Problem 91 - Elementary Algebra Within Reach

Question

Solve the percent equation. 150 is \(250 \%\) of what number?

Step-by-Step Solution

Answer

The required number is 60

Step 1: Convert Percentage to Decimal

First of all, convert the percentage (250%) into a decimal value. For this, divide the percentage number by 100. So, \(250\% = \frac{250}{100} = 2.5\)

Step 2: Write the Equation

Write an equation to represent the problem. Let the unknown be denoted by 'x'. Recall that '150 is 250% of number x' can be expressed in mathematical equation like \(150 = 2.5 * x\).

Step 3: Solve for x

Rearrange the equation to solve for x. Doing this gives \(x = \frac{150}{2.5}\)

Key Concepts

Solving Equations

Equations are mathematical statements where two expressions are set equal to each other. Solving these equations means finding the value of the unknown variable. In the exercise given, the problem revolves around finding an unknown, labeled as 'x', which is a common practice in mathematics.

When we set up an equation such as \(150 = 2.5 \times x\), we need to isolate 'x' to find its value. This means rearranging the equation so that 'x' stands alone on one side of the equation. In this scenario, it involves dividing both sides by 2.5, the coefficient of 'x'. This rearrangement allows us to find \(x = \frac{150}{2.5}\).

By applying basic algebraic operations like division, we transform and simplify the equation to solve it. Practicing solving equations helps in mastering problem-solving skills which are useful in various stages of math and real-life challenges.

Percentage Conversion

Percentage conversion is a key mathematical skill for understanding how parts relate to a whole. It involves changing a percentage to a decimal, which simplifies computations in problems like our example.

To convert a percentage to a decimal, simply divide by 100. This is because 'percent' means 'per hundred'. For instance, \(250\%\) is converted by calculating \(\frac{250}{100} = 2.5\). This decimal form allows us to easily use it in equations or further calculations.

• *Example*: \(75\% = \frac{75}{100} = 0.75\)
• *Example*: \(120\% = \frac{120}{100} = 1.2\)

Turning percentages into decimals is particularly useful in solving equations involving percentages, just like our given problem.

Algebraic Expressions

Algebraic expressions form the foundation of algebra and involve numbers, variables, and operations that collectively represent a value. An expression like \(2.5 \times x\) is termed algebraic because it involves both a number (2.5) and a variable (x).

In our context, setting up a problem as an algebraic expression involves writing out what the problem states in mathematical terms. '150 is 250% of what number?' translates and simplifies as the algebraic equation \(150 = 2.5 \times x\).

Understanding algebraic expressions enables breaking down real-world situations into manageable calculations. This understanding helps in writing equations which can easily be solved using learned techniques like isolating the variable, as demonstrated in this problem.