Number - Connecting Fractions, Decimals, and Percentages

Equivalence across Forms: Fractions, decimals, and percentages are three ways to express parts of a whole. Imagine a '100-grid' square consisting of 100 small boxes; if 50 boxes are shaded, it represents the fraction $\frac{50}{100}$ (or $\frac{1}{2}$), the decimal $0.50$, and the percentage $50\%$.

Converting Fractions to Decimals: To convert a fraction to a decimal, divide the numerator by the denominator ($numerator \div denominator$). On a vertical number line, the fraction $\frac{1}{4}$ would be positioned at the same point as $0.25$, exactly one-quarter of the way up from $0$ toward $1$.

Converting Decimals to Percentages: To turn a decimal into a percentage, multiply the value by $100$ and add the symbol $\%$. Visually, this is equivalent to moving the decimal point two places to the right. For example, $0.85$ becomes $85\%$.

Converting Percentages to Fractions: A percentage is always a part 'out of 100'. To convert, place the percentage value over a denominator of $100$ and simplify. For instance, $60\% = \frac{60}{100}$, which simplifies to $\frac{3}{5}$. In a pie chart, this would look like a circle divided into 5 equal sectors with 3 sectors colored in.

Place Value Relationship: Decimals rely on place value where the first digit after the point is the 'tenths' ($\frac{1}{10}$), the second is 'hundredths' ($\frac{1}{100}$), and the third is 'thousandths' ($\frac{1}{1000}$). $0.07$ is specifically $7$ hundredths, which is $7\%$.

Comparing and Ordering: To compare fractions, decimals, and percentages, it is most efficient to convert all numbers into decimal form first. Once converted, you can stack them vertically by their decimal points to easily see which value is larger or smaller.

Terminating vs. Recurring Decimals: Some fractions like $\frac{1}{5}$ result in a terminating decimal ($0.2$), while others like $\frac{2}{3}$ result in a recurring decimal ($0.666...$ or $0.\bar{6}$). A recurring decimal is indicated by a bar over the repeating digit or dots.