Problem 6 - Prealgebra

Question

Use the percent proportion to solve each problem What is \(74 \%\) of \(56 ?\)

Step-by-Step Solution

Answer

74% of 56 is 41.44.

Step 1: Identify the Proportion Formula

The percent proportion formula is \( \frac{x}{100} = \frac{a}{b} \), where \(x\) is the percentage, \(a\) is the part, and \(b\) is the whole.

Step 2: Set Up the Equation

We need to find what number is \(74\%\) of \(56\). Based on the problem, \(x = 74\) and \(b = 56\). Thus, the equation is \( \frac{74}{100} = \frac{a}{56} \).

Step 3: Cross-Multiply

Cross-multiply to solve for \(a\). This gives us \(74 \times 56 = 100 \times a\).

Step 4: Calculate Cross-Multiplication

Calculate the left side: \(74 \times 56 = 4144\).

Step 5: Solve for a

Now, solve for \(a\) by dividing both sides by \(100\): \(a = \frac{4144}{100} = 41.44\).

Key Concepts

Percentage Calculation

Percentage calculation is all about finding a part of a whole. In the context of the problem, we need to find 74% of 56. This means understanding that we are calculating 74 parts out of every 100 parts of 56.

• To calculate percentages, you multiply the whole number by the percentage expressed as a decimal.
• A percentage like 74% is converted to a decimal by dividing by 100, which means 74% becomes 0.74.

So, to find 74% of 56, you multiply 56 by 0.74, which ultimately gives you 41.44.
A precise understanding of this process aids in various situations, from determining discounts during shopping to calculating statistics.

Cross-Multiplication

Cross-multiplication is a mathematical technique mostly used to solve equations that involve two fractions set equal to each other. It helps to clear fractions quickly and allows us to solve linear equations more easily.

• In our example, we use the percent proportion, \[\frac{74}{100} = \frac{a}{56},\] to set up a situation where cross-multiplication can effectively eliminate the fractions.
• We multiply diagonally: 74 times 56 and 100 times \(a\).
• This gives the equation: \(74 \times 56 = 100 \times a\).

This step simplifies solving the equation, as it reduces the original fraction equation into a more straightforward multiplication one.

Equation Solving

Equation solving is the final step in problems involving percent proportions and cross-multiplication.
Once the cross-multiplication step has provided us with the equation \(74 \times 56 = 100 \times a\), our task is to isolate \(a\) (the unknown variable we want to find).
Here’s how we tackle it:

• Calculate the product of 74 and 56, resulting in 4144.
• Next, we divide both sides of the equation by 100 to solve for \(a\).
• Performing \(4144 \div 100\) provides the solution: \(a = 41.44\).

This stepwise isolation and simplification help in pinpointing the exact value of our unknown, ensuring accuracy and clarity in solutions.