OAT Quantitative Reasoning flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Order of Operations

Evaluate parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Example: 2+3×4 equals 14, not 20.

Divisibility Rule for 3

A whole number is divisible by 3 exactly when the sum of its digits is a multiple of 3. For 4, check whether the last two digits form a multiple of 4.

LCM versus GCF

Write each number's prime factorization. The least common multiple uses the highest power of every prime appearing; the greatest common factor uses the lowest power of only the shared primes.

Negative Exponent Rule

For any nonzero base a, a raised to a negative exponent n equals 1 divided by a to the positive n. Thus 2^(-3) equals 1/8, and any nonzero number to the zero power is 1.

Adding and Dividing Fractions

To add or subtract, rewrite over a common denominator, then combine numerators. To divide, multiply by the reciprocal of the divisor: (a/b)÷(c/d) = (a/b)×(d/c).

Percent Change Formula

Percent change equals (new value minus old value) divided by old value, times 100. A positive result is an increase; a negative result is a decrease.

Successive Percent Discounts

Stacked percent changes multiply rather than add. Taking 30% off and then another 10% off leaves 0.70×0.90 = 0.63 of the price, a 37% total discount.

Ratio to Fraction of Total

In a ratio a:b, the first quantity is a/(a+b) of the total and the second is b/(a+b). Use this to convert ratio parts into actual amounts.

Reversing a Percent Decrease

To undo a decrease of p percent, the required increase is p/(100−p)×100 percent. A 25% drop therefore needs a 33⅓% rise to return to the original amount.

Slope Between Two Points

Slope equals the change in y divided by the change in x: m=(y2−y1)/(x2−x1). Horizontal lines have slope 0; vertical lines have undefined slope.

Inequality Sign Reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. Adding or subtracting a negative does not reverse it.

Quadratic Formula and Discriminant

x = [−b ± √(b²−4ac)]/(2a). The discriminant b²−4ac gives two distinct real roots when positive, one repeated root when zero, and no real roots when negative.

Perpendicular Line Slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so their product equals −1. Parallel lines share the same slope and never intersect.

Difference of Squares

a²−b² factors as (a−b)(a+b). This pattern quickly factors expressions such as x²−49 = (x−7)(x+7) and 9y²−16 = (3y−4)(3y+4).

Average Speed for Round Trips

Average speed equals total distance divided by total time, not the average of the two speeds. Traveling 60 mph out and 40 mph back averages 48 mph over equal distances.

Combined Work Rate

If one worker finishes a job in a hours and another in b hours, their combined time t satisfies 1/t = 1/a + 1/b. Two 6-hour workers together finish in 3 hours.

Mixture Problem Setup

Track the pure substance: (concentration₁)(volume₁) + (concentration₂)(volume₂) = (final concentration)(total volume). Then solve the resulting linear equation for the unknown volume.

Simple versus Compound Interest

Simple interest is I = Prt. Compound interest gives A = P(1+r/n)^(nt), where n is the number of compounding periods per year and t is years.

Distance Rate Time

Distance equals rate multiplied by time, so time equals distance divided by rate. Convert units first so that rate and time use consistent measures before computing.

Consecutive Integer Expressions

Represent consecutive integers as n, n+1, n+2. Consecutive even or odd integers differ by 2, so use n, n+2, n+4 and note that n is odd or even as stated.

Triangle Angle Sum

The three interior angles of any triangle add to 180 degrees. An exterior angle equals the sum of the two remote, nonadjacent interior angles.

Pythagorean Theorem

In a right triangle, a²+b²=c², where c is the hypotenuse opposite the right angle. Common triples include 3-4-5, 5-12-13, and 8-15-17 with their multiples.

Circle Circumference and Area

Circumference equals 2πr and area equals πr². Arc length equals the central angle divided by 360, multiplied by the circumference.

Similar Figure Scaling

When two figures are similar with side scale factor k, corresponding lengths scale by k, areas by k², and volumes by k³. Apply the correct power for the quantity asked.

Trapezoid and Parallelogram Area

Trapezoid area equals ½(b₁+b₂)h, averaging the parallel bases times height. Parallelogram area equals base times perpendicular height, not slant side length.

Rectangular Solid Formulas

A rectangular box has volume lwh and surface area 2(lw+lh+wh). A cylinder instead has volume πr²h and lateral surface area 2πrh.

Mean Median Mode

The mean is the sum divided by the count. The median is the middle value of ordered data, averaged when the count is even. The mode is the most frequent value.

Weighted Mean

The weighted mean equals the sum of each value times its weight, divided by the sum of the weights. Use it for course grades and combined group averages.

Complement Rule

The probability that an event does not occur equals 1 minus the probability that it does. This is often faster than counting unfavorable outcomes directly.

Independent Events Multiplication

For independent events, P(A and B) = P(A)×P(B). For mutually exclusive events, P(A or B) = P(A)+P(B), since they cannot occur together.

Addition Rule with Overlap

When events can occur together, P(A or B) = P(A)+P(B)−P(A and B). Subtract the overlap so outcomes counted twice are counted only once.

Conditional Probability

P(A given B) equals P(A and B) divided by P(B), provided P(B) is greater than zero. It restricts the sample space to outcomes where B occurred.