Fractions, Decimals and Percents: Converting Between Forms

Fundamentals of Fractions and Rational Numbers

A rational number is any number that can be expressed as a ratio of two integers with a non-zero denominator. In mathematics, a fraction represents the division of the numerator by the denominator. Every fraction fundamentally conveys this operation, showing a specific quantity partitioned by another. For fractions with nonnegative numerators and positive denominators, a proper fraction has a numerator smaller than its denominator, whereas an improper fraction has a numerator greater than or equal to its denominator. An improper fraction with a nonnegative numerator and a positive denominator represents at least one whole.

When an improper fraction represents a value greater than a whole, it can be written as a mixed number, which consists of a whole number combined with a proper fraction. To convert an improper fraction to a mixed number, divide the numerator by the denominator to obtain the whole number quotient and remainder. In this improper-to-mixed-number conversion, the remainder becomes the numerator of the fractional part over the original denominator. For a nonnegative mixed number, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator; for a negative mixed number, apply the negative sign to the entire resulting fraction.

Simplifying fractions is an essential skill when working with rational expressions. A fraction is in lowest terms when the greatest common factor of the numerator and denominator is 1. To simplify a fraction, divide both the numerator and denominator by their greatest common factor. This simplification works because dividing the numerator and denominator by the same nonzero number does not change the value of the fraction. It is also critical to remember core division properties: any nonzero number divided by itself equals 1, zero divided by any nonzero number equals 0, and division of any number by zero is undefined.

Converting Between Fractions and Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. Carrying out this division will produce one of two outcomes: a terminating decimal or a repeating decimal. A terminating decimal has a finite number of digits after the decimal point. In contrast, a repeating decimal has one or more digits that repeat indefinitely after the decimal point. For instance, 1/2 equals the terminating decimal 0.5, whereas 1/3 equals the repeating decimal 0.333… and is approximately 0.333 when rounded to three decimal places.

To convert a terminating decimal to a fraction, write the number without its decimal point over 10 raised to the number of decimal places, then simplify. Decimal place values directly dictate the appropriate power of 10: one decimal place represents tenths and corresponds to a denominator of 10; two decimal places represent hundredths and correspond to a denominator of 100; and three decimal places represent thousandths and correspond to a denominator of 1000. Once the decimal is placed over the matching denominator, divide both the numerator and denominator by their greatest common factor to reduce the fraction to lowest terms.

Recognizing common benchmark fractions and their decimal equivalents allows you to solve conversion problems quickly. Memorizing standard values eliminates the need to perform long division repeatedly during timed testing sessions.

The following table displays standard benchmark fractions alongside their exact or approximate decimal and percentage counterparts.

Converting Percents with Decimals and Fractions

A percentage is inherently connected to parts per hundred. In fact, a fraction with a denominator of 100 has a numerator equal to its percentage value. To convert a fraction to a percentage, multiply the fraction by 100 and add the percent symbol. For example, (1/4) × 100 = 25, so 1/4 equals 25% when expressed as a percentage. Similarly, the fraction 3/4 corresponds to 75%, while eighths follow suit with 1/8 equal to 12.5%, 3/8 equal to 37.5%, 5/8 equal to 62.5%, and 7/8 equal to 87.5%. For thirds, 1/3 is approximately equal to 33.33% and 2/3 is approximately equal to 66.67%.

Converting between decimals and percentages involves moving the decimal point two places because percentages are based on hundreds. To convert a decimal to a percentage, move the decimal point two places to the right and attach a percent symbol. Moving the decimal point two places to the right is equivalent to multiplying by 100. For instance, the decimal 0.2 shifts two places to become 20%, 0.4 becomes 40%, 0.6 becomes 60%, and 0.8 becomes 80%.

Reversing the process allows you to translate percentages back into decimal and fractional expressions. To convert a percentage to a decimal, remove the percent symbol and move the decimal point two places to the left. Moving the decimal point two places to the left is equivalent to dividing by 100. To convert a percentage to a fraction, write its numerical value over 100; if that value is a decimal, first clear the decimal by multiplying the numerator and denominator by the same power of 10, then simplify.

Comparing and Ordering Rational Numbers

To compare fractions with the same positive denominator, compare their numerators; the fraction with the larger numerator is greater. For example, among fifths, 4/5 is greater than 3/5, which is greater than 2/5, which is greater than 1/5. Conversely, for positive fractions with the same numerator, the fraction with the smaller denominator is greater. This occurs because dividing an identical whole into fewer equal pieces yields larger individual parts; thus, 1/4 is greater than 1/8, and 1/2 is greater than 1/4.

When fractions have different denominators, other methods must be used. To compare fractions with different denominators, convert them to equivalent fractions with a least common denominator. Once the denominators are matched, compare the numerators directly. For fractions a/b and c/d with positive denominators b and d, cross-multiplication shows that a/b is greater than c/d if ad is greater than bc.

Questions often require arranging values in a designated order. Ascending order means arranging numbers from least to greatest. Descending order means arranging numbers from greatest to least. To compare and order a set containing fractions, decimals, and percents, convert all values into a single consistent format such as decimals. Because adding trailing zeros to the right of a decimal does not change its numerical value, expanding decimals to the same number of decimal places makes direct comparisons straightforward.

Number Line Placement and Signed Numbers

A number line provides a spatial model for ordering rational numbers. On a standard horizontal number line, numbers increase in value from left to right. Because of this structure, every positive rational number is strictly greater than every negative rational number. Any positive value, no matter how small, sits to the right of any negative value on the line.

Comparing negative numbers requires special attention to their values relative to zero. For two negative numbers, the number with the greater absolute value is the smaller number. Because numbers increase as you move toward the right, a negative number with a larger absolute value lies farther to the left, making it lesser in value. Understanding signed numbers and using consistent decimal formatting helps with ordering, but use exact values or sufficient precision when comparing repeating decimals or very close numbers.