Question

\(37.5\) is \(1 / 4\) of \(25 \%\) of what number?

Short answer

The unknown number is 600.

Step-by-step solution

- Understand the Problem

We are given that 37.5 is equal to one-quarter of 25% of some unknown number. Mathematically, this can be expressed as: \( 37.5 = \frac{1}{4} \times 25\% \times x \)

- Convert Percentage to Decimal

Convert 25% to a decimal for easier calculation. Since 25% is the same as 25 out of 100, we can convert it as follows: \( 25\% = 0.25 \)

- Set up the Equation

Substitute the decimal value of 25% into the equation: \( 37.5 = \frac{1}{4} \times 0.25 \times x \)

- Simplify the Right Side

Simplify the right side of the equation by first multiplying \( \frac{1}{4} \) and \( 0.25 \): \( \frac{1}{4} \times 0.25 = 0.0625 \) So the equation now looks like: \( 37.5 = 0.0625 \times x \)

- Solve for x

To solve for the unknown number \( x \), divide both sides of the equation by 0.0625: \( x = \frac{37.5}{0.0625} \) Calculate the division: \( x = 600 \)

- Verify the Solution

Verify the solution by plugging \( x = 600 \) back into the original context: \( 25\% \times 600 = 150 \) \( \frac{1}{4} \times 150 = 37.5 \). The solution is correct.

Key concepts

Percentage Calculations

Understanding percentages is key for solving GMAT problems. A percentage represents a part of a whole, where 100% stands for the entire amount. To convert a percentage to a decimal, divide by 100. For example, 25% becomes \( 0.25 \) because \( 25/100 = 0.25 \). This conversion is crucial for easier calculations in algebraic equations. In our exercise, we converted 25% to 0.25 to simplify the equation.

Algebra

Algebra involves working with variables and constants to find unknown values. It is all about setting up equations based on given information. In the exercise, the problem is expressed in an algebraic equation: \( 37.5 = \frac{1}{4} \times 25\% \times x \). Here, \( x \) is our unknown number. By simplifying the right side, we can isolate \( x \), making it easier to solve. Algebraic skills are essential for GMAT problem-solving.

Equation Solving

Solving equations involves finding the value of the unknown that makes the equation true. In our example, after converting 25% to 0.25, the problem looks like: \( 37.5 = \frac{1}{4} \times 0.25 \times x \). Simplify the equation step by step: First, multiply the constants: \( \frac{1}{4} \times 0.25 = 0.0625 \), so the equation becomes \( 37.5 = 0.0625 \times x \). Then, solve for \( x \) by dividing both sides by 0.0625: \( x = 37.5 / 0.0625 = 600 \).

Decimal Conversion

Decimal conversion is a technique used to simplify calculations. Converting percentages into decimals involves dividing by 100. In our problem, 25% was converted to 0.25 before substituting it into the equation. This step simplifies operations and reduces errors. To convert a fraction to a decimal, divide the numerator by the denominator. For instance, \( \frac{1}{4} = 0.25 \). Remember, mastering decimal conversions can make solving equations much more manageable.