Problem 61 - Elementary and Intermediate Algebra

Question

Between January 2004 and January \(2008,\) the south end of the Great Salt Lake dropped \(\frac{1}{2} \mathrm{ft},\) rose \(\frac{6}{5} \mathrm{ft},\) rose \(\frac{3}{4} \mathrm{ft},\) and dropped \(\frac{3}{2} \mathrm{ft} .\) By how much did the level change? Source: U.S. Geological Survey

Step-by-Step Solution

Answer

The lake level dropped by \(\frac{1}{20} \text{ft}\).

Step 1 - List the changes

Identify each change in the lake's level between January 2004 and January 2008. The changes are: dropped \(\frac{1}{2} \text{ft}\), rose \(\frac{6}{5} \text{ft}\), rose \(\frac{3}{4} \text{ft}\), and dropped \(\frac{3}{2} \text{ft}\).

Step 2 - Assign positive or negative values

Assign negative values to drops and positive values to rises: - Drop \(\frac{1}{2} \text{ft} = -\frac{1}{2} \text{ft}\) - Rose \(\frac{6}{5} \text{ft} = +\frac{6}{5} \text{ft}\) - Rose \(\frac{3}{4} \text{ft} = +\frac{3}{4} \text{ft}\) - Drop \(\frac{3}{2} \text{ft} = -\frac{3}{2} \text{ft}\)

Step 3 - Convert to common denominator

Convert all fractions to a common denominator to make addition and subtraction easier. The least common denominator for \(\frac{1}{2}\), \(\frac{6}{5}\), \(\frac{3}{4}\), and \( \frac{3}{2} \) is 20. - \(-\frac{1}{2} = -\frac{10}{20}\) - \(\frac{6}{5} = \frac{24}{20}\) - \(\frac{3}{4} = \frac{15}{20}\) - \(-\frac{3}{2} = -\frac{30}{20}\)

Step 4 - Sum the values

Add the values together: \(-\frac{10}{20} + \frac{24}{20} + \frac{15}{20} - \frac{30}{20}\).

Step 5 - Simplify the sum

Simplify the sum by combining the numerators: \(-10 + 24 + 15 - 30 = -1\). Thus, \(-\frac{1}{20} \text{ft}\).

Step 6 - Interpret the result

The result is \(-\frac{1}{20} \text{ft}\), which means that the lake level dropped by \(\frac{1}{20} \text{ft}\) over the given period.

Key Concepts

adding fractions

Adding fractions involves combining the numerators while keeping a common denominator. For example, if you have fractions like \(\frac{1}{4} + \frac{2}{4}\), you add the numerators (1 and 2) and keep the denominator (4). This gives you \(\frac{3}{4}\).

It's important to ensure that the fractions have the same denominator before you add them. If they don't, you'll need to convert them so they do. Take \(\frac{1}{3}\) and \(\frac{1}{6}\), for instance. You'll need to convert them to a common denominator like 6 to add them: \(\frac{1}{3} = \frac{2}{6}\), then \(\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\).

This concept is crucial when summing fractions in exercises and everyday math situations.

subtracting fractions

Subtracting fractions follows the same concept as adding fractions: having a common denominator is key. If you need to subtract \(\frac{5}{8}\) from \(\frac{3}{4}\), you first change \(\frac{3}{4}\) to \(\frac{6}{8}\) so both fractions share the denominator 8. Then, you subtract the numerators: \(\frac{6}{8} - \frac{5}{8} = \frac{1}{8}\).

When dealing with mixed numbers or improper fractions, you might need to convert them to improper fractions first or find equivalent fractions with a common denominator. For example, converting \(\frac{2}{3}\) and \(\frac{1}{6}\) into a common denominator like 6 gives you \(\frac{4}{6}\) and \(\frac{1}{6}\). Thus, subtracting \(\frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\).

This method simplifies handling fraction subtraction in various scenarios.

common denominator

Finding a common denominator is essential for adding or subtracting fractions. A common denominator is a shared multiple of the denominators of the fractions involved. This allows easy addition or subtraction.

For example, consider fractions \(\frac{1}{2}\) and \(\frac{2}{5}\). The denominators are 2 and 5, and the least common multiple (LCM) is 10. So, you convert \(\frac{1}{2}\) to \(\frac{5}{10}\) and \(\frac{2}{5}\) to \(\frac{4}{10}\). Now, both fractions have a denominator of 10, allowing you to add or subtract them as needed.

Another example: for \(\frac{3}{4}\) and \(\frac{7}{8}\), the LCM is 8, making the equivalent fractions \(\frac{6}{8}\) and \(\frac{7}{8}\). With a common denominator, it's straightforward to proceed with the operation.

Utilizing a common denominator ensures smooth and accurate fraction arithmetic.