Question

10 is \(1 / 2\) of \(5 \%\) of what number?

Short answer

The number is 400.

Step-by-step solution

Understanding the Problem

The problem states that 10 is one-half of 5% of a certain number. This means we need to find a number such that when we take 5% of it, and then one-half of that result, we get 10.

Set Up the Equation

Let the unknown number be represented as \( x \). According to the problem, \[10 = \frac{1}{2} \times 5\% \times x\].

Convert Percentage to Decimal

Convert 5% to its decimal form: \(5\% = 0.05\). Then rewrite the equation: \[10 = \frac{1}{2} \times 0.05 \times x\].

Eliminate the Fraction

Multiply both sides of the equation by 2 to eliminate the fraction: \[ 10 \times 2 = 0.05x\], which simplifies to \[20 = 0.05x\].

Solve for x

To solve for \(x\), divide both sides by 0.05: \[ x = \frac{20}{0.05} \]. Compute the division to find \( x = 400 \).

Key concepts

Percentage Calculations

Percentage calculations are important in many GMAT problems. A percentage represents a fraction of 100. For example, 5% means 5 out of every 100. To convert a percentage to a decimal, divide by 100. Thus, 5% becomes 0.05. This conversion is critical in simplifying equations where percentages are involved. It can be especially useful when dealing with multiple steps that include multiplication, division, or fractional components.

Algebraic Equations

Algebraic equations are mathematical statements that equate two expressions with one or more unknown variables. In the given problem, we represent the unknown number as x. To set up the equation, translate the given conditions into an equation form. This generally involves identifying variables and their relationships. For instance, $$10 = \frac{1}{2} \times 5\% \times x$$ This form helps systematically solve for the unknown value by performing arithmetic operations in steps. Doing this makes it easier to isolate the variable and find its value.

Mathematical Problem Solving

Mathematical problem solving involves understanding the problem, devising a plan, carrying out the plan, and then checking the results. For the given exercise, we first understood what the problem was asking. Here is how we systematically approached solving it:

• Understand that 10 is half of 5% of a number.
• Translate this understanding into an equation, 10 = \( \frac{1}{2}\) \( \times\) 5 % x.
• Convert percentage to decimals, resulting in 10 = \( \frac{1}{2}\) \(\times\) 0.05 x.
• Solve for the unknown by eliminating fractions and performing arithmetic operations.
  

This problem-solving methodology can be applied widely across different types of math problems, ensuring a logical approach to finding solutions.

Graduate Management Admission Test

The GMAT is a standardized exam used for admissions into graduate management programs worldwide. It tests various skills, such as quantitative reasoning, verbal reasoning, and analytical writing. The kind of problem presented here is typical of the quantitative section. By practicing problems that involve percentage calculations, algebraic equations, and logical problem-solving steps, you can improve your chances of scoring well on the quantitative section of the GMAT. Mastery of these concepts ensures you are well-prepared for similar questions.

Step-by-Step Solution

To solve complex problems effectively, follow a step-by-step approach. Take a problem one element at a time, as demonstrated in the current example:

• Understand the Problem: This involves decoding what the problem is asking.
• Set Up the Equation: Translating words into a math equation helps to solve it logically.
• Convert to Simple Forms: Where necessary, such as converting percentages to decimals.
• Eliminate Complications: Remove any fractions or complex parts by applicable arithmetic operations.
• Solve for the Variable: Isolate the variable and calculate its value.

This logical breakdown ensures simplicity and makes complex problems more manageable. Regular practice of this method will lead you to master GMAT mathematical problem-solving.