Problem 86 - Algebra for College Students

Question

Solve each problem. Hybrids. The ratio of the number of cars with gasoline engines to hybrids sold at a car dealer is 9 to \(2 .\) If the dealer sells 10 hybrids one month, then how many gasoline cars did the dealer sell?

Step-by-Step Solution

Answer

The dealer sold 45 gasoline cars.

Step 1: Understand the Ratio

The given ratio of gasoline cars to hybrids is 9:2. This means that for every 9 gasoline cars, there are 2 hybrids.

Step 2: Set Up the Proportion

Let the number of gasoline cars sold be represented by the variable \(x\). Since the ratio of gasoline cars to hybrids is 9:2, we can write the proportion as follows: \[\frac{9}{2} = \frac{x}{10}\].

Step 3: Solve the Proportion

To find the value of \(x\), we need to solve the proportion. Cross-multiply to get: \[9 \cdot 10 = 2 \cdot x\]. This simplifies to: \[90 = 2x\].

Step 4: Isolate the Variable

Solve for \(x\) by dividing both sides of the equation by 2: \[x = \frac{90}{2}\]. This gives: \[x = 45\].

Step 5: Interpret the Result

The dealer sold \(45\) gasoline cars in the month when 10 hybrids were sold.

Key Concepts

ratios and proportions

Ratios and proportions are fundamental concepts in mathematics. A ratio shows the relationship between two quantities. For example, the ratio 9:2 in the problem states that for every 9 gasoline cars sold, there are 2 hybrids.
In contrast, a proportion is an equation that expresses two ratios as equal. In this problem, the ratio of gasoline cars to hybrids can be set up as a proportion to find the unknown quantity. Here's how it works: if you know one part of the ratio and the total for the other, you can find the missing number by setting up a proportion.
For the given problem, the proportion can be written as: \(\frac{9}{2} = \frac{x}{10}\). Proportions are powerful tools that help in comparing different quantities and finding unknown values by maintaining the equality of ratios.

cross-multiplication

Cross-multiplication is a method used to solve proportions. It involves multiplying across the equals sign diagonally.
This method is particularly useful because it transforms the proportion into a simple linear equation, which is easier to solve. In the given exercise, we start with the proportion: \(\frac{9}{2} = \frac{x}{10}\).
Using cross-multiplication, we multiply 9 by 10 and 2 by x to get: \[9 \times 10 = 2 \times x\].
This simplifies to 90 = 2x.
Cross-multiplication helps eliminate the fractions, making it straightforward to isolate the variable and solve for it. It's a reliable method that ensures accuracy when dealing with ratios and proportions.

linear equations

A linear equation is an equation that makes a straight line when plotted on a graph. It has the general form: \ax + b = c\. Solving a linear equation involves isolating the variable. Upon cross-multiplying in the given problem, we obtained the linear equation: 90 = 2x.
To isolate the variable, we performed the following steps:

• Divide both sides of the equation by the coefficient of x. In this case, divide 90 by 2.
• This gives us: \[ x = \frac{90}{2} \]
• Which simplifies to x = 45.

Linear equations are straightforward and often used in various types of problem-solving, including ratio and proportion problems, like in this exercise. They help in finding precise values for unknown variables by using simple mathematical operations.