OAT Quantitative Reasoning cheat sheet

Order of operations: parentheses, exponents, multiplication/division left to right, addition/subtraction left to right - so 2+3×4=14, not 20.

Integer rules: same signs multiply/divide to positive, opposite signs to negative; subtracting a negative adds: a−(−b)=a+b.

Divisibility: by 3 if digit sum is a multiple of 3; by 4 if the last two digits form a multiple of 4; by 9 if digit sum is a multiple of 9.

Prime factorization via factor tree; LCM takes the highest power of each prime, GCF takes the lowest common power.

Absolute value |x| is distance from 0, so |−7|=7 and |x|=k gives x=k or x=−k for k>0.

Exponent rules: a^m·a^n=a^(m+n), a^m/a^n=a^(m−n), (a^m)^n=a^(mn), a^0=1 for a≠0, a^(−n)=1/a^n.

Square roots: √(ab)=√a·√b; estimate between perfect squares, e.g. √50 lies between 7 and 8 since 49<50<64.

Scientific notation: a×10^n with 1≤|a|<10; multiplying adds exponents, dividing subtracts them.

Fraction arithmetic: add/subtract over a common denominator; multiply straight across; divide by multiplying by the reciprocal.

Percent change = (new−old)/old × 100%; a 20% increase then 20% decrease leaves 0.96 of the original, a net 4% loss.

Percent of a number: part = percent × whole; convert percent to decimal by dividing by 100 before multiplying.

Successive percents multiply: 30% off then 10% off equals 0.70×0.90=0.63, i.e. 37% total discount.

Ratio a:b means a/(a+b) of the total; scale by finding the multiplier that maps ratio parts to actual values.

Proportions solve by cross-multiplication: a/b=c/d implies ad=bc.

Common conversions to memorize: 1/8=0.125, 1/3≈0.333, 3/8=0.375, 5/8=0.625, 1/6≈0.167, 2/3≈0.667.

Percent increase needed to reverse a decrease of p% is p/(100−p)×100%, e.g. a 25% drop needs a 33⅓% rise to recover.

Distribute then combine like terms: 3(x−4)+2x = 5x−12; like terms share the same variable and exponent.

Linear equation strategy: clear fractions by multiplying by the LCD, then isolate the variable using inverse operations.

Slope m=(y₂−y₁)/(x₂−x₁); point-slope form y−y₁=m(x−x₁); slope-intercept y=mx+b.

Parallel lines share slope; perpendicular slopes are negative reciprocals (product = −1).

Systems: substitution when one variable is isolated, elimination when coefficients can be matched; a unique solution requires non-parallel lines.

Multiplying or dividing an inequality by a negative number reverses the inequality sign.

Quadratic formula x=[−b±√(b²−4ac)]/(2a); discriminant b²−4ac>0 gives two real roots, =0 one, <0 none.

Factoring patterns: a²−b²=(a−b)(a+b); x²+(p+q)x+pq=(x+p)(x+q).

Rate equation: distance = rate × time; for round trips with equal distances, average speed = total distance / total time, not the mean of speeds.

Work problems add rates: combined time t satisfies 1/t = 1/a + 1/b for individual times a and b.

Mixture problems track the amount of solute: (concentration₁)(volume₁)+(concentration₂)(volume₂)=(final concentration)(total volume).

Age problems: set up both people's ages in the same year, then apply the stated offset (e.g. five years ago).

Consecutive integers n, n+1, n+2; consecutive odd/even differ by 2. Translate "is" to =, "of" to ×, "per" to division.

Simple interest I=Prt; compound amount A=P(1+r/n)^(nt).

Coin/count problems: value = (number of coins)(denomination); total value sums across denominations.

Check units before computing; convert to a single unit (minutes to hours, cents to dollars) to avoid scale errors.

Triangle area = ½bh; sum of interior angles = 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a²+b²=c² for right triangles; common triples: 3-4-5, 5-12-13, 8-15-17 and their multiples.

Circle: circumference C=2πr, area A=πr²; arc length = (central angle/360)×2πr.

Rectangle area lw, perimeter 2(l+w); trapezoid area = ½(b₁+b₂)h; parallelogram area = bh.

Rectangular solid volume lwh, surface area 2(lw+lh+wh); cylinder volume πr²h.

Similar figures: corresponding sides share one scale factor k; areas scale by k² and volumes by k³.

Complementary angles sum to 90°, supplementary to 180°; vertical angles are equal.

Unit conversions: 1 mile=5,280 ft, 1 yard=3 ft, 1 gallon=4 quarts, 1 kg=1,000 g, 1 hour=3,600 s.

Mean = sum/count; median is the middle value of ordered data (average the two middle values when n is even); mode is most frequent.

Range = max−min; standard deviation grows with spread and is unaffected by adding a constant to every value.

Weighted mean = Σ(value×weight)/Σweights; use it for grade and mixture averages.

Probability of an event = favorable outcomes / total equally likely outcomes; P(not A)=1−P(A).

Independent events multiply: P(A and B)=P(A)P(B); mutually exclusive events add: P(A or B)=P(A)+P(B).

Overlapping events: P(A or B)=P(A)+P(B)−P(A and B).

Conditional probability P(A|B)=P(A and B)/P(B).

Reading tables and graphs: verify axis scales and units before estimating; percent of total = part/total×100%.

Order of operations: parentheses, exponents, multiplication/division left to right, addition/subtraction left to right - so 2+3×4=14, not 20.

Integer rules: same signs multiply/divide to positive, opposite signs to negative; subtracting a negative adds: a−(−b)=a+b.

Divisibility: by 3 if digit sum is a multiple of 3; by 4 if the last two digits form a multiple of 4; by 9 if digit sum is a multiple of 9.

Prime factorization via factor tree; LCM takes the highest power of each prime, GCF takes the lowest common power.

Absolute value |x| is distance from 0, so |−7|=7 and |x|=k gives x=k or x=−k for k>0.

Exponent rules: a^m·a^n=a^(m+n), a^m/a^n=a^(m−n), (a^m)^n=a^(mn), a^0=1 for a≠0, a^(−n)=1/a^n.

Square roots: √(ab)=√a·√b; estimate between perfect squares, e.g. √50 lies between 7 and 8 since 49<50<64.

Scientific notation: a×10^n with 1≤|a|<10; multiplying adds exponents, dividing subtracts them.

Fraction arithmetic: add/subtract over a common denominator; multiply straight across; divide by multiplying by the reciprocal.

Percent change = (new−old)/old × 100%; a 20% increase then 20% decrease leaves 0.96 of the original, a net 4% loss.

Percent of a number: part = percent × whole; convert percent to decimal by dividing by 100 before multiplying.

Successive percents multiply: 30% off then 10% off equals 0.70×0.90=0.63, i.e. 37% total discount.

Ratio a:b means a/(a+b) of the total; scale by finding the multiplier that maps ratio parts to actual values.

Proportions solve by cross-multiplication: a/b=c/d implies ad=bc.

Common conversions to memorize: 1/8=0.125, 1/3≈0.333, 3/8=0.375, 5/8=0.625, 1/6≈0.167, 2/3≈0.667.

Percent increase needed to reverse a decrease of p% is p/(100−p)×100%, e.g. a 25% drop needs a 33⅓% rise to recover.

Distribute then combine like terms: 3(x−4)+2x = 5x−12; like terms share the same variable and exponent.

Linear equation strategy: clear fractions by multiplying by the LCD, then isolate the variable using inverse operations.

Slope m=(y₂−y₁)/(x₂−x₁); point-slope form y−y₁=m(x−x₁); slope-intercept y=mx+b.

Parallel lines share slope; perpendicular slopes are negative reciprocals (product = −1).

Systems: substitution when one variable is isolated, elimination when coefficients can be matched; a unique solution requires non-parallel lines.

Multiplying or dividing an inequality by a negative number reverses the inequality sign.

Quadratic formula x=[−b±√(b²−4ac)]/(2a); discriminant b²−4ac>0 gives two real roots, =0 one, <0 none.

Factoring patterns: a²−b²=(a−b)(a+b); x²+(p+q)x+pq=(x+p)(x+q).

Rate equation: distance = rate × time; for round trips with equal distances, average speed = total distance / total time, not the mean of speeds.

Work problems add rates: combined time t satisfies 1/t = 1/a + 1/b for individual times a and b.

Mixture problems track the amount of solute: (concentration₁)(volume₁)+(concentration₂)(volume₂)=(final concentration)(total volume).

Age problems: set up both people's ages in the same year, then apply the stated offset (e.g. five years ago).

Consecutive integers n, n+1, n+2; consecutive odd/even differ by 2. Translate "is" to =, "of" to ×, "per" to division.

Simple interest I=Prt; compound amount A=P(1+r/n)^(nt).

Coin/count problems: value = (number of coins)(denomination); total value sums across denominations.

Check units before computing; convert to a single unit (minutes to hours, cents to dollars) to avoid scale errors.

Triangle area = ½bh; sum of interior angles = 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a²+b²=c² for right triangles; common triples: 3-4-5, 5-12-13, 8-15-17 and their multiples.

Circle: circumference C=2πr, area A=πr²; arc length = (central angle/360)×2πr.

Rectangle area lw, perimeter 2(l+w); trapezoid area = ½(b₁+b₂)h; parallelogram area = bh.

Rectangular solid volume lwh, surface area 2(lw+lh+wh); cylinder volume πr²h.

Similar figures: corresponding sides share one scale factor k; areas scale by k² and volumes by k³.

Complementary angles sum to 90°, supplementary to 180°; vertical angles are equal.

Unit conversions: 1 mile=5,280 ft, 1 yard=3 ft, 1 gallon=4 quarts, 1 kg=1,000 g, 1 hour=3,600 s.

Mean = sum/count; median is the middle value of ordered data (average the two middle values when n is even); mode is most frequent.

Range = max−min; standard deviation grows with spread and is unaffected by adding a constant to every value.

Weighted mean = Σ(value×weight)/Σweights; use it for grade and mixture averages.

Probability of an event = favorable outcomes / total equally likely outcomes; P(not A)=1−P(A).

Independent events multiply: P(A and B)=P(A)P(B); mutually exclusive events add: P(A or B)=P(A)+P(B).

Overlapping events: P(A or B)=P(A)+P(B)−P(A and B).

Conditional probability P(A|B)=P(A and B)/P(B).

Reading tables and graphs: verify axis scales and units before estimating; percent of total = part/total×100%.