Number - Converting between Fractions, Decimals, and Percentages

The Interconnectedness of FDP: Fractions, decimals, and percentages are three different ways of expressing the same part of a whole. Imagine a 'Conversion Triangle' where you can move between any two forms. For example, half of a pizza can be visualized as the fraction $\frac{1}{2}$, the decimal $0.5$, or the percentage $50\%$.

Converting Fractions to Decimals: To convert any fraction $\frac{a}{b}$ to a decimal, perform the division $a \div b$. Visually, if you have a fraction like $\frac{3}{4}$, you are dividing 3 units into 4 equal parts, resulting in $0.75$. If the division ends, it is a 'terminating decimal'; if a digit or pattern repeats forever, it is a 'recurring decimal' (e.g., $\frac{1}{3} = 0.333...$).

Converting Decimals to Percentages: Percent means 'per hundred'. To convert a decimal to a percentage, multiply the value by $100$. Visually, this is represented by moving the decimal point two places to the right. For example, $0.345$ becomes $34.5\%$. You can imagine the decimal point 'jumping' over two digits to the right.

Converting Percentages to Fractions: Write the percentage as a fraction with a denominator of $100$. If you visualize a $10 \times 10$ grid (100 squares total) and $25$ squares are shaded, the percentage is $25\%$, which is written as $\frac{25}{100}$. Always simplify the fraction by dividing the numerator and denominator by their greatest common factor (e.g., $\frac{25}{100} = \frac{1}{4}$).

Converting Percentages to Decimals: To convert a percentage to a decimal, divide the number by $100$. Visually, this involves moving the decimal point two places to the left. For instance, $85\%$ (which is $85.0\%$) becomes $0.85$. If the percentage is a single digit like $7\%$, you must add a placeholder zero to get $0.07$.

Converting Fractions to Percentages via Equivalent Fractions: If a fraction's denominator is a factor of $100$ (like $2, 4, 5, 10, 20, 25, 50$), multiply both the numerator and denominator by the same number to make the denominator $100$. For example, to convert $\frac{4}{5}$, multiply both by $20$ to get $\frac{80}{100}$, which is $80\%$.

Recurring Decimal Notation: When a decimal repeats, we use a bar or a dot over the repeating digit(s) to show it continues infinitely. For example, $\frac{2}{3} = 0.666...$, which is written as $0.\bar{6}$ or $0.\dot{6}$. This represents a value that is slightly more than $0.66$ but less than $0.67$ on a number line.