Problem 81 - Introductory and Intermediate Algebra for College Students 4th

Question

14 is \(25 \%\) of what number? (Section \(2.4,\) Example 6 )

Step-by-Step Solution

Answer

The number is 56.

Step 1: Convert the percentage

We first convert the 25% to a decimal by dividing by 100. Hence, \(25 \% = 0.25\).

Step 2: Formulate an equation

If 14 is 25% of a number (say \(x\)), it means \(0.25 \times x = 14\). So we simply solve for \(x\) to find the number.

Step 3: Solve the equation

Divide both sides of the equation \(0.25x = 14\) by 0.25 to isolate \(x\). You will find that \(x = \frac{14}{0.25} = 56\).

Key Concepts

Percentage Conversion

Understanding percentage conversion is vital when working with percent problems in algebra. A percentage is a way of expressing a number as a fraction of 100. For example, saying '25%' is essentially the same as saying '25 out of 100' or '25 per 100'.

To convert a percentage to a decimal, which is a necessary step in formulating algebraic equations, you divide the percentage by 100. This is because 'per cent' means 'per hundred'. So, converting 25% to a decimal involves simply performing the division: \(25 \div 100 = 0.25\). This decimal form can then be used in algebraic expressions and equations.

Formulating Algebraic Equations

When faced with word problems involving percentages, the ability to formulate algebraic equations is key to finding solutions. You start by identifying the quantities involved and then translate the relationships stated in the problem into an arithmetic expression using algebraic symbols.

For instance, if the problem states that '14 is 25% of what number?', you are required to label the unknown number, often with the variable \(x\). The phrase '25% of' translates to multiplying the decimal equivalent of 25%, which is 0.25, by the variable \(x\). This results in an algebraic equation: \(0.25 \times x = 14\). The creation of such equations converts word problems into solvable mathematical formats.

Solving Linear Equations

Solving linear equations is a fundamental algebraic skill. Once the algebraic equation is formulated, finding the solution often involves isolating the variable. This usually entails performing the same operation on both sides of the equation to maintain the equality balance.

In the given example, the equation we have is \(0.25x = 14\). To solve for \(x\), you divide both sides by the coefficient of \(x\), which is 0.25 in this case. By doing so, you're left with \(x\) on one side of the equation: \(x = \frac{14}{0.25}\). When you carry out this division, you find that \(x = 56\), which is the answer to the problem. This process of solving is applicable to all linear equations, regardless of their complexity.