Question

Weather The normal high and low temperatures for four cities for January are given in degrees Celsius. Find the measure of the difference between the two temperatures. a. Boston, \(2^{\circ} \mathrm{C},-6^{\circ} \mathrm{C}\) b. San Francisco, \(13^{\circ} \mathrm{C}, 6^{\circ} \mathrm{C}\) c. Chicago, \(-2^{\circ} \mathrm{C},-11^{\circ} \mathrm{C}\) d. Houston, \(16^{\circ} \mathrm{C}, 4^{\circ} \mathrm{C}\)

Short answer

Boston: 8°C, San Francisco: 7°C, Chicago: 9°C, Houston: 12°C.

Step-by-step solution

Understand the Problem

We need to find the difference between the high and low temperatures for each city. The difference between two numbers is calculated by subtracting the smaller number from the larger one.

Analyze for Boston

For Boston, the high temperature is \(2^{\circ} \mathrm{C}\) and the low temperature is \(-6^{\circ} \mathrm{C}\). To find the difference, subtract the low from the high: \(2 - (-6)\). Simplifying this gives \(2 + 6 = 8^{\circ} \mathrm{C}\).

Analyze for San Francisco

For San Francisco, the high temperature is \(13^{\circ} \mathrm{C}\) and the low temperature is \(6^{\circ} \mathrm{C}\). The difference is \(13 - 6 = 7^{\circ} \mathrm{C}\).

Analyze for Chicago

For Chicago, the high temperature is \(-2^{\circ} \mathrm{C}\) and the low temperature is \(-11^{\circ} \mathrm{C}\). The difference is \(-2 - (-11)\). Simplifying this gives \(-2 + 11 = 9^{\circ} \mathrm{C}\).

Analyze for Houston

For Houston, the high temperature is \(16^{\circ} \mathrm{C}\) and the low temperature is \(4^{\circ} \mathrm{C}\). The difference is \(16 - 4 = 12^{\circ} \mathrm{C}\).

Key concepts

Mathematical Subtraction

Understanding subtraction is key to finding the difference between two numbers. Subtraction involves taking one number away from another. But it's not just limited to whole numbers; it also applies to negative numbers and temperatures.
To subtract two numbers:

• Identify which number is larger (especially important when dealing with negative numbers)
• Subtract the smaller number from the larger number
• If subtracting a negative number, change the minus (\(-\)) to a plus (\(+\))

For example, in mathematical subtraction for a temperature difference, subtracting a negative number results in addition, such as \(2 - (-6)\), which becomes \(2 + 6\). This will give you the difference between the two temperatures.

Temperature Conversion

Temperature conversion is not directly needed in this particular exercise as it deals with Celsius alone. However, it can often be useful, particularly when converting temperatures between Fahrenheit and Celsius or Kelvin.
Remember these formulas for conversion:

• From Celsius to Fahrenheit: \(F = \frac{9}{5}C + 32\)
• From Fahrenheit to Celsius: \(C = \frac{5}{9}(F - 32)\)
• From Celsius to Kelvin: \(K = C + 273.15\)

These conversions are useful when you're dealing with temperatures from different scales, and can help contextualize temperatures in problem-solving settings, even if not required for the current subtraction exercise.

Integer Operations

Integer operations include the addition, subtraction, multiplication, and division of whole numbers, which include both positive and negative numbers. In the exercise, we focus on subtraction, but understanding how integers work is crucial across all operations.
Key things to know about integers:

• Positive numbers are greater than zero (e.g., 13, 16)
• Negative numbers are less than zero (e.g., -6, -11)
• When adding two negative numbers, the result is more negative (e.g., \(-2 + (-9)\) yields \(-11\))
• Subtracting a negative integer is the same as adding its positive counterpart (e.g., \(7 - (-3)\) becomes \(7 + 3\))

In solving for each city’s temperature difference, you'll use the principle of subtracting integers, particularly when both involve negative numbers as in Chicago's case with \(-2 - (-11)\). Simplifying this reveals how these integer rules apply and result in a positive difference of \(9\).