Problem 14 - Prealgebra

Question

Find the percent of each number mentally. $$87 \frac{1}{2} \% \text { of } 56$$

Step-by-Step Solution

Answer

87 1/2% of 56 is 49.

Step 1: Understand the Problem

We need to find 87 1/2% of 56, which means we will calculate what fraction of 56 corresponds to 87 1/2%.

Step 2: Convert Percentage to Fraction

The percentage 87 1/2% can be written as a mixed number: \( 87.5\% = 87 + \frac{1}{2}\% \). Converting this to an improper fraction gives us \( \frac{175}{2}\% \).

Step 3: Convert Fraction to Decimal

Now, convert \( \frac{175}{2}\% \) into a decimal by dividing 175 by 200 (since percent means out of 100, we need to divide by 100 and then consider the fractional part). Hence, \( \frac{175}{200} = 0.875 \).

Step 4: Apply the Decimal to the Number

Multiply 56 by the decimal found in Step 3 to find the result. \( 0.875 \times 56 = 49 \).

Step 5: Conclude the Calculation

Therefore, 87 1/2% of 56 is 49.

Key Concepts

Fraction to Decimal Conversion

When tackling percentages like 87 1/2%, one key skill is converting fractions to decimals. Start by expressing the mixed percentage as a fraction: 87 1/2% becomes 87 + \( \frac{1}{2} \)%, which simplifies to \( \frac{175}{2} \)%. To convert this fraction into a decimal, first understand that percent means "per hundred." So, you're looking at \( \frac{175}{2} \) out of 200. Divide 175 by 200 to get \( 0.875 \). This decimal form is vital because it allows direct multiplication with other numbers, particularly when calculating percentages of quantities.

Basic Arithmetic

Basic arithmetic underpins almost all calculations, including percentage problems. Here, you'll use addition, subtraction, multiplication, and division. Converting \( 87 \frac{1}{2} \)% into a fraction showcases addition by combining whole numbers with fractions. Multiplying the decimal result \( 0.875 \) by 56 utilizes your multiplication skills. Division was integral in obtaining \( 0.875 \) from \( \frac{175}{200} \).These arithmetic basics make percentage calculations straightforward, enabling you to solve problems with minimal error.

Mental Math Strategies

Mental math can simplify percentage calculations considerably. Begin by breaking numbers into more manageable parts. In this exercise, fraction and decimal conversion allow you to handle complex percentages without extensive calculation tools.Mental conversion steps:

• Recognize 50% as half.
• Acknowledge 25% as a quarter.
• Add smaller parts incrementally.

For example, for \( 87 \frac{1}{2} \)% of 56:

• 50% of 56 is 28.
• 25% of 56 is 14.
• 10% of 56 is 5.6.
• 2.5% of 56 is 1.4.

Combine parts: \( 28 + 14 + 5.6 + 1.4 = 49 \).Using mental math strategies can help verify results quickly and build intuition over time.