GMAT (Graduate Management Admission Test) Quantitative Foundations: Arithmetic & Number Properties Syllabus

How do you solve an equation of the form |x| = k (for k > 0)?

Split into two cases: x = k or x = -k. If k = 0, then x = 0; if k < 0, there is no solution.

For n consecutive integers, when is their sum divisible by n?

The sum of n consecutive integers is divisible by n when n is odd. (The average is the middle term, an integer.) When n is even, it is not.

How do you find the sum of a set of consecutive integers (or any evenly spaced set)?

Sum = average x number of terms, where average = (first term + last term) / 2.

How many integers are there from a to b inclusive?

Count = b - a + 1 (the 'add one before you're done' rule for inclusive counting).

How do you add or subtract fractions with different denominators?

Convert to a common denominator (e.g., the LCD), then add or subtract the numerators and keep the common denominator. Simplify the result.

How do you multiply two fractions?

Multiply the numerators together and the denominators together: (a/b) x (c/d) = ac/bd. Simplify, ideally by cross-canceling first.

How do you divide by a fraction?

Multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b) x (d/c) = ad/bc.

What does each place represent to the right of the decimal point?

The first place is tenths (1/10), the second is hundredths (1/100), the third is thousandths (1/1000), and so on—each place is one-tenth of the place to its left.

How do you convert a decimal to a fraction?

Write the decimal digits over the corresponding power of 10 (based on the number of decimal places), then simplify. E.g., 0.36 = 36/100 = 9/25.

How do you find 'P percent of a number N'?

Multiply: (P/100) x N. For example, 15% of 80 = 0.15 x 80 = 12.

What is the percent change formula?

Percent change = (new value - original value) / original value x 100. A positive result is an increase; a negative result is a decrease.

How do you handle two successive percent changes (e.g., +20% then -10%)?

Multiply the multipliers, do not add the percents. +20% then -10% = 1.20 x 0.90 = 1.08, a net 8% increase.

What single multiplier represents increasing a quantity by P percent?

Multiply by (1 + P/100). For example, a 25% increase means multiplying by 1.25; a 25% decrease means multiplying by 0.75.

Give the fraction, decimal, and percent equivalents of common benchmarks: 1/2, 1/4, 3/4, 1/5.

1/2 = 0.5 = 50%; 1/4 = 0.25 = 25%; 3/4 = 0.75 = 75%; 1/5 = 0.2 = 20%.

Give the decimal and percent equivalents of 1/3, 2/3, and 1/8.

1/3 ≈ 0.333... ≈ 33.3%; 2/3 ≈ 0.667... ≈ 66.7%; 1/8 = 0.125 = 12.5%.

How do you convert a fraction to a percent?

Divide the numerator by the denominator to get a decimal, then multiply by 100 (move the decimal point two places right). E.g., 3/8 = 0.375 = 37.5%.

What are benchmark values, and how are they used in estimation?

Benchmark values are easy reference numbers (like 1/2, 1/4, 10%, round numbers) used to approximate quickly. You round messy numbers to nearby benchmarks to estimate answers.

What is the difference between a part-to-part ratio and a part-to-whole ratio?

Part-to-part compares one group to another (e.g., boys:girls = 3:2). Part-to-whole compares one group to the total (e.g., boys:total = 3:5). The whole is the sum of the parts.

In a ratio a:b, what is the 'unknown multiplier' method?

Write the actual quantities as ax and bx, where x is a common scaling factor (the multiplier). Use given information to solve for x, then find the real amounts.

How do you scale a ratio to find actual quantities given the total?

Add the ratio parts to get total parts, divide the actual total by total parts to find the value of one part, then multiply each ratio term by that value.

What characterizes direct proportion, and what stays constant?

In direct proportion, as one quantity increases the other increases at the same rate; their ratio y/x stays constant (y = kx).

What characterizes inverse proportion, and what stays constant?

In inverse proportion, as one quantity increases the other decreases; their product xy stays constant (y = k/x).

What is a unit rate?

A unit rate expresses a quantity per single unit of another (e.g., miles per 1 hour, price per 1 item). Find it by dividing so the denominator equals 1.

How do you convert units using conversion factors (dimensional analysis)?

Multiply by fractions equal to 1 that have the desired unit on top and the unit to cancel on the bottom, so unwanted units cancel and the target unit remains.

How do you combine two groups with different ratios into one overall ratio?

Scale both ratios to share a common term (often a common total of parts), then add the corresponding quantities. Express the combined amounts as a new ratio.

State the laws of exponents for multiplying and dividing like bases.

x^a x x^b = x^(a+b); x^a / x^b = x^(a-b). When multiplying, add exponents; when dividing, subtract them.

State the power-of-a-power and power-of-a-product rules, plus x^0 and negative exponents.

(x^a)^b = x^(ab); (xy)^a = x^a y^a; x^0 = 1 (x ≠ 0); x^(-a) = 1/x^a.

What is the principal square root, and what does the radical sign denote?

The principal (non-negative) square root of n, written √n, is the non-negative value that squares to n. For example, √25 = 5, not -5.

How do you express and evaluate a fractional (rational) exponent like x^(m/n)?

x^(m/n) = the nth root of x raised to the m power = (n-th root of x)^m = n-th root of (x^m). For example, 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.

What is scientific notation, and what rules govern its form?

Scientific notation writes a number as a x 10^n, where 1 <= |a| < 10 and n is an integer. Positive n means a large number; negative n means a small number (e.g., 4500 = 4.5 x 10^3).

What is the order of operations (PEMDAS), and how are equal-priority operations handled?

Parentheses/grouping, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Operations of equal priority are done left to right.