Problem 11 - Prealgebra

Question

Find the percent of each number mentally. $$75 \% \text { of } 16$$

Step-by-Step Solution

Answer

75% of 16 is 12.

Step 1: Understand Percent Calculation

Percent means 'per hundred'. Calculating 75% of 16 is the same as finding \( \frac{75}{100} \times 16 \). It's useful to break this operation into simpler steps that can be done mentally. We will break it into easier fractions of the number.

Step 2: Find 50% of 16

First, find 50%, which is the same as half of 16. Calculate \( \frac{1}{2} \times 16 = 8 \). 50% of 16 is 8.

Step 3: Find 25% of 16

Next, find 25%, which is the same as one-quarter of 16. Calculate \( \frac{1}{4} \times 16 = 4 \). 25% of 16 is 4.

Step 4: Combine the Results

To find 75% of 16, we add the results of 50% and 25% of 16 together. So, we compute \( 8 + 4 = 12 \).

Step 5: Verify the Solution

Verify by recalculating: 75% of 16 is \( \frac{75}{100} \times 16 = 12 \). Thus, the calculations were done correctly.

Key Concepts

Mental Math

When it comes to calculating percentages in your head, mental math becomes a valuable skill. Instead of relying on calculators or lengthy formulas, you can simplify and solve complex calculations quickly and efficiently. The key to effective mental math lies in breaking down the problem into smaller, manageable parts. For instance, if you need to determine what 75% of 16 is, remember that percentages can be reconstructed as fractions. Instead of directly calculating it, consider it as a combination of easier fractions:

• 50% (which is half)
• 25% (which is one quarter)

By adding up these simpler fractions, you achieve the desired result without intensive calculations. Regular practice with similar problems will increase both your speed and accuracy, making you proficient in mental gymnastics involving numbers.

Fractional Percentages

Understanding fractional percentages is essential when breaking down more complex percentage calculations mentally. Fractional percentages represent portions of the whole, making calculations more intuitive. When calculating something like 75% of a number, you can think of it in terms of 50% and 25%, which are fractions of 100% (half and one-quarter, respectively). To find 50% of any number, simply divide the number by 2. Finding 25% can be done by dividing the number by 4. These fundamental fractions are the building blocks of many percentage problems:

• 50%: Is equivalent to one-half, divide by 2
• 25%: Is one-fourth, divide by 4
• 75%: Sum of 50% and 25%

With this approach, fractional percentages transform potentially tedious calculations into straightforward and easily manageable mental steps.

Step-by-Step Solution

Having a step-by-step solution provides a structured method to tackle percentage problems with confidence. Starting with the essence of the problem, like understanding that 'percent' means out of 100, helps anchor our calculations. Breaking a complex percentage into component parts, as with 75% of 16, allows for easier mental processing. The step-by-step solution is as follows:

• Step 1: Recognize that 75% means \( \frac{75}{100} \) or \( \frac{3}{4} \)
• Step 2: Calculate 50% of 16, which is \( 8 \)
• Step 3: Calculate 25% of 16, which is \( 4 \)
• Step 4: Add the two results: \( 8 + 4 = 12 \)
• Step 5: Verify by recalculating to ensure accuracy

Each step is an opportunity to ensure that understanding and accuracy go hand in hand, making the process less daunting and more approachable.