GMAT

The Quantitative Reasoning section contains 21 Problem Solving questions in 45 minutes, a little over two minutes per question. Every question has five answer choices and draws on arithmetic and algebra: operations with fractions, percents, ratios, exponents, and roots; number properties such as divisibility, primes, and remainders; equations, inequalities, and functions; algebraic expressions; rate, work, mixture, and set word problems; and statistics, counting, and probability. No calculator is permitted, so the section rewards clean setup, number sense, and shortcuts that avoid long computation. The section is computer-adaptive at the question level: each answer influences the difficulty of the next question.

Arithmetic is the foundation of every Quantitative Reasoning question. Even problems that look algebraic usually end in a fraction, percent, or exponent computation that must be done quickly and accurately by hand. This chapter reviews the rules and shortcuts that keep that computation short.

Order of Operations

Evaluate grouping symbols first, then exponents and roots, then multiplication and division from left to right, then addition and subtraction from left to right (PEMDAS). Multiplication and division share one level, as do addition and subtraction; neither member of a pair automatically comes first.

3 + 4 × 2² − (6 ÷ 2)

= 3 + 4 × 2² − 3 parentheses

= 3 + 4 × 4 − 3 exponents

= 3 + 16 − 3 multiplication

= 16 addition and subtraction, left to right

18 − 12 ÷ 4 × 3 = 18 − 3 × 3 = 18 − 9 = 9 (not 18 − 1 = 17)

An exponent applies only to what it touches: −3² = −(3²) = −9, but (−3)² = 9. A fraction bar acts as a grouping symbol for everything above it and everything below it.

Properties of Operations

Property	Rule	Useontheexam
Commutative	a+b=b+a;a×b=b×a	Reordertermstopairfriendlynumbers:25×17×4=
		(25×4)×17=1700 ,
Associative	(a+b)+c=a+(b+c);(ab)c=a(bc)	Regroupsumsandproductsfreely;subtractionand
		divisionareneithercommutativenorassociative
Distributive	a(b + c) = ab + ac; a(b − c) = ab − ac	1 Mentalmath:7×98=7(100−2) =700−14=686; factoringreversesit

Property	Rule	Useontheexam
Identity	a+0=a;a×1=a	Multiplybyaformof1(suchas√3/√3)tochange
		appearancewithoutchangingvalue
Inverse	a + ( −a) = 0; a × (1/a) = 1 for a ≠ 0	Divisionbyzeroisundefined;aconditionsuchasx≠0
		isoftenthekeytoaproblem

Signed Numbers and Absolute Value

Operation	Rule	Example
Add samesigns ,	Addtheabsolutevalues;keepthecommonsign	( −7)+( −5) =−12
Add differentsigns ,	Subtracttheabsolutevalues;takethesignofthelarger	( −7)+5=−2
	absolutevalue
Subtract	Add the opposite: a − b = a + ( −b)	4 − ( −9) = 13
Multiplyordivide	Samesignsgiveapositiveresult;differentsignsgivea	( −6)( −3) =18;( −6)÷3=−2
	negativeresult
Manyfactors	Anoddnumberofnegativefactorsgivesanegative	( −1)( −2)( −3) =−6;( −2)⁴=16
	product;anevennumbergivesapositiveproduct

The absolute value |x| is the distance from x to zero on the number line, so it is never negative: |5| = |−5| = 5 and |0| = 0. More generally, |a − b| is the distance between a and b, which is the key to absolute value equations and inequalities in Chapter 1.3. Two identities appear often: |ab| = |a| × |b|, and √(x²) = |x| (not x, because x may be negative).

Fractions

Operation	Rule	Example
Simplify	Dividenumeratoranddenominatorbytheirgreatest	12/18=(12÷6)/(18÷6) =2/3
	commonfactor
Addorsubtract	Useacommondenominator:a/b+c/d=(ad+	2/3+3/4=(8+9)/12=17/12
	bc)/bd
Multiply	Multiplystraightacross;cancelcommonfactors	(4/9)×(15/8) =(1/3)×(5/2) =5/6
	first
Divide	Multiplybythereciprocalofthedivisor	(2/5)÷(4/15) =(2/5)×(15/4) =3/2
Compare	Cross-multiply: a/b > c/d when ad > bc (b d ,	5/8vs7/11:55>56?No so5/8<7/11 ,
	positive)
Complexfraction	MultiplytopandbottombytheLCDofallthesmall	(1/2+1/3)/(1/6) =(3+2)/1=5
	fractions

Fractions between 0 and 1 shrink when squared: (1/2)² = 1/4, so for 0 < x < 1, x² < x < √x.

Adding the same positive number to the top and bottom moves a positive fraction toward 1: 2/5 < 3/6 < 4/7.

Dividing by a fraction less than 1 makes a positive number larger: 6 ÷ (1/3) = 18.

Decimals, Fractions, and Percents

Each decimal place is a power of ten: 0.375 = 3/10 + 7/100 + 5/1000 = 375/1000 = 3/8. To convert a fraction to a decimal, divide; to convert a decimal to a fraction, write it over the matching power of ten and simplify. To multiply decimals, multiply as whole numbers and count the total decimal places; to divide, shift both decimal points the same number of places until the divisor is a whole number (0.36 ÷ 0.004 = 360 ÷ 4 = 90). Memorizing the common equivalents below eliminates most long division.

Fraction	Decimal	Percent	Fraction	Decimal	Percent
1/2	05 .	50%	1/8	0125 .	125% .
1/3	0333 . …	33⅓%	3/8	0375 .	375% .
2/3	0666 . …	66⅔%	5/8	0625 .	625% .
1/4	025 .	25%	1/6	01666 . …	16⅔%
3/4	075 .	75%	1/9	0111 . …	111% .
1/5	02 .	20%	1/12	00833 . …	8⅓%

Percents

Every basic percent problem is one equation: part = percent × whole. Identify which of the three quantities is missing and solve for it. "Of" signals multiplication and "is" signals the equals sign.

Questiontype	Setup	Example
Findthepart	part=percent×whole	30%of80=03×80=24 .
Findthepercent	percent=part÷whole	24iswhatpercentof80?24/80=03=30% .
Findthewhole	whole=part÷percent	24is30%ofwhat?24÷03=80 .

percent change = (new − original) ÷ original × 100%

From 80 to 100 is a 25% increase (20/80); from 100 back to 80 is a 20% decrease (20/100). The base is always the original value, so the same difference gives different percents in the two directions.

Successive percent changes multiply; they do not add. Convert each change to a multiplier (an increase of r% is × (1 + r/100); a decrease of r% is × (1 − r/100)) and multiply the multipliers. A 20% increase followed by a 25% decrease gives 1.2 × 0.75 = 0.90, a net 10% decrease. A 10% increase followed by a 10% decrease gives 1.1 × 0.9 = 0.99, a net 1% decrease, not zero. For two changes of a% and b%, the net change is a + b + ab/100 percent (use negative values for decreases).

Percentchangemultipliersandreversals(Continued)

Change	Multiplier	Changethatreversesit	Check
Increase25%	×125 .	Decrease20%	125×08=1 . .
Increase50%	×150 .	Decrease33⅓%	1 .5 × (2/3) = 1
Increase100%(doubling)	×2	Decrease50%	2×05=1 .
Decrease20%	×080 .	Increase25%	08×125=1 . .
Decrease25%	×075 .	Increase33⅓%	0 .75 × (4/3) = 1
Decrease50%	×050 .	Increase100%	05×2=1 .

Percent versus percentage points. If an interest rate rises from 10% to 12%, it rises by 2 percentage points (a difference of two rates), but by 20 percent (2 is 20% of 10). Read which one the question asks for, and expect the other value among the wrong answers.

Ratios and Proportions

A ratio compares quantities: 12 men and 8 women give a ratio of men to women of 12 : 8 = 3 : 2. A part-to-part ratio of 3 : 2 means the whole contains 3 + 2 = 5 parts, so men are 3/5 of the group; this parts method converts any ratio into actual amounts once the total is known. A proportion states that two ratios are equal, a/b = c/d, and is solved by cross- multiplying: ad = bc.

a : b = 2 : 3 and b : c = 4 : 5

Make the shared term b equal: b = LCM(3, 4) = 12

a : b = 8 : 12 b : c = 12 : 15

a : b : c = 8 : 12 : 15

Scale each ratio so the common quantity matches, then read the three-term ratio. Ratios never give actual amounts on their own; a total or one actual value is needed.

Relationship	Equation	Whatstaysconstant	Example
Directvariation	y=kx	Theratioy/x=k	Costofpensatafixedprice:3pens
			for$450 so10penscost$15 . ,
Inversevariation	y=k/x	Theproductxy=k	6workerstake10days so4workers ,
			take6×10÷4=15days

Exponents

Rulesofexponents(Continued)

Rule	Form	Example
Powerofaproduct	(ab)ⁿ=aⁿbⁿ	(2 × 5)³ = 2³ × 5³ = 1 ,000
Powerofaquotient	(a/b)ⁿ=aⁿ/bⁿ	(2/3)³ =8/27
Zeroexponent	a⁰=1(a≠0)	( −7)⁰=1
Negativeexponent	a⁻ⁿ=1/aⁿ	4⁻² =1/16;(2/3)⁻² =9/4
Fractionalexponent	a^(m/n)=(ⁿ√a)ᵐ	8^(2/3) =(∛8)² =4;27^( −1/3) =1/3

No rule for sums: 2³ + 2⁴ is not 2⁷. Sums of powers are handled by factoring (Chapter 1.4).

Negative bases: an even power is positive and an odd power is negative: (−2)⁴ = 16, (−2)⁵ = −32.

Size of powers: for x > 1, higher powers are larger; for 0 < x < 1, higher powers are smaller.

Compare powers by rewriting with a common base or a common exponent: 2³⁰ = 8¹⁰ and 3²⁰ = 9¹⁰, so 3²⁰ is larger.

Roots and Radicals

The square root symbol √a always means the nonnegative root, so √25 = 5, even though x² = 25 has two solutions, 5 and −5. Roots multiply and divide like exponents, but they do not add: √9 + √16 = 7, while √(9 + 16) = 5. Simplify a root by pulling out perfect-square factors: √72 = √(36 × 2) = 6√2, and like radicals combine like like terms: √12 + √27 = 2√3 + 3√3 = 5√3.

√a × √b = √(ab) √a ÷ √b = √(a/b) √(x²) = |x|

1/√3 = (1 × √3)/(√3 × √3) = √3/3

1/(√3 + 1) = (√3 − 1)/((√3 + 1)(√3 − 1)) = (√3 − 1)/(3 − 1) = (√3 − 1)/2

2/(√5 − √3) = 2(√5 + √3)/(5 − 3) = √5 + √3

To clear a single root from a denominator, multiply by that root over itself. To clear a binomial with a root, multiply by its conjugate (the same two terms with the opposite sign between them); the difference of squares removes the roots.

Estimation

Without a calculator, estimation is often the fastest route to the answer, especially when the choices are spread apart. Round to friendly numbers, keep track of whether each rounding pushes the result up or down, and stop as soon as only one choice fits.

Technique	How	Example
Roundtofriendlyvalues	Replaceawkwardnumberswithnearbyroundones	049×812÷198≈05×800÷20=20 . . .
		(exact:about2009) .
Benchmarkfractions	Compareto1/2 1/3 1/4 or1 , , ,	7/15isjustunder1/2;9/17isjustover1/2
Scientificnotation	Trackthepowersoftenseparately	(4 × 10⁵)(3 × 10⁻²) = 12 × 10³ = 1 .2 × 10⁴
Bounds	Findavaluetheanswermustexceedandoneit muststaybelow	1 √90isbetween√81=9and√100=10 and , closerto95 .

Multiplication and division share a level in PEMDAS; work left

Percentage points measure a difference between two

to right.

percents; percent change measures relative change.

|a − b| is the distance between a and b; √(x²) = |x|.

Combine ratios by making the shared term equal; use parts to convert ratios into amounts.

Percent change uses the original value as the base.

Exponent rules apply to products and quotients, never to

Successive percent changes multiply: +20% then −25% is ×

sums; roots do not add.

0.9.

Rationalize with the conjugate; estimate whenever the answer choices are far apart.