The recurring arithmetic principles that make GMAT quant questions easier to reason through.
Decimals hide the relationships in a calculation; fractions reveal them. Most ratio, percentage, and probability questions get faster the moment you convert. The enabling move: multiplying the top and bottom of a fraction by the same number doesn't change its value — which means you can clear any decimal without ever doing long division.
GMAT quant is multiple choice, and the answer choices are usually far enough apart that estimation alone resolves the question. Rounding 324 × 571 to 300 × 600 ≈ 180,000 gets you to the right ballpark without any real arithmetic. The discipline is to estimate first and only commit to exact computation if two answers survive the rough cut.
When estimation leaves two answers close, the last digit of the product breaks the tie. The final digit of any product depends only on the final digits of the factors — so 324 × 571 must end in 4, regardless of the rest. Paired with rough estimation, most calculations resolve without doing the full multiplication.
The GMAT never asks for genuinely ugly multiplication. If you're computing a product as large as 244 × 371, you have almost certainly missed a cancellation. Train the instinct to scan for shared factors across numerators and denominators before any number gets multiplied out. What looked like a wall of digits collapses to a single clean fraction.
When multiplication is genuinely unavoidable, the order in which you pair factors matters. Pairing numbers that combine into round intermediates — 5 with 20 into 100, 4 with 6 into 24 — keeps the working numbers small and the arithmetic tractable. Scan the full product first to find the easiest matches; never multiply left to right.
Combining fractions starts with a common denominator — the lowest common multiple of the existing denominators. Once every fraction sits over the same bottom, you simply add or subtract the numerators. Choosing the smallest common denominator keeps the arithmetic light.
Walkthrough video coming soon.
Multiplying fractions is mechanically simple: numerators across the top, denominators across the bottom. The catch is that you should almost never multiply first. Cancel shared factors anywhere on the top against anywhere on the bottom before computing, and the result usually emerges with only minor arithmetic left.
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Division by a fraction is the same as multiplication by its reciprocal — flip the divisor and multiply. The same logic extends to complex fractions: a factor stuck in the denominator of a denominator effectively rises to the top, and vice versa. Once you see this, nested fractions stop being intimidating.
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A square root in the denominator can almost always be removed by multiplying the top and the bottom by that same square root. The denominator then becomes a clean integer, and the expression takes the form a calculator-free question expects. This is the basic move; the conjugate version handles harder cases.
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When the denominator has the form a + √b or a − √b, multiplying by its conjugate — the same expression with the opposite sign — clears the radical. The mechanism is the difference of two squares: (a + √b)(a − √b) = a² − b. Recognise the pattern, swap the sign, multiply.
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Most square roots in GMAT questions are not in their simplest form. The technique: factor the number under the radical and pull out any perfect-square factors. √98 becomes √(49 × 2) becomes 7√2. The work is mechanical once you know your perfect squares cold.
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Radicals add and subtract like variables: only like terms combine. 2√2 + 3√2 collapses to 5√2, but √2 + √3 does not. When an expression mixes several radicals, simplify each one first — terms that looked unlike often share a radical once simplified and combine cleanly.
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The product of two square roots equals the square root of the product: √a × √b = √(ab). The same identity works in reverse — √6 splits into √2 × √3 — which lets you align radicals that would otherwise not combine. This unlock is what makes harder radical simplifications possible.
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Expressions with sums or differences of powers — 5¹⁰ + 5¹¹ − 5⁹, for example — almost never need expanding. Pull out the lowest power as a common factor, and what remains in the brackets is small and easy to evaluate. The same instinct applies wherever a common factor lurks across all terms.
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Five rules govern almost every exponent question on the GMAT. Multiplying powers with the same base adds exponents; dividing subtracts them. Anything nonzero to the zero is one. Anything to the one is itself. A negative exponent is the reciprocal of the positive one. Memorise to instinct.
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Any expression of the form a² − b² factors into (a + b)(a − b). The pattern shows up constantly in GMAT problems, often in disguise — 5¹⁶ − 3¹⁶ is a typical example, and once factored, the same identity often applies again to one of the resulting pieces. Spot it once, apply it repeatedly.
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When a number in a calculation sits just past a round value — 51, 98, 199 — splitting it into the round part plus or minus a small adjustment makes the arithmetic mental. 24 × 51 becomes 24 × 50 plus one more 24. Use this whenever exact computation is unavoidable.
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Recognising 144, 256, or 625 as 12², 4⁴, and 5⁴ is the difference between a question taking thirty seconds and three minutes. Squares through about 20, cubes through 10, and a handful of higher powers should be in instant memory. They unlock cancellations and rewrites that are otherwise hidden.
Walkthrough video coming soon.