Problem 72 - Algebra 1

Question

Evaluate the expression. Use estimation to check your answer. GASOLINE PRICES Last month the price of gasoline was \(\$ 1.19\) per gallon. This month the price of gasoline is \(\$ 1.13\) per gallon. What was the change in the price per gallon of gasoline?

Step-by-Step Solution

Answer

The change in the price per gallon of gasoline from last month to this month is -\$0.06.

Step 1: Identify Initial and Final Prices

Firstly, we have to identify the initial price of gasoline, which was \$1.19 per gallon last month. Then, identify the final price of gasoline, which is \$1.13 per gallon this month.

Step 2: Calculate the Change

To calculate the change in the price, subtract the final price from the initial price. That is, \( \$1.19 - \$1.13 = \$0.06 \). The change in the price per gallon of gasoline is therefore \( \$0.06 \).

Step 3: Verify with Estimation

In this step, round off the initial and final prices to integers, say \$1.00 and \$1.00 respectively. The estimated change in price, therefore, is \( \$1.00 - \$1.00 = \$0.00 \). This estimated change is near to our calculated change in the price per gallon of gasoline.

Key Concepts

Estimation Techniques

Estimation is a valuable skill in mathematics as it allows you to quickly check the reasonableness of your answers. In our exercise, we estimated the change in gasoline prices to see if our subtraction was correct. To estimate, you can round numbers to make calculations easier. For example, rounding both \\(1.19 and \\)1.13 to \\(1.00 gives us an easy subtraction to perform mentally:\[ \\)1.00 - \\(1.00 = \\)0.00 \]. This shows the change is very small.Another method is to round to the nearest tenth if that provides more insight. If we rounded \\(1.19 to \\)1.20 and \\(1.13 to \\)1.10, we would subtract like so: \[ \\(1.20 - \\)1.10 = \$0.10 \].Estimation doesn’t aim for precision but for an answer that is close enough to help you verify your detailed calculation. It acts as a sanity check, ensuring your calculations are on the right track. Use it as a way to quickly catch errors.

Understanding Decimal Subtraction

Decimal subtraction works similarly to whole number subtraction. However, managing the decimal point is crucial. Let's break down this process:- Line up the numbers by the decimal points.- If necessary, add zeroes to make the number of decimal places equal.- Subtract as if they were whole numbers, starting from the rightmost digit.In our example, we subtracted \\(1.13 from \\)1.19. By aligning the decimal points: \[ \begin{array}{c} 1.19 \ -1.13 \ \hline 0.06 \end{array} \]After aligning, we subtract the digits from right to left, borrowing from the next left digit if needed. This process helps ensure accuracy in calculations involving money and prevents common mistakes that arise when the decimals are not correctly managed. Always double-check that the decimal point in the answer lines up with those in the numbers above.

Steps to Effective Problem-Solving

Problem-solving in math requires a clear approach to avoid confusion and errors. Here's a breakdown:1. **Understand the Problem:** Clearly define what is asked. In our case, it's the change in gasoline price per gallon.2. **Plan the Approach:** Decide the method to solve it. For our problem, subtraction is chosen.3. **Execute the Calculation:** Perform the subtraction of \\(1.19 and \\)1.13, resulting in \\(0.06.4. **Verify the Result:** Use estimation to check if the result feels right, which we've done by rounding the numbers.5. **Review and Reflect:** Ensure the answer makes sense within the context. Is a \\)0.06 change reasonable given your estimation?Each step builds on the previous, ensuring your understanding and calculations are robust. This methodical process helps when tackling not just arithmetic problems but various mathematical challenges.