Problem 60 - Prealgebra

Question

Body Temperature Suppose your normal body temperature is \(98 \frac{3}{5} \circ\) Fahrenheit. If your temperature goes up \(3 \frac{1}{5} \circ\) on Monday and then down \(1 \frac{4}{5} \circ\) on Tuesday, what is your temperature on Tuesday?

Step-by-Step Solution

Answer

Your temperature on Tuesday is \(99 \frac{3}{5} \degree F\).

Step 1: Convert All Mixed Numbers to Improper Fractions

To make calculations easier, convert the mixed numbers to improper fractions. For the normal body temperature, convert \(98 \frac{3}{5}\) to an improper fraction: \(98 \frac{3}{5} = \frac{491}{5}\). For the increase, convert \(3 \frac{1}{5}\) to an improper fraction: \(3 \frac{1}{5} = \frac{16}{5}\). For the decrease, convert \(1 \frac{4}{5}\) to an improper fraction: \(1 \frac{4}{5} = \frac{9}{5}\).

Step 2: Add the Increase to the Starting Temperature

On Monday, your temperature increased by \(3 \frac{1}{5}\) degrees. To find the new temperature, add the improper fractions: \(\frac{491}{5} + \frac{16}{5} = \frac{507}{5}\).

Step 3: Subtract the Decrease from Monday's Temperature

On Tuesday, your temperature decreased by \(1 \frac{4}{5}\) degrees. Subtract this from Monday's result: \(\frac{507}{5} - \frac{9}{5} = \frac{498}{5}\).

Step 4: Convert the Final Result Back to a Mixed Number

Convert the improper fraction \(\frac{498}{5}\) back to a mixed number. Divide 498 by 5 to get 99 with a remainder of 3. So, \(\frac{498}{5} = 99 \frac{3}{5}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They are used frequently in everyday scenarios, such as describing measurements or temperatures. When you see a mixed number, like \(98 \frac{3}{5}\), it means you have \(98\) whole units plus an additional \(\frac{3}{5}\) of a unit.

Mixed numbers are useful because they can give a clearer picture of a value, especially when dealing with measurements that are more than one whole but less than two or three. However, they can complicate arithmetic operations, so they must often be converted into improper fractions for calculations.

• Example: To convert the mixed number \(3 \frac{1}{2}\) into an improper fraction, multiply the whole number \(3\) by the denominator \(2\), giving you \(6\). Then add the numerator \(1\) to get \(7\), resulting in the fraction \(\frac{7}{2}\).

Understanding mixed numbers is crucial for real-life situations where precise measurements are used. They also play a key role in performing operations like addition, subtraction, and conversion between different forms of numbers.

Improper Fractions

Improper fractions occur when the numerator, the top number, is greater than the denominator, the bottom number. This means the value of the fraction is actually greater than \(1\). Improper fractions, like \(\frac{7}{3}\), are easier to use in calculations than mixed numbers.

• Converting to Improper Fractions: To convert a mixed number into an improper fraction, multiply the whole number by the fraction’s denominator and add the numerator. For instance, converting \(4 \frac{2}{5}\) involves multiplying \(4\) by \(5\) and adding \(2\). This results in the fraction \(\frac{22}{5}\).
• Why Use Them: Improper fractions simplify mathematical operations since they eliminate the step of dealing with both whole and fractional parts separately.

Converting back to mixed numbers can be important once calculations are complete, as it presents the result in a more interpretable format. Use division: divide the numerator by the denominator to retrieve the whole number and remainder. The whole number becomes the new whole number, and the remainder over the original denominator becomes the fraction part of the mixed number.

Temperature Calculation

Calculating temperature changes using fractions involves understanding both addition and subtraction of these numbers. When dealing with mixed numbers, it’s often necessary to convert them into improper fractions first to simplify the calculating process. In temperature calculations, like other applications, precision is key.

• Steps to Calculate: When calculating a change in temperature, first convert any mixed numbers into improper fractions. Next, perform the necessary addition or subtraction, keeping the denominators the same.
• Example Calculation: Suppose you want to determine a net temperature change after a rise and a fall. First, combine the initial temperature with any rise in temperature by adding the fractions. Then, subtract any decreases.

Finally, the resulting improper fraction can be converted back to a mixed number to provide an easily understandable result. This process helps in maintaining accurate values, especially when used in contexts like medical data or scientific measurements, where precision is essential.