Problem 19 - Prealgebra

Question

Use the percent proportion to solve each problem. Round to the nearest tenth. 27 is \(90 \%\) of what number?

Step-by-Step Solution

Answer

27 is 90% of 30.

Step 1: Understand the Proportion

We need to find what number 27 is 90% of. We can use the percent proportion formula, which is \( \frac{part}{whole} = \frac{percent}{100} \). In this problem, 27 is the 'part' and we are solving for the 'whole'.

Step 2: Write the Proportion Equation

Substitute the given values into the proportion: \( \frac{27}{x} = \frac{90}{100} \). Our goal is to solve for \(x\), which represents the 'whole'.

Step 3: Cross Multiply

Cross-multiply to eliminate the fraction: \(27 \times 100 = 90 \times x\). This simplifies to \(2700 = 90x\).

Step 4: Solve for the Whole

Divide both sides by 90 to isolate \(x\): \(x = \frac{2700}{90}\). Calculating this gives \(x = 30\).

Step 5: Conclusion

We found that 27 is 90% of 30. Hence, the original number is 30.

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to solve equations that involve fractions. It is especially useful in percent problems when dealing with proportions. The basic premise is to eliminate the fraction by multiplying diagonally across the equals sign.

In our example, we started with the proportion \( \frac{27}{x} = \frac{90}{100} \). To cross-multiply, you multiply the numerator of each fraction by the denominator of the other fraction:

• First, multiply the numerator of the first fraction (27) by the denominator of the second fraction (100), which gives 2700.
• Next, multiply the numerator of the second fraction (90) by the denominator of the first fraction (x), which gives 90x.

This results in the equation: \( 2700 = 90x \). Now, you have removed the fractions, making the equation easier to solve.

Understanding cross-multiplication sets the foundation for solving complex problems involving proportions efficiently.

Solving Equations

After cross-multiplication, solving the equation becomes straightforward. Solving equations involves isolating the variable to find its value. Here, we derived the equation \( 2700 = 90x \). The goal is to isolate \(x\).

To do this, divide both sides of the equation by 90 to solve for \(x\):

• \( x = \frac{2700}{90} \)
• Calculating the division, we get \( x = 30 \)

By dividing both sides by the same number, we maintain the integrity of the equation. This is a critical step in ensuring we solve for the correct value of the unknown variable.

The process of solving equations is central to problem-solving in mathematics. It is the method we apply to find unknown values in a variety of mathematical scenarios, especially when interpreting data from word problems.

Percents in Math

Understanding percents is vital for solving real-world problems, and they often appear in equations and proportions. A percent represents a part out of 100, making it an essential concept for comparing ratios and proportions.

In our problem, we identified that 27 is 90% of some unknown number. We used the percent proportion formula to set up our equation: \( \frac{part}{whole} = \frac{percent}{100} \). Here's how we approached it:

• "Part" is the portion in question, which is 27.
• "Percent" is given as 90%.
• The "whole" is the value we are solving for to understand the total amount that 27 represents 90% of.

By establishing these relationships, we use the formula \( \frac{27}{x} = \frac{90}{100} \) to direct our solution process.

Percents not only help in solving problems like these, but they are also crucial in fields such as finance, statistics, and everyday decision-making. Mastery of percentages allows one to interpret and analyze data efficiently.