Problem 7 - Prealgebra

Question

Use the percent proportion to solve each problem 21 is \(30 \%\) of what number?

Step-by-Step Solution

Answer

21 is 30% of 70.

Step 1: Understand the Percent Proportion Formula

The percent proportion formula is \( \frac{part}{whole} = \frac{percent}{100} \). We use this formula to find the unknown number when given a part and its percentage.

Step 2: Identify the Given Values

In this problem, we know `21` is the part and `30%` is the percent. We need to find the whole, which is the unknown number.

Step 3: Set Up the Proportion

Set up the proportion using the values: \( \frac{21}{x} = \frac{30}{100} \). Here, `x` represents the unknown whole number we want to find.

Step 4: Solve for the Unknown Number

Cross-multiply to solve the proportion: \( 21 \times 100 = 30 \times x \). This simplifies to \( 2100 = 30x \). Now, divide each side by `30`: \( x = \frac{2100}{30} \).

Step 5: Calculate the Value of x

Divide 2100 by 30 to find `x`: \( x = 70 \). Therefore, `21` is `30%` of `70`.

Key Concepts

Understanding the Percent Formula

The percent formula is a key concept in mathematical calculations involving percentages. It helps us determine the relationship between a part and the whole. This relationship is typically represented by the formula \( \frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100} \). This equation allows you to find one missing component if you know the other three. In the formula, the "part" is the specific portion or quantity you know, and the "whole" is the total quantity that you want to find out. The "percent" is the given percentage that the part represents of the whole, expressed as a fraction of 100. Used correctly, this formula is a powerful tool for solving problems involving real-life applications like discounts, interest rates, and statistical data.

Solving for Unknown Values Using Proportions

To solve for an unknown value using a proportion, you need to set up an equation where two ratios are equivalent. The common method involves cross-multiplication, which enables one to isolate the unknown variable. Let's walk through the steps with our original exercise involving the percent formula.- First, write the known values in a proportion: \( \frac{21}{x} = \frac{30}{100} \). Here, 21 is the part, 30 is the percent, and \( x \) is the unknown whole.- Next, perform cross-multiplication: \( 21 \times 100 = 30 \times x \). This rearranges the equation to \( 2100 = 30x \).- Finally, solve for \( x \) by dividing both sides of the equation by 30 to isolate \( x \): \( x = \frac{2100}{30} \).By dividing, you find that \( x = 70 \). Thus, you discover that 21 is 30% of 70. Remember, cross-multiplication is a reliable step when solving for one unknown in proportional relationships.

Prealgebra Problem Solving Techniques

Prealgebra problem solving often involves basic arithmetic and algebraic techniques to find unknown values. The focus is mainly on understanding how numbers relate to one another through various mathematical models like proportion, addition, subtraction, multiplication, and division. Common approaches include:

• Identifying given information and what needs to be found.
• Set up equations using these known values, such as writing proportions and equations.
• Utilize algebraic manipulation, such as cross-multiplying, to simplify equations.
• Perform calculations carefully, double-checking work for accuracy.

These steps allow a problem to be broken down into manageable parts, making the entire process more systematic and predictable. Strong problem-solving skills in prealgebra pave the way for more advanced mathematics studies, building a solid foundation of understanding that is essential for tackling more complex problems confidently.