Problem 81 - Prealgebra

Question

Find each of the following sums and differences. (Add or subtract.) $$5 \frac{1}{10}+6 \frac{2}{100}+7 \frac{3}{1,000}$$

Step-by-Step Solution

Answer

The sum is \(18 \frac{123}{1000}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Start by converting each mixed number into an improper fraction. For \(5 \frac{1}{10}\), convert it as follows:For the whole number part, \(5\), multiply it by the denominator of the fraction part \((10)\) giving \(50\). Add the numerator \(1\) to get \(51\), therefore, \(5 \frac{1}{10} = \frac{51}{10}\).Similarly, for \(6 \frac{2}{100}\), \(6 \times 100 + 2 = 602\), so \(6 \frac{2}{100} = \frac{602}{100}\).Finally, for \(7 \frac{3}{1,000}\), \(7 \times 1000 + 3 = 7003\), so \(7 \frac{3}{1000} = \frac{7003}{1000}\).

Step 2: Find the Common Denominator

The fractions have denominators of 10, 100, and 1000. The least common denominator (LCD) is 1000. We will convert each fraction to have the denominator of 1000:For \(\frac{51}{10}\), multiply both the numerator and denominator by 100 to get \(\frac{5100}{1000}\).For \(\frac{602}{100}\), multiply both the numerator and denominator by 10 to get \(\frac{6020}{1000}\).The last fraction, \(\frac{7003}{1000}\), is already with the denominator 1000.

Step 3: Add the Fractions

Now that all fractions have the same denominator, add them together:\[\frac{5100}{1000} + \frac{6020}{1000} + \frac{7003}{1000} = \frac{5100 + 6020 + 7003}{1000} = \frac{18123}{1000}\].

Step 4: Convert the Result Back to a Mixed Number

Convert \(\frac{18123}{1000}\) back to a mixed number. Divide 18123 by 1000, which gives 18 with a remainder of 123. So, \(\frac{18123}{1000} = 18 \frac{123}{1000}\).

Key Concepts

Mixed Numbers

Mixed numbers are expressions that combine a whole number and a fraction. They are used to represent quantities greater than one with a whole and fractional part. For example, in the mixed number \(5 \frac{1}{10}\), the 5 is the whole number, and \(\frac{1}{10}\) is the fractional part. This format helps when visualizing measurements like feet and inches or pounds and ounces, where both the whole and part measure something tangible.

When working with mixed numbers, especially calculating, you'll frequently convert them to improper fractions. Doing so makes addition, subtraction, multiplication, and division often simpler. The conversion process involves multiplying the whole number by the fraction's denominator, then adding the numerator to this product. The result becomes the new numerator of the improper fraction with the original denominator.

Improper Fractions

Improper fractions are those where the numerator (top number) is greater than or equal to the denominator (bottom number). For example, \(\frac{51}{10}\) is an improper fraction. Improper fractions are key when dealing with operations like addition and subtraction of mixed numbers: they're easier to manipulate in equations.

To convert a mixed number to an improper fraction, as seen in the exercise solution, multiply the whole number by the denominator, add the numerator, and write this sum over the original denominator. This conversion helps streamline arithmetic operations, allowing for simpler computation and, ultimately, easier understanding and manipulation of fractions.

Least Common Denominator

The least common denominator (LCD) of a set of fractions is the smallest number that all the denominators divide into evenly. It's vital for adding and subtracting fractions with different denominators, as fractions must first convert to equivalent fractions with a common denominator.

In the exercise, the denominators 10, 100, and 1000 result in an LCD of 1000. This means converting each fraction so that the denominator is 1000, facilitating simple addition. This conversion is done by multiplying the numerator and denominator of each fraction by the same number until the denominators match the LCD.

This step ensures accuracy and coherence when you add or subtract fractions without errors that arise when ignoring differing denominators.

Addition of Fractions

Adding fractions involves ensuring they have the same (common) denominator, accomplished by finding the least common denominator if necessary. After denominators match, you can add the numerators together while keeping the common denominator, simplifying further if necessary.

Using the exercise's example: once each fraction was rewritten to have 1000 as the denominator, the addition process was straightforward. The numerators \(5100\), \(6020\), and \(7003\) were added together to get \(18123\), then placed over the common denominator of 1000, yielding \(\frac{18123}{1000}\).

Understanding this addition process ensures one can handle more complex fraction operations confidently. After addition, the fraction can often convert back to a mixed number to express the result plainly.