GACE Mathematics (6–12) (711)

A fraction represents a part of a whole and is written as a/b where a is the numerator and b is the denominator (b ≠ 0).

Proper fraction: Numerator is less than denominator (3/7). Value is between 0 and 1. Improper fraction: Numerator is greater than or equal to denominator (9/4). Value is 1 or greater. Mixed number: Combination of a whole number and a proper fraction (2 1/4).

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator over the original denominator. Example: 17/5 = 3 2/5 (since 17 ÷ 5 = 3 remainder 2).

To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place over the original denominator. Example: 4 3/8 = (4 × 8 + 3)/8 = 35/8.

Fractions are equivalent if they represent the same value. Multiply or divide both numerator and denominator by the same nonzero number to create equivalent fractions. Example: 2/3 = 4/6 = 6/9 = 8/12.

To reduce a fraction to lowest terms, divide both numerator and denominator by their GCD. Example: 24/36 = 24÷12 / 36÷12 = 2/3.

Decimals represent fractions with denominators that are powers of 10.

Divide the numerator by the denominator. Example: 3/8 = 0.375. Some fractions produce repeating decimals: 1/3 = 0.333... = 0.3̄.

For terminating decimals, write the decimal over the appropriate power of 10 and simplify. Example: 0.625 = 625/1000 = 5/8.

For repeating decimals, use algebra. Let x = 0.272727.... Then 100x = 27.272727.... Subtracting: 99x = 27, so x = 27/99 = 3/11.

Each position represents a power of 10. Moving left, values increase by factors of 10. Moving right of the decimal point, values decrease by factors of 10: tenths (10⁻¹), hundredths (10⁻²), thousandths (10⁻³), and so on.

A percentage is a ratio expressed as a fraction of 100. The symbol % means "per hundred."

Percent to decimal: Divide by 100 (move decimal two places left). 45% = 0.45. Decimal to percent: Multiply by 100 (move decimal two places right). 0.08 = 8%. Percent to fraction: Write over 100 and simplify. 75% = 75/100 = 3/4. Fraction to percent: Convert to decimal, then to percent. 3/5 = 0.6 = 60%.

Three basic types: Finding a percent of a number: What is 30% of 80? Calculate 0.30 × 80 = 24. Finding what percent one number is of another: 12 is what percent of 48? Calculate 12/48 = 0.25 = 25%. Finding the whole when a percent is known: 15 is 20% of what number? Calculate 15/0.20 = 75.

Percent change = (New Value - Original Value) / Original Value × 100.

Percent increase example: Price rises from $40 to $52. Change = (52 - 40)/40 × 100 = 30% increase.

Percent decrease example: Population drops from 5000 to 4200. Change = (4200 - 5000)/5000 × 100 = -16% (16% decrease).

When applying multiple percent changes, calculate each step sequentially. A 20% increase followed by a 20% decrease does NOT return to the original value. Example: Starting with 100, a 20% increase gives 120. A 20% decrease of 120 gives 120 × 0.80 = 96, not 100.

A ratio compares two quantities. The ratio of a to b can be written as a:b, a/b, or "a to b."

A proportion is an equation stating that two ratios are equal: a/b = c/d.

In a proportion a/b = c/d, the cross products are equal: ad = bc. This technique solves for unknowns. Example: If 3/x = 12/20, then 3 × 20 = 12x, giving 60 = 12x, so x = 5.

Direct proportion: As one quantity increases, the other increases proportionally. y = kx where k is constant. Example: If 5 items cost $15, then 8 items cost $24 (maintaining $3 per item).

Inverse proportion: As one quantity increases, the other decreases proportionally. xy = k where k is constant. Example: If 4 workers complete a job in 6 hours, then 8 workers complete it in 3 hours.

Scientific notation expresses numbers as a product of a coefficient (between 1 and 10) and a power of 10.

Standard form: a × 10ⁿ where 1 ≤ a < 10.

Large numbers: 45,000,000 = 4.5 × 10⁷. Small numbers: 0.00032 = 3.2 × 10⁻⁴.

Multiplication: Multiply coefficients and add exponents. (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹. Division: Divide coefficients and subtract exponents. (8 × 10⁶)/(2 × 10²) = 4 × 10⁴.

Adjust the coefficient to maintain proper form. If (5 × 10³)(4 × 10²) = 20 × 10⁵, rewrite as 2.0 × 10⁶.

The absolute value of a number is its distance from zero on the number line, always nonnegative. |5| = 5 and |-5| = 5.

Properties: |a| ≥ 0 for all real a. |a| = 0 if and only if a = 0. |ab| = |a| × |b|. |a/b| = |a|/|b| for b ≠ 0. |a + b| ≤ |a| + |b| (triangle inequality).

The distance between two numbers a and b on the number line is |a - b| = |b - a|.