GMAT Arithmetic: Integers, Fractions, Decimals, Percentages
Last updated: May 2, 2026
Arithmetic: Integers, Fractions, Decimals, Percentages questions are one of the highest-leverage areas to study for the GMAT. This guide breaks down the rule, the elements you need to recognize, the named traps that catch most students, and a memory aid that scales to test day. Read it once, then practice the same sub-topic adaptively in the app.
The rule
On the GMAT, arithmetic problems hinge on fluently converting between four representations of the same number: integer, fraction, decimal, and percentage. Choose the representation that makes the operation cheapest — fractions for multiplication and division of "clean" parts, decimals for addition and comparison, percentages for change and "of" language. Always anchor a percent to its base, and never mix a percent change with a percent total without converting one of them.
Elements breakdown
Conversion Mechanics
Move between the four forms without losing precision.
- Fraction to decimal: divide numerator by denominator
- Decimal to percent: multiply by 100
- Percent to fraction: write over 100, then reduce
- Integer divisibility check before reducing fractions
Common examples:
- $\frac{3}{8} = 0.375 = 37.5\%$
- $0.04 = \frac{4}{100} = \frac{1}{25} = 4\%$
Percent of a Base
Every percent must point to a specific base quantity.
- Identify the base before computing
- "Percent of" means multiply by $\frac{p}{100}$
- Successive percents multiply, never add
- Percent change uses original value as base
Common examples:
- $30\%$ of $80$ is $0.30 \times 80 = 24$
- Up $20\%$ then down $20\%$ gives $0.96$ of original, not $1.00$
Percent Change
Compute increase or decrease relative to the starting value.
- Formula: $\frac{\text{new} - \text{old}}{\text{old}} \times 100$
- Increase by $p\%$: multiply by $1 + \frac{p}{100}$
- Decrease by $p\%$: multiply by $1 - \frac{p}{100}$
- Reverse a change: divide, do not subtract
Common examples:
- From $50$ to $65$: $\frac{15}{50} = 30\%$ increase
- To undo a $25\%$ increase, divide by $1.25$, not subtract $25\%$
Fraction Arithmetic
Add, subtract, multiply, divide fractions cleanly.
- Common denominator for addition and subtraction
- Multiply numerators and denominators directly
- Divide by multiplying by the reciprocal
- Cancel common factors before multiplying
Common examples:
- $\frac{2}{3} + \frac{1}{4} = \frac{8 + 3}{12} = \frac{11}{12}$
- $\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{5}{4}$
Decimal Place Discipline
Track decimal places when multiplying, dividing, or rounding.
- Count total decimal places when multiplying
- Shift decimal point when dividing by powers of 10
- Round only at the final step
- Estimate magnitude before computing
Common examples:
- $0.4 \times 0.05 = 0.020$ (three decimal places total)
- $\frac{0.6}{0.02} = \frac{60}{2} = 30$
Common patterns and traps
The Successive-Percent Trap
Two consecutive percent changes are presented, and the trap answer adds them as if they were independent. The correct approach multiplies the change factors. A $10\%$ increase followed by a $10\%$ decrease is not zero net change — it is a $1\%$ net decrease because $1.10 \times 0.90 = 0.99$.
A choice that simply adds or subtracts the two given percents (e.g., $0\%$ change, $20\%$ change) when the actual answer requires the product of the change factors.
The Wrong-Base Switch
The problem changes the reference value mid-question, and a wrong answer applies the percent to the original base when it should be the new value (or vice versa). For instance, after a $25\%$ markup, a $10\%$ discount is taken on the marked-up price, not the original. Identify the base for each percent before computing.
A choice that uses the original price as the base for the second discount, producing a value that is a few dollars off from the correct answer.
The Fraction Shortcut
Decimal-heavy problems often hide a clean fraction. Recognize $0.125 = \frac{1}{8}$, $0.375 = \frac{3}{8}$, $0.625 = \frac{5}{8}$, $0.875 = \frac{7}{8}$, $0.\overline{3} = \frac{1}{3}$, $0.\overline{6} = \frac{2}{3}$. Converting to a fraction often replaces a long multiplication with a single division.
A problem with $87.5\%$ of $560$ that becomes $\frac{7}{8} \times 560 = 490$ in one step.
The Reverse-Percent Trap
The problem gives the result after a percent change and asks for the original. Students subtract the percent from the result instead of dividing. If a price after a $25\%$ increase is $\$60$, the original is $\frac{60}{1.25} = \$48$, not $60 - 15 = \$45$.
A choice that subtracts the percent of the new value from the new value, producing an answer slightly larger than the correct one.
Magnitude Sanity Check
Before selecting an answer, estimate. If the question asks for a small percent of a small number and you compute a value larger than the original base, you have multiplied where you should have divided. The GMAT routinely seeds answer choices that are off by a factor of 10 or 100 to catch decimal errors.
Two choices identical except for a decimal shift (e.g., $0.45$ vs $4.5$ vs $45$), exploiting decimal-place miscounting.
How it works
Treat the four forms as the same number wearing different clothes. If a question asks "What is $\frac{5}{8}$ of $240$?", convert mentally to the cheapest form: $\frac{5}{8} \times 240 = 5 \times 30 = 150$. If it asks "$62.5\%$ of $240$", recognize $62.5\% = \frac{5}{8}$ and do the same calculation. The trap is reaching for the calculator-style approach ($0.625 \times 240$) when a fraction shortcut exists. Anchor every percent to its base — when a price rises $20\%$ then falls $20\%$, the second $20\%$ is taken from the new, larger price, so the final value is $0.8 \times 1.2 = 0.96$ of the original, a $4\%$ net decrease. Always estimate the magnitude before committing to a choice; if you compute a $400\%$ change but the answers cluster around $40\%$, you confused a percent with a percent change.
Worked examples
A small bookstore raised the price of a hardcover novel by $20\%$ in January. In March, the bookstore offered a $15\%$ discount off the January price. If the original price in December was $\$40$, what is the price after the March discount?
What is the price after the March discount?
- A $\$34.00$
- B $\$40.80$ ✓ Correct
- C $\$42.00$
- D $\$44.00$
- E $\$48.00$
Why B is correct: The January price is $40 \times 1.20 = 48$. The March price applies the $15\%$ discount to the January price: $48 \times 0.85 = 40.80$. The base for the discount is the marked-up price, not the original.
Why each wrong choice fails:
- A: This subtracts $15\%$ of the original $\$40$ from the original price ($40 - 6 = 34$), ignoring the January markup entirely. (The Wrong-Base Switch)
- C: This adds the two percents naively ($+20\% - 15\% = +5\%$) and computes $40 \times 1.05 = 42$, treating successive changes as additive. (The Successive-Percent Trap)
- D: This applies a $10\%$ net change ($40 \times 1.10 = 44$), miscombining the two percents and using the wrong base. (The Successive-Percent Trap)
- E: This is the January price after the markup but before the March discount; it ignores the second step of the problem.
In a survey of $480$ commuters, $\frac{3}{8}$ reported using public transit, $37.5\%$ reported driving alone, and the rest carpooled or biked. How many commuters carpooled or biked?
How many commuters carpooled or biked?
- A $60$
- B $120$ ✓ Correct
- C $180$
- D $240$
- E $300$
Why B is correct: Recognize that $\frac{3}{8} = 37.5\%$, so transit and drive-alone each account for $\frac{3}{8}$ of $480 = 180$ commuters. Together they total $360$, leaving $480 - 360 = 120$ commuters who carpooled or biked.
Why each wrong choice fails:
- A: This computes $\frac{1}{8}$ of $480 = 60$, mistakenly subtracting $\frac{3}{8} + \frac{3}{8} + \frac{1}{8}$ from a misremembered total. (The Fraction Shortcut)
- C: This is $\frac{3}{8}$ of $480$, the count for either transit or drive-alone alone, not the remainder.
- D: This is half of $480$, perhaps from confusing $37.5\%$ with $50\%$ or doubling one category instead of subtracting both. (Magnitude Sanity Check)
- E: This computes $480 - 180 = 300$, subtracting only one of the two known categories from the total.
After a $30\%$ increase, the population of a town reached $26{,}000$. What was the population before the increase?
What was the population before the increase?
- A $18{,}200$
- B $19{,}500$
- C $20{,}000$ ✓ Correct
- D $22{,}100$
- E $23{,}400$
Why C is correct: To reverse a $30\%$ increase, divide the new value by $1.30$, not subtract $30\%$. The original population is $\frac{26000}{1.3} = 20000$. Verify: $20000 \times 1.30 = 26000$.
Why each wrong choice fails:
- A: This subtracts $30\%$ of $26{,}000$ from $26{,}000$ ($26000 - 7800 = 18200$), the classic reverse-percent error. (The Reverse-Percent Trap)
- B: This applies a $25\%$ reduction to $26{,}000$ ($26000 \times 0.75 = 19500$), misreading the percent or guessing a midpoint. (The Reverse-Percent Trap)
- D: This subtracts $15\%$ of $26{,}000$ ($26000 - 3900 = 22100$), halving the percent change rather than reversing it. (The Wrong-Base Switch)
- E: This subtracts $10\%$ of $26{,}000$ ($26000 \times 0.90 = 23400$), an arbitrary partial reversal that does not undo a $30\%$ increase. (The Reverse-Percent Trap)
Memory aid
BCC: Base, Convert, Check. Identify the **B**ase, **C**onvert to the easiest form, then **C**heck the magnitude before circling.
Key distinction
A percent without a base is meaningless. "$20\%$ more" and "$20\%$ of" produce different numbers because they point to different reference values.
Summary
Master arithmetic by fluently converting between forms, always identifying the base, and multiplying — never adding — successive percent changes.
Practice arithmetic: integers, fractions, decimals, percentages adaptively
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Start your free 7-day trialFrequently asked questions
What is arithmetic: integers, fractions, decimals, percentages on the GMAT?
On the GMAT, arithmetic problems hinge on fluently converting between four representations of the same number: integer, fraction, decimal, and percentage. Choose the representation that makes the operation cheapest — fractions for multiplication and division of "clean" parts, decimals for addition and comparison, percentages for change and "of" language. Always anchor a percent to its base, and never mix a percent change with a percent total without converting one of them.
How do I practice arithmetic: integers, fractions, decimals, percentages questions?
The fastest way to improve on arithmetic: integers, fractions, decimals, percentages is targeted, adaptive practice — working questions that focus on your specific weak spots within this sub-topic, getting immediate feedback, and revisiting items you missed on a spaced-repetition schedule. Neureto's adaptive engine does this automatically across the GMAT; start a free 7-day trial to see your sub-topic mastery climb in real time.
What's the most important distinction to remember for arithmetic: integers, fractions, decimals, percentages?
A percent without a base is meaningless. "$20\%$ more" and "$20\%$ of" produce different numbers because they point to different reference values.
Is there a memory aid for arithmetic: integers, fractions, decimals, percentages questions?
BCC: Base, Convert, Check. Identify the **B**ase, **C**onvert to the easiest form, then **C**heck the magnitude before circling.
What's a common trap on arithmetic: integers, fractions, decimals, percentages questions?
Confusing percent change with percent of total
What's a common trap on arithmetic: integers, fractions, decimals, percentages questions?
Adding successive percent changes instead of multiplying
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