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Chapter 8: Problem 34

Translate to an equation and solve. 2000 is \(6 \frac{1}{4} \%\) of what number?

Short Answer

Expert verified
Answer: 32,000

Step by step solution

01

Convert the percentage into a decimal

To make the percentage easier to work with, we will convert it to a decimal by dividing it by 100. So, \(6 \frac{1}{4} \% = 6.25\%= \frac{6.25}{100}= 0.0625\).
02

Set up the equation

Let \(x\) be the number we are trying to find. The problem states that \(2000\) is \(6.25\%\) of that number, so we will write this relationship as an equation: \(2000 = 0.0625x\).
03

Solve for the number x

Now, we need to isolate \(x\) to find its value. To do this, divide both sides of the equation by \(0.0625\): $$x=\frac{2000}{0.0625}=32000$$
04

State the answer

The number we are looking for is 32,000. So, 2000 is \(6 \frac{1}{4}\)% of 32,000.

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Key Concepts

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

Decimal Conversion
When dealing with percentages greater than 1, we often need to convert them into decimal form to simplify mathematical calculations. This conversion is straightforward: divide the percentage by 100. For example, the mixed number percentage \(6 \frac{1}{4} \%\) needs special attention. First, convert the fraction \(\frac{1}{4}=0.25\), then combine it with the whole number 6 to form 6.25. Finally, convert 6.25% to a decimal by dividing by 100, giving \(0.0625\). This decimal form is much easier to work with in an equation because it is compatible with multiplication and division operations.
Equation Setup
Setting up an equation from a word problem is a crucial skill in problem solving. It involves translating words into mathematical language. In this problem, we know that 2000 is \(6 \frac{1}{4} \%\) of some unknown number, which we'll call \(x\). The keyword 'is' in percentage problems often signals an equation relationship. Here, it helps us set up the equation \(2000 = 0.0625x\). This equation reflects that 2000 is a part of number \(x\), quantified by the percentage \(0.0625\). Setting up this equation correctly is the foundation for finding the solution.
Problem Solving
Solving an equation is a step-by-step process where we aim to find the unknown variable. Here, we need to solve the equation \(2000 = 0.0625x\) to find \(x\). To isolate \(x\), we divide both sides by \(0.0625\), which simplifies to \(x=\frac{2000}{0.0625}\). This operation involves basic division, which is straightforward once we've set up the equation properly. Performing the division gives us \(x=32000\). Hence, we conclude that 32,000 is the original number, confirming that 2000 is \(6 \frac{1}{4} \%\) of it. Breaking down the problem into these manageable parts ensures we solve it systematically and accurately.

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Most popular questions from this chapter

Translate to an equation and solve. Rosita sells real estate and receives a commission of \(\$ 750 .\) If her commission rate is \(6 \%\) of the seller's net, what is the seller's net?

Solve. Hannah purchased a curio stand discounted \(35 \%\). If the price after the discount was \(\$ 246.68\), what was the initial price?

Write each decimal number as a percent. $$0.01$$

Write each fraction as a percent. $$\frac{4}{9}$$

Write each percent as a decimal number. $$53 \frac{2}{5} \%$$
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