator is larger than the denominator. This can seem confusing at first because we’re used to the numerator being smaller, but they have clear advantages.
Improper fractions make addition and subtraction more straightforward compared to mixed numbers. For example, with an improper fraction like \(\frac{9}{8}\), all the components are expressed as parts of the same whole. This is particularly useful because it allows for more straightforward operations when you have multiple fractions with a common denominator.
Why convert back to a mixed number at all? Well, while improper fractions are great for calculations, mixed numbers often make more sense when interpreting results. For instance, \(\frac{35}{4}\) is an unacceptable fraction at first glance, but converting it to 8 3/4 gives a clearer idea of the size of the number. Here’s the conversion process:

• Divide the numerator (35) by the denominator (4): 35 ÷ 4 = 8 with a remainder of 3
• The whole number part is 8, and the fraction is \(\frac{3}{4}\)

Thus, \(\frac{35}{4}\) is equivalent to 8 3/4 as a mixed number.

The Importance of a Common Denominator

When you're adding fractions, especially multiple ones, it’s crucial to have a common denominator. This ensures that all fractions are talking about the same sized parts of their whole. Think of it as counting apples, but all apples have to be the same size to count them properly. So let's dig deeper into why a common denominator is necessary.
With fractions like 5/8, 9/8, and the whole number 7 converted to 56/8, they all have the same denominator, which is 8. This denominator represents the division of a whole into 8 parts, and each fraction refers to how many parts are being considered.

• When adding, like in this exercise, these fractions: \(\frac{5}{8} + \frac{9}{8} + \frac{56}{8}\), they each contain pieces of size 1/8.
• You simply add the numerators: 5, 9, and 56 to get 70 since they are all parts of the same whole.

By ensuring all fractions have a common denominator before adding, subtracting, or comparing them, we maintain consistency and accuracy in our calculations. It’s one of the foundational principles of dealing with fractions efficiently.