Problem 21 - Prealgebra

Question

Use the percent proportion to solve each problem. Round to the nearest tenth. $$ 80 \% \text { of what number is } 12 ? $$

Step-by-Step Solution

Answer

100% of a number is 15.

Step 1: Set up the Percent Proportion

The percent proportion is given by \( \frac{part}{whole} = \frac{percent}{100} \). In this problem, the 'part' is 12, the 'percent' is 80, and the 'whole' is what we are trying to find. Therefore, the equation becomes: \[ \frac{12}{\text{whole}} = \frac{80}{100} \]

Step 2: Solve for the Whole

To find the 'whole', we cross-multiply the proportion equation: \[ 12 \times 100 = 80 \times \text{whole} \] Simplifying gives us: \[ 1200 = 80 \times \text{whole} \]

Step 3: Isolate the Whole

To find the value of the 'whole', divide both sides by 80: \[ \text{whole} = \frac{1200}{80} \]

Step 4: Calculate the Division

Perform the division to find the value of the 'whole': \[ \text{whole} = 15 \]

Step 5: Round to the Nearest Tenth

The division results in a whole number, so there's no need to round. However, if needed, 15.0 would be considered rounding to the nearest tenth.

Key Concepts

Understanding Percent Problems

Percent problems are quite common in math and everyday life. They typically involve finding the part, whole, or percentage value in a situation. In our problem, we are asked to determine the whole when given a part and a percentage. These types of problems allow us to apply mathematical reasoning to convert real-world situations into mathematical expressions. When dealing with percent problems, always remember that

• 'part' is the portion of interest or the result of applying the percent to the 'whole',
• 'percent' is the fraction of the whole that the part represents, expressed as a percentage of 100, and
• 'whole' is the entire amount or quantity to which the percent applies.

By setting up an equation using these three components, you can solve percent problems using proportions.

Proportional Reasoning Explained

Proportional reasoning is a critical mathematical skill. It helps in solving equations that involve ratios and proportions. We use it in our percent problem to find unknown values. Understanding proportions means you can look at two ratios, compare them, and make logical mathematical deductions. In our problem, we represent the relationship between the part, whole, and percent in proportion form:

• Our equation is written as \( \frac{part}{whole} = \frac{percent}{100} \)
• This form shows the equality of two ratios
• Proportional reasoning helps to determine the unknown by maintaining this balance

It allows us to simplify complex problems into a solvable equation by understanding how the quantities relate to each other. The key is maintaining the consistency of the ratio on both sides of the equation.

Basic Algebra for Solving Problems

At the core of solving percent problems is a good understanding of basic algebra. Algebra involves manipulating equations to find unknown values. Here, we use a simple equation derived from a proportion to find our unknown 'whole'. First, we set up our proportion \( \frac{12}{\text{whole}} = \frac{80}{100} \). Then we turn to algebra to

• Cross-multiply to eliminate the fraction: \( 12 \times 100 = 80 \times \text{whole} \)
• Simplify to get \( 1200 = 80 \times \text{whole} \)
• Isolate the variable by dividing both sides: \( \text{whole} = \frac{1200}{80} \)

This method helps break down the problem into manageable parts, allowing us to isolate and find the desired value using algebraic techniques.

Mastering Cross-Multiplication

Cross-multiplication is a handy tool when dealing with proportions like in our percent problem. It's a method used to solve equations involving two fractions set equal to each other. This technique simplifies fractions and solves for unknown variables effectively. By cross-multiplying, we have:

• First, multiply the numerator of one fraction by the denominator of the other: \( 12 \times 100 \)
• Then, do the same for the other pair of numbers: \( 80 \times \text{whole} \)
• Set the products equal to form a new equation: \( 1200 = 80 \times \text{whole} \)

This step is crucial for clearing fractions and reducing the equation to a simpler form for solving. Cross-multiplication is especially valuable for those just beginning to grapple with algebra and proportions as it provides a straightforward path to the solution.