Problem 24 - Prealgebra

Question

Should be set up the same way. Then the actual calculations should be done on a calculator. Manufacturing A car manufacturer estimates that \(25 \%\) of the new cars sold in one city have defective engine mounts. If \(2,136\) new cars are sold in that city, how many will have defective engine mounts?

Step-by-Step Solution

Answer

534 cars will have defective engine mounts.

Step 1: Understand the Problem

We need to determine the number of cars with defective engine mounts. We are given that 25% of the cars sold are defective, and the total number of cars sold is 2,136.

Step 2: Convert Percentage to Decimal

To work with the percentage mathematically, convert 25% to a decimal by dividing it by 100. This gives us 0.25.

Step 3: Calculate the Number of Defective Cars

Multiply the total number of cars, 2,136, by the decimal representation of the percentage, 0.25, to find the number of defective cars.\[ ext{Number of defective cars} = 2,136 \times 0.25\]

Step 4: Perform the Calculation

Perform the multiplication to find:\[2,136 \times 0.25 = 534\]Thus, there are 534 defective cars.

Key Concepts

Problem Solving Steps

Solving percentage problems requires a systematic approach to ensure accuracy and understanding. It begins with a clear understanding of the problem you are attempting to solve. In this context, we are tasked with determining how many cars out of a lot sold have defective engine mounts based on a given percentage.

The first step is identifying the relevant numbers and percentages in the problem. This involves point out that 25% of the cars have defects and that the total number of cars sold is 2,136.

Next, break the problem into smaller parts:

• Recognizing the percentage that needs conversion.
• Understanding the total number involved.
• Applying mathematical operations to these values.

Finally, ensure you review and check your work after calculation. This involves revisiting each step to confirm that operations were performed correctly and logically follow the problem's requirements.

Decimal Conversion

Whenever you're dealing with percentages in algebra or other mathematical problems, converting the percentage to a decimal form is crucial. This step simplifies later calculations.

To convert a percentage to a decimal, divide by 100. For example, converting 25% involves calculating:

\[ 25\% = \frac{25}{100} = 0.25 \]

This decimal representation allows for straightforward multiplication with the total, giving a portion of the total represented by that percentage.

Remember these key points about decimal conversion:

• The decimal will always be smaller than the original percentage.
• Ensure precision by keeping every step clear and methodical.

Proper conversion ensures that the mathematical operations that follow are correct and logical.

Mathematical Operations

Understanding which mathematical operations to apply is essential in solving percentage-related problems. From our problem, after converting the percentage to a decimal, next is the multiplication.

The equation for finding how many cars are defective is:

\[ \text{Number of defective cars} = 2,136 \times 0.25 \]

Step through this calculation by considering what each number represents—2,136 as the total, and 0.25 as the slice of that total which is defective.

Multiplying gives:

• It simplifies the relation between percentages and totals.
• This operation clearly defines the defective portion of the total.

Checking the multiplication ensures accuracy: performing it accurately shows there are 534 defective cars out of the total sold, as calculated through careful application of each step.