ASTB-E

Factors of a number are integers that divide evenly into that number with no remainder. To find factors, systematically test integers starting from 1.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Prime factorization expresses a number as a product of prime numbers. Use a factor tree or repeated division.

Prime factorization of 60: 60 = 2 × 30 30 = 2 × 15 15 = 3 × 5 Therefore, 60 = 2² × 3 × 5

The GCF is the largest factor shared by two or more numbers. Find it by listing factors or using prime factorization.

GCF of 24 and 36:

GCF = 12

Using prime factorization: 24 = 2³ × 3 36 = 2² × 3²

GCF = 2² × 3 = 12 (take lowest power of each common prime)

Multiples of a number are products of that number and positive integers.

Multiples of 6: 6, 12, 18, 24, 30, 36, ...

The LCM is the smallest multiple shared by two or more numbers.

LCM of 8 and 12:

LCM = 24

Using prime factorization:

8 = 2³ 12 = 2² × 3

LCM = 2³ × 3 = 24 (take highest power of each prime)

Quick tests to determine if a number is divisible by another:

Divisible by 10: ends in 0

A fraction represents a part of a whole, written as a/b where a is the numerator (part) and b is the denominator (whole). The denominator cannot be zero.

Proper fractions have numerators smaller than denominators (3/4). Improper fractions have numerators equal to or larger than denominators (5/3). Mixed numbers combine whole numbers and fractions (1 2/3).

Converting improper fractions to mixed numbers: Divide numerator by denominator. The quotient is the whole number; the remainder is the new numerator. 17/5 = 3 2/5 (17 ÷ 5 = 3 remainder 2)

Converting mixed numbers to improper fractions: Multiply the whole number by the denominator, add the numerator, and place over the original denominator. 3 2/5 = (3 × 5 + 2)/5 = 17/5

Equivalent fractions represent the same value. Create equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number. 2/3 = 4/6 = 6/9 = 8/12

To simplify (reduce) a fraction, divide both numerator and denominator by their GCF. 18/24: GCF of 18 and 24 is 6. 18/24 = 3/4

With like denominators, add or subtract numerators and keep the denominator: 3/8 + 2/8 = 5/8

With unlike denominators, first find a common denominator (preferably the LCM): 2/3 + 1/4: LCM of 3 and 4 is 12 2/3 = 8/12 and 1/4 = 3/12 8/12 + 3/12 = 11/12

Multiply numerators together and denominators together: 2/3 × 4/5 = 8/15

Simplify before multiplying when possible by canceling common factors: 3/4 × 8/9 = (3 × 8)/(4 × 9) = 1/1 × 2/3 = 2/3

To multiply mixed numbers, first convert to improper fractions.

To divide fractions, multiply by the reciprocal of the divisor: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6

Decimals represent fractions with denominators that are powers of 10. Place values to the right of the decimal point are tenths, hundredths, thousandths, and so on.

Converting fractions to decimals: Divide numerator by denominator. 3/4 = 0.75 1/3 = 0.333... (repeating)

Converting decimals to fractions: Place the decimal digits over the appropriate power of 10 and simplify. 0.75 = 75/100 = 3/4 0.125 = 125/1000 = 1/8

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

Converting percentages to decimals: Divide by 100 (move decimal point two places left). 75% = 0.75 8% = 0.08 150% = 1.50

Converting decimals to percentages: Multiply by 100 (move decimal point two places right). 0.45 = 45% 0.08 = 8% 1.25 = 125%

Converting percentages to fractions: Place over 100 and simplify. 25% = 25/100 = 1/4 60% = 60/100 = 3/5

Finding a percentage of a number: Convert percentage to decimal and multiply. 30% of 80 = 0.30 × 80 = 24

Finding what percentage one number is of another: Divide the part by the whole and multiply by 100. 15 is what percent of 60? (15/60) × 100 = 25%

Finding the whole when given a part and percentage: Divide the part by the percentage as a decimal. 24 is 40% of what number? 24 ÷ 0.40 = 60

Percent increase or decrease = (Amount of change / Original amount) × 100

Original price: $80. New price: $100. Percent increase = (20/80) × 100 = 25%

Original price: $100. New price: $75. Percent decrease = (25/100) × 100 = 25%

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

A proportion states that two ratios are equal: a/b = c/d

Cross-multiplication solves proportions: If a/b = c/d, then ad = bc.

x = 15

Rates compare quantities with different units (miles per hour, dollars per pound).

Distance = Rate × Time

If traveling at 60 mph for 2.5 hours: Distance = 60 × 2.5 = 150 miles

Work rate problems: If A completes a job in x hours and B completes it in y hours, their combined rate is 1/x + 1/y jobs per hour.

A linear equation in one variable has the form ax + b = c. To solve, isolate the variable using inverse operations.

x = 5

A system of equations contains two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.

Solve one equation for one variable

Substitute that expression into the other equation

Solve for the remaining variable

Substitute back to find the other variable

Equation 1: y = 2x + 1 Equation 2: 3x + y = 11

Substitute y = 2x + 1 into Equation 2: 3x + (2x + 1) = 11 5x + 1 = 11 5x = 10

x = 2

Substitute x = 2 into y = 2x + 1: y = 2(2) + 1 = 5

Solution: (2, 5)