Problem 1 - Prealgebra

Question

Find the percent of each number mentally. $$75 \% \text { of } 64$$

Step-by-Step Solution

Answer

75% of 64 is 48.

Step 1: Concept Understanding

To solve this problem, we need to find what 75% of 64 is. This means we are looking for a proportion of 64.

Step 2: Convert Percent to Fraction or Decimal

75% can be converted to the fraction \(\frac{75}{100}\) or the decimal 0.75. This conversion lets us multiply with 64 to find the desired percentage value.

Step 3: Multiply by the Fraction

Multiply the fraction \(\frac{75}{100}\) by 64. You can simplify the fraction first if desired: \(\frac{3}{4}\). Then, calculate \(\frac{3}{4} \times 64 = 48\).

Step 4: Multiply by the Decimal

Alternatively, multiply the decimal 0.75 by 64 directly. Calculate 0.75 \(\times\) 64 = 48.

Key Concepts

Mental Math

Mental math is a skill that allows you to perform calculations in your head without using a calculator or pen and paper. When it comes to percent calculations, mental math can be particularly useful.

• To find 75% of a number, break it down mentally into easier chunks. For instance, 75% can be seen as three-fourths of the number.
• If you know that 50% of a number is half of it, then another 25% (which is half of 50%) can help you form the 75% figure.
• To quickly calculate 75% of 64, you can compute one-half of 64, which is 32, and then figure out one-fourth of 64, which is 16. Add those results together to get 48.
• This step-by-step breakdown helps you see how larger percent values get divided into manageable parts, making mental calculations more approachable.

Fractions to Decimals

Understanding how to convert fractions to decimals is a key skill in percent calculations. When you convert a fraction, like \(\frac{75}{100}\), to a decimal, it simplifies many math problems.

• Start by dividing the numerator by the denominator. So, divide 75 by 100 to get 0.75.
• Converting fractions to decimals is especially helpful because most calculators and digital devices work seamlessly with decimals.
• It also aids in making the multiplication process straightforward, simplifying the task of finding any percentage of a number.

Decimals might seem complex, but once you understand their connection with fractions, they open a wide range of calculation opportunities.

Proportions

Proportions are relationships that express how one quantity relates to another. In percent calculations, this relationship is crucial.

• Consider the proportion set by 75% of any number. It indicates that out of every 100 units, 75 are needed.
• This understanding of proportion is the foundation of percent problems. It tells you directly how much of one part relates to the whole.
• Using proportions, like \(\frac{3}{4}\), simplifies large problems. For example, knowing that 75% is the same as three-fourths helps you reduce complex tasks into bite-sized portions.

When you recognize percentages as proportions, it becomes much easier to handle them, be it in math class, shopping, or even cooking.

Simplifying Fractions

Simplifying fractions makes them easier to work with, especially when dealing with percentages. It involves reducing fractions to their simplest form.

• Take \(\frac{75}{100}\). By finding common factors (in this case, 25), you can simplify it to \(\frac{3}{4}\).
• Simplifying fractions before multiplying can save time and reduce errors, particularly in mental math.
• It also makes numbers more digestible and provides a clearer understanding of the relationship between parts.

Learning to simplify fractions helps in not just mathematical contexts but also in practical, everyday situations where proportional reasoning is required.