Problem 11
Question
The highest recorded temperature in Hawaii is \(100^{\circ} \mathrm{F}\). The highest recorded temperature in Colorado is \(18^{\circ} \mathrm{F}\) higher than that of Hawaii. What is the highest recorded temperature in Colorado?
Step-by-Step Solution
Verified Answer
The highest recorded temperature in Colorado is \(118^{\circ} \mathrm{F}\)
1Step 1: Identify the Highest Recorded temperature in Hawaii
The Problem statement says that the highest recorded temperature in Hawaii is 100 degrees Fahrenheit.
2Step 2: Identify the Increase in Temperature
The problem states that the highest recorded temperature in Colorado is 18 degrees Fahrenheit higher than in Hawaii.
3Step 3: Calculate the Highest Recorded Temperature in Colorado
Add the identified increase in temperature to the highest recorded temperature in Hawaii. \( 100^{\circ} \mathrm{F} + 18^{\circ} \mathrm{F} = 118^{\circ} \mathrm{F}\) .
Key Concepts
Problem-Solving Strategies in AlgebraUnderstanding Algebraic ExpressionsArithmetic Operations in Word Problems
Problem-Solving Strategies in Algebra
When it comes to solving word problems in algebra, it’s essential to simplify the process by breaking it down into smaller, manageable steps.
The first step is to carefully read the problem to understand what is being asked. Identify the knowns and the unknowns, and look for any keywords that signal specific operations. In the exercise provided, the problem states the highest recorded temperature in Hawaii and asks for the temperature in Colorado, which is higher by a given amount.
Next, translate the words into an algebraic expression. This requires recognizing that 'higher than' refers to an addition operation. Once the operation is determined, we move to forming an equation and solving for the unknown. In the given exercise, we devise the equation as the temperature in Hawaii plus 18 degrees.
Finally, it is crucial to check your answer by plugging the result back into the word problem to see if it makes sense. Reviewing the problem to ensure that you haven't missed any details that might change the equation is also a good practice. Following these strategies will lead students to logically conclude that the highest temperature in Colorado is 118 degrees Fahrenheit.
The first step is to carefully read the problem to understand what is being asked. Identify the knowns and the unknowns, and look for any keywords that signal specific operations. In the exercise provided, the problem states the highest recorded temperature in Hawaii and asks for the temperature in Colorado, which is higher by a given amount.
Next, translate the words into an algebraic expression. This requires recognizing that 'higher than' refers to an addition operation. Once the operation is determined, we move to forming an equation and solving for the unknown. In the given exercise, we devise the equation as the temperature in Hawaii plus 18 degrees.
Finally, it is crucial to check your answer by plugging the result back into the word problem to see if it makes sense. Reviewing the problem to ensure that you haven't missed any details that might change the equation is also a good practice. Following these strategies will lead students to logically conclude that the highest temperature in Colorado is 118 degrees Fahrenheit.
Understanding Algebraic Expressions
Algebraic expressions are the building blocks of algebra and are used to represent real-world situations mathematically. An algebraic expression consists of numbers, variables (which can represent unknowns or variables in word problems), and arithmetic operations.
When solving word problems, it's important to convert verbal statements into algebraic expressions. This often involves recognizing the implied multiplication (e.g., 'three times a number' becomes '3x') or addition/subtraction. In the case of our temperature problem, '18 degrees Fahrenheit higher' is effectively translated into the algebraic expression '100 + 18', as no variable is needed for the known values.
If variables were involved, we would represent them with letters and formulate an equation that we could solve. This translation process is fundamental in algebra, as it allows us to work with numerical and abstract concepts fluidly.
When solving word problems, it's important to convert verbal statements into algebraic expressions. This often involves recognizing the implied multiplication (e.g., 'three times a number' becomes '3x') or addition/subtraction. In the case of our temperature problem, '18 degrees Fahrenheit higher' is effectively translated into the algebraic expression '100 + 18', as no variable is needed for the known values.
If variables were involved, we would represent them with letters and formulate an equation that we could solve. This translation process is fundamental in algebra, as it allows us to work with numerical and abstract concepts fluidly.
Arithmetic Operations in Word Problems
Arithmetic operations, including addition, subtraction, multiplication, and division, are often used to solve algebraic word problems. The ability to correctly identify and apply these operations is crucial for finding the right solution.
In our textbook example, we faced a simple arithmetic operation—addition—which we applied by adding 18 to the highest temperature of Hawaii, represented by the number 100. This involved a straightforward computational step: \( 100^{\circ} \mathrm{F} + 18^{\circ} \mathrm{F} = 118^{\circ} \mathrm{F} \).
However, in more complex problems, you might need to use multiple operations, set up proportions, or work with expressions that include variables. It's good practice to write down each operation step by step, as this can help prevent errors and make the process more understandable. Remember to follow the order of operations—parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right)—to ensure accurate results.
In our textbook example, we faced a simple arithmetic operation—addition—which we applied by adding 18 to the highest temperature of Hawaii, represented by the number 100. This involved a straightforward computational step: \( 100^{\circ} \mathrm{F} + 18^{\circ} \mathrm{F} = 118^{\circ} \mathrm{F} \).
However, in more complex problems, you might need to use multiple operations, set up proportions, or work with expressions that include variables. It's good practice to write down each operation step by step, as this can help prevent errors and make the process more understandable. Remember to follow the order of operations—parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right)—to ensure accurate results.
Other exercises in this chapter
Problem 10
Use the subtraction rule to rewrite the subtraction expression as an equivalent addition expression. Then evaluate the expression. $$ \frac{1}{2}-\frac{1}{4} $$
View solution Problem 10
Graph the numbers on a number line. Then write the numbers in increasing order. $$-\frac{1}{2},-\frac{3}{4}, \frac{1}{4}$$
View solution Problem 11
Find the quotient. $$10 \div\left(-\frac{1}{5}\right)$$
View solution Problem 11
Find the product. $$-5 \cdot 2 \cdot(-7)$$
View solution