Arithmetic: complete GMAT guide

Arithmetic on the GMAT Focus Edition includes:

• Order of operations and signed numbers
• Fractions, decimals, and percent conversions
• Remainders and integer division
• Estimation and approximation
• Ratios of quantities and unit conversions
• Properties of even and odd integers

Most questions are designed so that the arithmetic can be simplified before you calculate. If you find yourself multiplying three-digit numbers by three-digit numbers, look for a factor, a common denominator, or an estimate that eliminates choices.

Work in this order: parentheses, exponents and roots, multiplication and division from left to right, addition and subtraction from left to right. A common error is treating multiplication as always before division. They have equal priority, so move left to right.

For negative numbers, remember that $(-2)^2=4$ while $-2^2=-4$. The parentheses decide whether the negative sign is part of the base. Also remember that subtracting a negative is addition: $5-(-3)=8$.

To add or subtract fractions, use a common denominator. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal:

Simplify before multiplying when possible. For $\frac{14}{15} \times \frac{25}{21}$, cancel 7 from 14 and 21, and cancel 5 from 15 and 25, leaving $\frac{2}{3} \times \frac{5}{3}=\frac{10}{9}$. The numbers stay small and the arithmetic is easier to check.

To compare fractions, cross-multiply or convert to decimals. For $\frac{7}{12}$ and $\frac{5}{9}$, cross-multiply: $7 \times 9=63$ and $5 \times 12=60$, so $\frac{7}{12}$ is larger.

Common decimal equivalents are worth memorizing:

When a decimal repeats, treat it as a fraction. For example, $0.\overline{3}=1/3$ and $0.\overline{6}=2/3$. This is often faster than long division.

For positive integers $a$ and $b$, division with remainder can be written as

where $q$ is the quotient and $r$ is the remainder, with $0 \le r \lt b$. The remainder is always less than the divisor.

Remainder questions often use this substitution technique. Represent the number as $divisor \times quotient + remainder$, then perform the operation and reduce again.

A special case is divisibility by 9 or 3 using digit sums. A number leaves the same remainder as its digit sum when divided by 9 or 3. For example, 472 has digit sum 13, which leaves remainder 4 when divided by 9, so 472 leaves remainder 4 when divided by 9.

Many GMAT arithmetic questions do not need exact values. They test whether you can identify the range of the answer or eliminate choices that are clearly too large or too small. Round numbers to convenient values, but keep track of whether you rounded up or down.

Estimation is risky when answer choices are close together. In that case, use exact arithmetic or compare differences. A good rule is to estimate first to eliminate, then calculate exactly only if two choices remain.

Unit conversions are arithmetic questions in disguise. Write the conversion as a fraction and cancel units. For example, to convert 45 miles per hour to feet per second, use $1$ mile $=5280$ feet and $1$ hour $=3600$ seconds:

Keep the units in the calculation until the end. If the units do not cancel to the unit the question asks for, you have set up the conversion backward.

Multiplying or dividing by a power of ten moves the decimal point. For $10^3=1000$, multiplying by $10^3$ moves the decimal three places to the right. Dividing moves it three places to the left. This rule helps you handle numbers such as $0.00042 \times 10^5$ without writing every zero: move the decimal five places right to get 42.

Scientific notation writes a number as $a \times 10^n$, where $1 \le a \lt 10$ and $n$ is an integer. To multiply two numbers in scientific notation, multiply the coefficients and add the exponents. To divide, divide the coefficients and subtract the exponents. Then adjust the coefficient so it remains between 1 and 10.

Questions about place value and digit changes also fall here. If a decimal is multiplied by 10, each digit moves one place to the left. If it is divided by 100, each digit moves two places to the right. Check the direction against the size of the result: multiplying by a number greater than 1 should make the result larger.

Arithmetic questions often rely on parity. The rules are simple:

• even plus even is even
• odd plus odd is even
• even plus odd is odd
• even times any integer is even
• odd times odd is odd

These rules can answer questions without testing numbers. If a question asks whether a sum is even, check the parity of each term rather than computing the sum.

1. Scan for parentheses, exponents, and negative signs before calculating.
2. Convert fractions and decimals into the form that makes the operation easiest.
3. Simplify or cancel before multiplying large numbers.
4. For remainder questions, write the number as $divisor \times quotient + remainder$.
5. Use estimation to eliminate choices, then calculate exactly if needed.
6. Check the units and the sign of the final answer.

• Doing multiplication before division when division appears first from left to right.
• Squaring a negative number without parentheses.
• Forgetting that a remainder must be smaller than the divisor.
• Rounding too early and losing the distinction between close choices.
• Inverting the wrong fraction in a division problem.
• Converting units in the wrong direction.
• Assuming a number is positive when the question allows negatives.