Problem 99 - Elementary Algebra

Question

After 3 months on a diet, Lisa had lost \(12 \%\) of her original weight. She lost 21 pounds. What was Lisa's original weight?

Step-by-Step Solution

Answer

Lisa's original weight was 175 pounds.

Step 1 - Understand the Problem

Lisa lost 12% of her original weight, and this loss amounts to 21 pounds. The goal is to find her original weight before the diet.

Step 2 - Set Up the Equation

Let Lisa's original weight be denoted as \text{W} pounds. According to the problem, 12% of her original weight is 21 pounds. This can be written as the equation: \[ 0.12 \times W = 21 \]

Step 3 - Solve for W

To find Lisa's original weight, solve the equation for \text{W} by dividing both sides by 0.12: \[ W = \frac{21}{0.12} \]

Step 4 - Perform the Division

Calculate the value: \[ W = \frac{21}{0.12} = 175 \]

Step 5 - Verify the Solution

Multiply 175 by 12% (or 0.12) to check the result: \[ 0.12 \times 175 = 21 \]. The calculation is correct, so Lisa's original weight was indeed 175 pounds.

Key Concepts

percentages

Percentages are a way of expressing a number as a fraction of 100. They are often used to describe proportions or comparisons. In this exercise, Lisa lost 12% of her original weight. This means if her original weight is treated as 100%, she retained 88% of her weight. To understand percentages better, remember:

• Percent means 'per hundred'.
• It can be written using the symbol '%'.
• For example, 12% of something is \( 12/100 \times $\text{total value} \).

Using percentages in word problems helps to make comparisons and determine parts of a whole.

algebraic equations

Algebraic equations are mathematical statements that show the equality of two expressions. They often include variables, which are symbols representing unknown values. In Lisa's weight loss problem, the unknown is her original weight, denoted by \(\text{W}\). Here's how the algebraic equation was set up:

• We start from the given information: 12% of her original weight equals 21 pounds.
• This can be written as an equation: \(0.12 \times W = 21\).
• To isolate \(\text{W}\), we divide both sides of the equation by 0.12: \( W = 21 / 0.12 \).

Solving this equation gives us the original weight. Understanding how to form and solve algebraic equations is crucial for tackling word problems.

problem-solving steps

Problem-solving steps provide a structured approach to tackle math problems. Here's a breakdown of the steps used in Lisa's weight loss exercise:

• Understand the Problem: Determine what is given and what needs to be found. Lisa lost 21 pounds, which is 12% of her original weight. We need to find her original weight.
• Set Up the Equation: Translate the word problem into a mathematical equation: 0.12 \( \times \) \( W = 21 \).
• Solve for the Variable: Isolate the variable by performing operations on both sides of the equation: \( W = 21 / 0.12 = 175 \).
• Perform the Calculation: Do the math to find the answer.
• Verify the Solution: Always check your work. Multiply 175 by 12% to ensure it equals 21 pounds.

Following these structured steps can make problem-solving more manageable and accurate.