Problem 88 - Elementary Algebra Within Reach

Question

Decide whether the statement is true or false. Justify your answer. $$\frac{1}{2} \%=50 \%$$

Step-by-Step Solution

Answer

False. The statement \(\frac{1}{2} \% = 50 \%\), is false because when 1/2 is converted into a percentage it equals 50% not \(\frac{1}{2}\%\). Therefore, 1/2% is significantly smaller than 50%.

Step 1: Making sense of the problem

The exercise misrepresents a fraction as a percentage. 1 divided by 2 is indeed 0.5 when represented as a first decimal and 50% when transformed into a percentage. However, the left side of the equation is incorrect as it states that 1/2 is 50 hundredths of a percent, which is an error.

Step 2: Convert Fraction into Percentage

Converting 1/2 (0.5) into a percentage gives 50% and not 0.5% because the percentage is calculated as (Fraction)*(100)

Step 3: Comparison

Now compare 1/2% with 50%. We quickly recognize that 1/2% is equivalent to 0.005 in decimal format, and is not the same as 50% which is 0.5 in decimal, so the provided statement is false.

Key Concepts

Percentages

Understanding percentages is important in math and everyday life. A percentage represents a fraction out of 100. It allows us to easily compare different values or changes over time. When we say 50%, it means 50 out of 100 or half of something. Think of percentages as a handy way to express large or complex numbers in a simple form that is easy to understand.

To convert a decimal number to a percentage, multiply it by 100 and add the percentage sign (%). For example, converting 0.5 into a percentage would involve multiplying by 100, which results in 50%. This is a straightforward yet powerful way to look at numbers and their relationships.

Fractions

Fractions are used to describe parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). The fraction \(\frac{1}{2}\) defines a half, meaning one part out of two equal parts.

Fractions can be converted to decimals and percentages for easier comparisons. For example, \(\frac{1}{2}\) can be converted to 0.5 as a decimal, and 50% as a percentage, by multiplying 0.5 by 100.

• Converting a fraction to a decimal involves dividing the numerator by the denominator.
• For instance, for \(\frac{1}{2}\), dividing 1 by 2 equals 0.5.
• After getting the decimal, the next step is to multiply this by 100 to obtain the percentage format.

Fractions offer a precise method for describing exactly how much of something there is.

Decimal Conversion

Decimal conversion is the process of transforming fractions and percentages into decimal form. This is useful for calculations that require more precision or when using digital tools like calculators.

Converting \(\frac{1}{2}\) into a decimal involves simple division: 1 divided by 2 results in 0.5. The decimal format is often more intuitive for performing arithmetic operations like addition and subtraction.

When converting percentages to decimals, you divide by 100. So, 50% becomes 0.5. This allows for easier mathematical operations and comparisons.

• To go from a percentage to a decimal, perform the operation by removing the % sign and dividing by 100.
• This rule helps in seamlessly transitioning between different number formats.

Decimal conversion turns complex numbers into simpler forms, aiding quick understanding and efficient calculations.