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Chapter 7: Problem 64

Solve the problem using a percent proportion. \(23 \%\) of what number is \(27.6 ?\)

Short Answer

Expert verified
The number is 120.

Step by step solution

01

Identify Known Values

The problem states that 23% of some number is 27.6. This means that the part is 27.6 and the percentage is 23%.
02

Set Up the Proportion

We use the percent proportion \(\frac{part}{whole} = \frac{percent}{100}\). Here, the part is 27.6, the percent is 23, and we need to find the whole.
03

Substitute the Known Values

Substitute the values into the proportion: \[ \frac{27.6}{x} = \frac{23}{100} \]
04

Solve for the Unknown

Cross multiply to solve for \(x\): \[ 27.6 \times 100 = 23 \times x \] \[ 2760 = 23x \] Solve for \(x\): \[ x = \frac{2760}{23} \] \[ x = 120 \]
05

Conclusion

23% of 120 is 27.6, therefore the unknown number is 120.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Percentage Calculations
Percentage calculations are used to express a number as a fraction of 100. When we say 23%, it means 23 out of every 100 units. Calculating percentages involves breaking down complex relations into a manageable format. Here are the basic components you need to understand:
  • Part: This is the portion or piece of the whole.
  • Whole: This is the total or complete amount we鈥檙e dealing with.
  • Percent: This is a way of expressing a number as a fraction of 100, represented with a % symbol.
In the given exercise, 23% of a number is equal to 27.6. We have to determine the 'Whole' when the 'Part' is 27.6 and the 'Percent' is 23%. This understanding sets the groundwork for solving the rest of the problem.
Cross Multiplication
Cross multiplication is a helpful technique when dealing with proportions. It allows us to find an unknown value by leveraging the equality of two ratios. In our exercise, the proportion is set up as: \[ \frac{27.6}{x} = \frac{23}{100} \] To find the unknown number, we use cross multiplication. Here鈥檚 how it works step-by-step:
  • Multiply the numerator of the first fraction by the denominator of the second fraction: \(27.6 \times 100\)
  • Multiply the denominator of the first fraction by the numerator of the second fraction: \(23 \times x\)
By equating these two products, we form a simpler equation that is easier to solve.
Solving Proportions
Solving proportions involves finding an unknown variable in an equation of two ratios. Let's break down the process: We start from the equation obtained from cross multiplication: \[ 27.6 \times 100 = 23 \times x \]
Solving for \(x\) is straightforward:
  • First, perform the multiplication on the left-hand side: \(27.6 \times 100 = 2760\)
  • Next, isolate \(x\) by dividing both sides of the equation by 23: \[ x = \frac{2760}{23} \]
  • Simplify the division to find \(x\): \[ x = 120 \]
This indicates that 23% of 120 is 27.6, so the number we were solving for is 120. Following this step-by-step resolves the proportion quickly and efficiently, making complex problems simple and accessible.

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Most popular questions from this chapter

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