Fractions, decimals, and percents

Fractions, decimals, and percents can name the same number. A decimal uses powers of ten, while a percent states how many hundredths. Converting representations should preserve the number-line location.

To convert a fraction to a decimal, divide numerator by denominator. To convert a decimal to a percent, multiply by $100$ and attach the percent symbol; reversing that move divides by $100$. Terminating decimals arise when a reduced denominator has only factors $2$ and $5$.

Representation is a strategic choice. Percents communicate relative change, decimals support calculator work and money, and fractions often preserve exact values and multiplicative structure.

Fractions, decimals, and percents are coordinate systems for the same value. A fraction emphasizes exact part-whole structure, a decimal emphasizes base-ten place value, and a percent compares with a standard whole of $100$. Converting notation should not change the number. For example, $\frac{3}{8} = 0.375 = 37.5\%$. A number-line check is useful: all three representations must land at the same point between $0$ and $1$.

To convert a fraction to a decimal, divide numerator by denominator. A terminating decimal occurs when the simplified denominator contains only factors $2$ and $5,$ because powers of ten contain only those primes. Other rational numbers produce repeating decimals. The bar or repeated block matters: $0.3$ is not equal to $\frac{1}{3,}$ but $0.333$… is. When an exact fraction is available, retain it through calculations and round only when the context requests an approximation.

Percent means “per hundred.” Converting a decimal to a percent multiplies by $100$ and attaches the percent unit; converting a percent to a decimal divides by $100$. The movement of a decimal point is a consequence of this scaling, not a rule detached from meaning. A value may exceed $100\%$ or be negative: $125\%$ represents $1.25$ times the reference amount, and $-8\%$ can represent an eight-percent change in the negative direction.

Percent-of problems contain three roles: part, whole, and rate. The relationship part $= rate\cdot whole$ can be rearranged only after the roles are identified. In “$18$ is what percent of $60,$” $18$ is the part and $60$ is the whole, so rate $= \frac{18}{60} = 0.30 = 30\%$. In “$45\%$ of what number is $27,$” $27$ is the part and the whole is unknown. Labelling the roles prevents the common mistake of dividing in the wrong order.

Choose a representation for the work it supports. Fractions preserve exact ratios and simplify multiplicative structure. Decimals align with money, measurement, and calculator input. Percents make comparisons with different-sized wholes easier. A final answer should name the reference whole and any rounding. Saying “the rate is $20\%$” is incomplete if it is unclear whether $20\%$ describes a discount, tax, growth, or a share of which total.

Repeating decimals can be converted back to fractions with place-value subtraction. If $x = 0.272727…,$ then $100x = 27.272727$…. Subtracting the first equation from the second gives $99x = 27,$ so $x = \frac{27}{99} = \frac{3}{11}$. The subtraction removes the infinite repeating tails because they align exactly. This method shows that a repeating decimal is not an approximation; it is another exact notation for a rational number. A rounded decimal such as $0.27,$ by contrast, equals exactly $\frac{27}{100}$ and is only near $\frac{3}{11}$.

Percent language can hide a multiplicative factor. Increasing by $12\%$ means multiplying by $1 + 0.12 = 1.12,$ while decreasing by $12\%$ means multiplying by $1 - 0.12 = 0.88$. These operations are not inverses: $a 12\%$ increase followed by $a 12\%$ decrease multiplies by $1.12\cdot 0.88 = 0.9856,$ leaving $98.56\%$ of the start. Whenever several percent changes occur, use a separate multiplier for each stage and keep the changing reference amount visible. Adding the percent labels usually gives the wrong model.

When data are rounded, a percent calculated from them is also approximate. Report enough digits to support the decision but not more precision than the inputs justify. In money contexts, keep full calculator precision during the computation and round the final currency amount to cents; in survey contexts, name the sample or reference population. A percent is never self-explanatory: $40\%$ of $50$ and $40\%$ of $500$ have the same rate but very different parts.