Question

Translate to an equation and solve. \(70 \%\) of what number is \(45.5 ?\)

Short answer

Answer: 65

Step-by-step solution

Set up the equation

Since 70% of the number is equal to 45.5, we can represent the given information as an equation: \(0.7x = 45.5\)

Solve the equation

To find the value of x, we need to divide both sides of the equation by 0.7: \(x = \frac{45.5}{0.7}\)

Compute the result

Now we compute the value of x: \(x \approx 65\) So, 70% of 65 is approximately 45.5.

Key concepts

Translating Text to Equation

Understanding how to convert a word problem into an algebraic expression is a critical skill in mathematics. When dealing with percentage issues, the first task is to identify the 'part' and the 'whole'. For example, if a problem states that '70% of a number is 45.5,' we interpret '70%' as the 'part' (percentage) and 'a number' as the 'whole' we are trying to find.

To translate this to an equation, we understand that 'of' usually means multiplication in mathematics. Therefore, '70% of a number' translates to '0.7 times a number,' with the number being our unknown quantity, usually represented by a variable such as 'x'. The complete equation then becomes \(0.7x = 45.5\) . This direct translation is crucial in setting up the foundation for problem-solving in algebra.

Percentage Equation

With percentage problems, the percentage equation is a fundamental tool. This equation links the part, the whole, and the percentage. The formula is often presented as \( \text{percentage} = \frac{\text{part}}{\text{whole}} \times 100\) . However, when solving for an unknown, it’s helpful to use the percentage in its decimal form to simplify calculations.

In the given example, 70%, when converted to a decimal, is 0.7. Thus, when you see 70% of some number, you replace it with 0.7 times that number. By setting the equation as \(0.7x = 45.5\) , we create a clear pathway to find 'x,' our unknown whole, by using basic algebraic operations.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. In the context of our example, \(0.7x = 45.5\) , the goal is to isolate 'x' to determine its value.

To do so, one basic technique is to perform the 'inverse operation.' Since 'x' is being multiplied by 0.7, the inverse operation is division. By dividing both sides of the equation by 0.7, we are left with \(x = \frac{45.5}{0.7} \) . This step cancels out the 0.7 on the left side, effectively isolating 'x' on one side of the equation, which is instrumental in solving for the unknown.

Basic Algebra

Basic algebra focuses on the use of letters to represent numbers and the rules for manipulating these symbols to solve equations. It's the foundation for all higher-level math courses and problem-solving. The concepts within basic algebra, such as adding, subtracting, multiplying, and dividing variables, as well as using inverse operations to solve equations, are pivotal.

In percentage problems, after translating words to an equation, we use basic algebra to simplify and solve the equation. Persistence in practicing basic algebra will inevitably lead to proficiency, making complex problem-solving accessible and manageable. Accurately setting up the equation, as seen in the example, and applying algebraic rules will eventually lead to finding the correct value of the unknown variable.