ACCUPLACER Arithmetic cheat sheet

Place values from right to left: ones, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions.

Order of operations: evaluate parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

Addition and multiplication are commutative (a+b=b+a, a×b=b×a); subtraction and division are not.

Multiplication distributes over addition: a×(b+c)=a×b+a×c.

Divisibility: by 2 if last digit is even; by 3 if digit sum is divisible by 3; by 5 if last digit is 0 or 5; by 9 if digit sum is divisible by 9; by 10 if last digit is 0.

A number is prime if its only positive divisors are 1 and itself; 1 is not prime and 2 is the only even prime.

Rounding: locate the target place, then round up if the digit to its right is 5 or greater, keep it if 4 or less.

Long division steps cycle: divide, multiply, subtract, bring down the next digit.

Remainder relationship: dividend = divisor × quotient + remainder, and the remainder is always less than the divisor.

To add or subtract fractions, rewrite over a common denominator, then combine numerators and keep the denominator.

To multiply fractions, multiply numerators together and denominators together; simplify before multiplying when possible.

To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c.

A fraction a/b equals 1 when a=b, is less than 1 when a<b, and is greater than 1 when a>b.

Simplify a fraction by dividing numerator and denominator by their greatest common factor.

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.

To compare fractions, cross-multiply: a/b > c/d exactly when a×d > b×c (with positive denominators).

Multiplying or dividing numerator and denominator by the same nonzero number produces an equivalent fraction.

Any whole number n can be written as the fraction n/1.

Align decimal points vertically when adding or subtracting decimals, then bring the point straight down into the answer.

When multiplying decimals, multiply as with whole numbers, then place the decimal point so the answer has as many decimal places as the two factors combined.

When dividing decimals, move the divisor's decimal point to make it a whole number and move the dividend's point the same number of places.

Multiplying by 10, 100, or 1000 shifts the decimal point right by 1, 2, or 3 places; dividing shifts it left the same amount.

To compare decimals, line up place values and compare digits from left to right, annexing zeros as needed (0.5 = 0.50).

A decimal terminates or repeats according to its denominator; e.g., denominators with only factors 2 and 5 terminate.

To round a decimal, find the target place and round up if the next digit is 5 or more.

Decimal place values to the right of the point: tenths, hundredths, thousandths, ten-thousandths.

To convert a fraction to a decimal, divide the numerator by the denominator.

Percent means per hundred: p% = p/100, so 45% = 0.45 = 45/100 = 9/20.

Percent equation: part = percent × whole, written as P = r × W where r is the decimal form of the percent.

To find what percent one number is of another, divide the part by the whole and multiply by 100.

Percent change = (new value − old value) ÷ old value × 100; a positive result is an increase, negative a decrease.

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

A proportion states two ratios are equal: a/b = c/d; solve by cross-multiplying, ad = bc.

In a proportion a/b = c/d, the cross products ad and bc are equal, so any missing term can be found from the other three.

To convert a decimal to a percent, multiply by 100 and append the percent sign; to convert a percent to a decimal, divide by 100.

Unit rate is a ratio with denominator 1, found by dividing the first quantity by the second.