MTEL (77)

A fraction a/b represents a parts of a whole divided into b equal parts. The numerator indicates the count of parts taken; the denominator indicates the total parts in the whole. Proper fractions have numerators less than denominators; improper fractions have numerators greater than or equal to denominators.

Multiplying or dividing both numerator and denominator by the same nonzero number produces an equivalent fraction: a/b = (a×k)/(b×k). Finding equivalent fractions with common denominators enables addition and subtraction.

Fractions with common denominators add or subtract numerators directly: a/c + b/c = (a+b)/c. Unlike denominators require finding the least common denominator (LCD), converting fractions, then operating. The LCD is the least common multiple of the denominators.

Fraction multiplication follows the rule: (a/b) × (c/d) = (ac)/(bd). Simplifying before multiplying reduces computational complexity. Cross-cancellation divides common factors from any numerator with any denominator.

Dividing by a fraction equals multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c). This rule derives from the definition of division as multiplication by the multiplicative inverse.

Decimal notation extends place value to represent parts of wholes. Moving right from the decimal point, places represent tenths, hundredths, thousandths, and so on. Each place is one-tenth the value of the place to its left.

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a terminating decimal to a fraction, write the decimal digits as the numerator over the appropriate power of 10, then simplify. Repeating decimals require algebraic techniques: let x equal the repeating decimal, multiply to shift the repeat, subtract to eliminate the repeat, then solve for x.

Addition and subtraction require aligning decimal points, then operating as with whole numbers. Multiplication proceeds without alignment; the product has as many decimal places as the total in both factors. Division may require moving the decimal point in the divisor to create a whole number, with corresponding movement in the dividend.

Percent means "per hundred." The symbol % represents division by 100. Thus 45% = 45/100 = 0.45. Percent provides a standardized way to compare ratios with different denominators.

To convert percent to decimal, divide by 100 (move decimal two places left). To convert decimal to percent, multiply by 100 (move decimal two places right). To convert percent to fraction, write over 100 and simplify. To convert fraction to percent, first convert to decimal, then to percent.

Three types of percent problems exist: finding the part (what is 25% of 80?), finding the percent (15 is what percent of 60?), and finding the whole (30 is 20% of what number?). The proportion part/whole = percent/100 solves all three types. Alternatively, use the equation: part = percent × whole.

Percent change equals (new value: original value)/original value × 100. Positive results indicate percent increase; negative results indicate percent decrease. Percent change allows meaningful comparison of changes across different scales.