Problem 75 - Introductory Algebra for College Students

Question

Solve by writing a sum of signed numbers and adding. The temperature at 8: 00 A.M. was \(-7^{\circ} \mathrm{F}\). By noon it had risen \(15^{\circ} \mathrm{F},\) but by \(4: 00 \mathrm{P.M.}\) it had fallen \(5^{\circ} \mathrm{F}\). What was the temperature at 4: 00 P.M.?

Step-by-Step Solution

Answer

The temperature at 4:00 PM is \(3^\circ F\).

Step 1: Represent the temperature rise

By noon, the temperature rises by 15 degrees. This is represented mathematically as \( -7 + 15 \).

Step 2: Represent the temperature fall

By 4:00 PM, the temperature falls by another 5 degrees. This is represented mathematically as \( (-7 + 15) - 5 \).

Step 3: Perform the operations

Perform the operations in the correct order. \( -7 + 15 = 8^\circ F \), then \( 8 - 5 = 3^\circ F \).

Key Concepts

Signed Numbers in Math

When dealing with temperature changes, particularly in disciplines like meteorology or daily life activities, we often encounter signed numbers. Signed numbers are simply numbers with a positive or negative sign, indicating direction or opposition.

For instance, a temperature of \( -7^\circ \mathrm{F} \) suggests it's 7 degrees below some reference point, such as freezing point. In mathematics, understanding signed numbers is crucial since they form the basis of arithmetic operations that involve both positive and negative quantities.

While adding signed numbers, similar signs mean you add the absolute values of the numbers and keep the sign. For opposite signs, subtract the smaller absolute value from the larger one, taking the sign of the larger absolute value. For example, combining a temperature rise of \( 15^\circ \mathrm{F} \) with a starting temperature of \( -7^\circ \mathrm{F} \) results in \( -7 + 15 = 8^\circ \mathrm{F} \) because the absolute value of 15 is greater than 7.

Performing Arithmetic Operations

Successfully performing arithmetic operations with signed numbers is fundamental to solving temperature change problems. As in our exercise, it involves two main steps: adding and subtracting. It's important to add or subtract numbers in the right order, especially when the equations get more complicated.

In our case, the first operation is to add the temperature rise by noon: \( -7 + 15 \). After that, we subtract the temperature drop by 4:00 PM, resulting in \( (-7 + 15) - 5 \). It might seem trivial with small numbers, but proper operation order becomes vital when dealing with larger values or when temperatures fluctuate frequently.

One way to ensure accuracy is by using parentheses to group numbers that should be added or subtracted first. It is also essential to simplify expressions step by step, as was done in the problem: first finding the sum of \( -7 + 15 \) and then subtracting 5 from the result.

Understanding Temperature Fluctuations

Grasping the concept of temperature fluctuations goes beyond just knowing the current temperature—it requires an understanding of how temperature changes over time and how these changes can be quantified using arithmetic operations.

In our exercise, we observe a temperature increase and then a decrease within an 8-hour period. Keeping a chronological order of these changes is important, as it reflects the natural progression of temperature during the day. Starting at a cold \( -7^\circ \mathrm{F} \) in the morning, the air warms by \( 15^\circ \mathrm{F} \) and cools down again by \( 5^\circ \mathrm{F} \) in the afternoon.

This teaches us that when we articulate temperature variations in signed numbers and perform the correct arithmetic operations, we can accurately determine subsequent temperatures at different points in time, leading us to the final temperature of \( 3^\circ \mathrm{F} \) by 4:00 PM in this scenario.