Question

Translate to an equation and solve. \(90 \%\) of what number is \(133.2 ?\)

Short answer

Answer: The number is 148.

Step-by-step solution

Set up the equation

We want to find the number X such that 90% of X is equal to 133.2; this can be represented mathematically as: \[ 0.90 X = 133.2\]

Solve for X

In order to find X, we need to divide both sides of the equation by 0.90, as shown below: \[ X = \frac{133.2}{0.90} \]

Calculate the result

Now, we can find the value of X by dividing: \[ X = \frac{133.2}{0.90} = 148 \]

State the answer

The number we were looking for is 148, as 90% of 148 equals 133.2.

Key concepts

Translating Word Problems into Equations

Turning a word problem into a mathematical equation is the first step in solving many math puzzles. In word problems, we often deal with percentages, quantities, and relationships that require translation into numbers and symbols. For example, in the given exercise, the phrase "90% of what number is 133.2" hints at a relationship between these quantities. Here, "what number" becomes our unknown variable, typically denoted as \(X\). The phrase "90% of" is translated into multiplication by 0.90. Thus, the entire sentence becomes the equation \(0.90X = 133.2\). This simple transformation opens the door to applying mathematical processes to find a solution. Once you grasp the concept of translating words into symbols, you can tackle more complex problems with confidence and precision.

Solving Equations

Solving equations involves finding the value of the unknown variable that satisfies the equation. In our example equation \(0.90X = 133.2\), the goal is to isolate \(X\) on one side of the equation. This usually involves applying operations such as addition, subtraction, multiplication, or division to both sides strategically. Here, the equation requires dividing both sides by 0.90 to solve for \(X\), because multiplication by 0.90 "hides" the true value of \(X\). By dividing, you undo this multiplication, revealing \(X\) on its own on one side: \(X = \frac{133.2}{0.90}\). Mastering this process of isolating variables allows you to unravel any equation effectively.

Mathematical Calculations

Once you have an equation ready to solve, it's time for the mathematical calculations. This is a straightforward but crucial part of solving any math problem. With our equation \(X = \frac{133.2}{0.90}\), you perform the division to determine the value of \(X\). Using a calculator can simplify this task:

• Divide 133.2 by 0.90.
• This calculation yields the solution: \(X = 148\).

This step confirms whether the manipulated equation accurately represents the word problem initially presented. Calculations serve as a bridge between the abstract world of algebra and real-life scenarios. Ensure accuracy in these operations to verify your final results.

Basic Algebra

Basic algebra is the backbone of translating and solving word problems like the one we have explored. It involves understanding how to manipulate numbers and variables to put the pieces of a puzzle together. At its core, algebra aims to define relationships and find unknowns. When working with algebra, remember these key ideas:

• Identify the unknown variable and represent it with a symbol, often \(X\).
• Apply arithmetic operations to both sides of the equation to maintain balance.
• Simplify equations wherever possible.

By following these principles, algebra allows us to systematically break down complex problems into manageable steps.