GRE Arithmetic: Integers, Fractions, Percentages
Last updated: May 2, 2026
Arithmetic: Integers, Fractions, Percentages questions are one of the highest-leverage areas to study for the GRE. 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
GRE arithmetic problems test whether you can move fluently between integer properties, fraction operations, and percentage relationships under time pressure. The right answer almost always rewards translating the words into a clean equation or ratio first, then computing — never the reverse. What students miss is that these problems are engineered around predictable slips: confusing percent OF with percent CHANGE, treating consecutive percent changes as additive, or forgetting that 'integer' rules out fractions and decimals (and includes negatives and zero).
Elements breakdown
Integer property checks
When a problem says 'integer,' immediately list which integers are in scope and which arithmetic facts apply.
- Confirm whether negatives and zero are allowed
- Check parity rules (even $\pm$ even = even, etc.)
- Apply divisibility rules for 2, 3, 5, 9
- Track sign behavior under multiplication
- For 'consecutive integers,' write $n, n+1, n+2$
Common examples:
- If $n$ is a positive integer, $n \ge 1$ — not $n > 0$ allowing $0.5$
- $0$ is even and is a multiple of every nonzero integer
Fraction manipulation
Translate every fraction operation into a common-denominator or cross-multiplication move before estimating.
- Find LCD before adding or subtracting
- Multiply: numerators times numerators, denominators times denominators
- Divide: invert the second fraction and multiply
- Reduce before multiplying to avoid huge numbers
- Convert mixed numbers to improper fractions first
Percent translation
Rewrite every percent statement as either a multiplier or an equation in decimals before computing.
- $x\%$ of $y$ becomes $\frac{x}{100} \cdot y$
- Increase by $x\%$ becomes multiplier $(1 + \frac{x}{100})$
- Decrease by $x\%$ becomes multiplier $(1 - \frac{x}{100})$
- Percent change = $\frac{\text{new} - \text{old}}{\text{old}} \times 100$
- Chain percent changes by multiplying, never adding
Base-of-percent discipline
Always identify which quantity is the base (the 'of' value) before computing a percent.
- Underline the word after 'of' or 'than'
- $A$ is $20\%$ more than $B$ means base is $B$
- $A$ is $20\%$ of $B$ also means base is $B$
- Reversed-base questions need $B$ in terms of $A$
- Recompute base when comparing two percent statements
Common patterns and traps
The Additive Percent Trap
When a problem chains two or more percent changes, the wrong answer adds them. The right answer multiplies the corresponding decimal multipliers. This trap is especially seductive when the percentages are small, because the additive answer is close enough to feel plausible. Always convert each change to a multiplier and multiply them in sequence.
An answer choice that equals the simple sum or difference of the stated percents (e.g., $5\%$ when the real compounded answer is $0\%$ or $-1\%$).
The Wrong-Base Trap
Percent problems hinge on which quantity is the base — the 'of' or 'than' value. Wrong answers compute the right percent on the wrong base, typically using the new value when the original was intended. Reversing 'A is 25% more than B' to 'B is 25% less than A' produces a slightly off answer that the test-makers reliably include.
An answer that uses the post-change value as the denominator in the percent-change formula.
The Integer-Scope Oversight
When a problem specifies 'integer,' it rules out fractions and decimals but admits negatives and zero unless further restricted. Wrong answers either treat 'integer' as 'positive integer' or treat 'positive' as 'nonnegative.' Read constraints literally: 'positive integer' means $\ge 1$, 'nonnegative integer' means $\ge 0$, and 'integer' alone covers all of $\dots, -2, -1, 0, 1, 2, \dots$
An answer count that excludes $0$ or excludes negative integers when the constraint didn't forbid them.
The Fraction-of-Fraction Slip
When the problem asks for a fraction of a fraction of a quantity, students often add the fractions or apply only one. The correct move is to multiply the fractions together, then apply the result to the base. This pattern hides inside problems about 'one-third of the remaining two-fifths' kinds of phrasing.
An answer equal to the sum of the two fractions applied to the base, rather than their product.
The Reciprocal-Recovery Pattern
If a value is multiplied by $k$, recovering the original requires dividing by $k$, i.e., multiplying by $\frac{1}{k}$ — which is generally NOT 'subtracting the same percent.' For instance, after a $20\%$ increase, you must decrease by $\frac{1}{6} \approx 16.67\%$ to return, not $20\%$. Recognizing this saves you from the most common percent-reversal trap.
An answer that mirrors the original percent change (e.g., '$20\%$') instead of computing the true reciprocal recovery.
How it works
Suppose a shirt is marked up $25\%$, then discounted $20\%$. The instinct is to say the net change is $+5\%$. It isn't. Translate to multipliers: $1.25 \times 0.80 = 1.00$, so the final price equals the original. That single example captures the whole sub-topic: words become multipliers, multipliers compose, and the answer falls out. Now suppose the question instead asks 'by what percent must the discounted price be raised to return to the marked-up price?' The base shifts from the original to the discounted price, and you must recompute: $\frac{1.25 - 1.00}{1.00} = 25\%$. Same numbers, different base, different answer. Train yourself to underline the base every time.
Worked examples
A regional bookstore raises the price of a hardcover novel by $30\%$ for the holiday season, then offers a $30\%$ discount on the new price during a January clearance. If the original price was $\$40$, what is the clearance price?
What is the clearance price of the novel?
- A $\$36.40$ ✓ Correct
- B $\$38.00$
- C $\$40.00$
- D $\$42.00$
- E $\$43.20$
Why A is correct: Convert each change to a multiplier: $1.30$ for the increase and $0.70$ for the decrease. Apply both: $40 \times 1.30 \times 0.70 = 40 \times 0.91 = \$36.40$. The two percent changes do NOT cancel because they apply to different bases — the discount applies to the marked-up price, not the original.
Why each wrong choice fails:
- B: This answer treats the $-30\%$ as if it applied to the original $\$40$ rather than to the marked-up price, giving $40 - 12 + 10 = \$38$ via flawed reasoning. (The Wrong-Base Trap)
- C: This is the additive-percent trap: $+30\%$ and $-30\%$ feel like they cancel, but multiplicatively $1.30 \times 0.70 = 0.91$, not $1.00$. (The Additive Percent Trap)
- D: This adds $\$2$ to the original price, possibly from misreading the discount as $25\%$ off the new price. It doesn't follow from any correct multiplier chain. (The Additive Percent Trap)
- E: This is just $40 \times 1.30 \times 0.83$ or similar — it omits the $30\%$ discount entirely or applies a partial discount. It treats the discount as smaller than stated. (The Wrong-Base Trap)
$n$ is an integer such that $-3 \le n \le 4$.
Compare Quantity A and Quantity B.
- A Quantity A is greater.
- B Quantity B is greater.
- C The two quantities are equal. ✓ Correct
- D The relationship cannot be determined from the information given.
Why C is correct: The integers from $-3$ to $4$ inclusive are $-3, -2, -1, 0, 1, 2, 3, 4$. Test each: $n^2 < 5$ holds for $n = -2, -1, 0, 1, 2$ (since $(-2)^2 = 4 < 5$ but $(-3)^2 = 9 \not< 5$). That's $5$ values, equal to Quantity B.
Why each wrong choice fails:
- A: To get more than $5$ values, you'd have to include $n = -3$ or $n = 3$, but $(-3)^2 = 9$ and $3^2 = 9$, both $\ge 5$. (The Integer-Scope Oversight)
- B: This trap omits $n = 0$ from the count, treating 'integer' as 'positive or negative integer.' But $0$ is an integer and $0^2 = 0 < 5$ qualifies. (The Integer-Scope Oversight)
- D: Both quantities are fully determined: the range of $n$ is given explicitly and $n^2 < 5$ is a definite condition. Nothing is variable.
At a manufacturing plant, $\frac{2}{5}$ of the workers are assigned to assembly. Of those assembly workers, $\frac{3}{8}$ work the night shift. If $42$ assembly workers work the night shift, how many workers does the plant employ in total?
What is the total number of workers at the plant?
- A $112$
- B $140$
- C $210$
- D $280$ ✓ Correct
- E $336$
Why D is correct: Let $T$ be the total. Assembly workers: $\frac{2}{5}T$. Night-shift assembly workers: $\frac{3}{8} \cdot \frac{2}{5}T = \frac{6}{40}T = \frac{3}{20}T$. Set $\frac{3}{20}T = 42$, so $T = 42 \cdot \frac{20}{3} = 14 \cdot 20 = 280$.
Why each wrong choice fails:
- A: This is $42 \div \frac{3}{8} = 112$, which gives the number of assembly workers, not the total. It stops one step short. (The Fraction-of-Fraction Slip)
- B: This adds the fractions $\frac{2}{5} + \frac{3}{8} = \frac{31}{40}$ or makes a similar additive error, giving a total inconsistent with the multiplicative structure of the problem. (The Fraction-of-Fraction Slip)
- C: This is $42 \cdot 5$, which would be the total only if $\frac{1}{5}$ of workers were night-shift assembly — but the actual fraction is $\frac{3}{20}$, not $\frac{1}{5}$. (The Wrong-Base Trap)
- E: This is $42 \cdot 8$, which incorrectly inverts only the $\frac{3}{8}$ fraction without accounting for the $\frac{2}{5}$ assembly fraction. (The Fraction-of-Fraction Slip)
Memory aid
MBC: Multiplier, Base, Check. Convert percents to multipliers, underline the base, and check whether the answer is a percent, a percent change, or a raw value.
Key distinction
'Percent of' gives you a value; 'percent change' gives you a ratio of difference to the original. The wrong answers in percent problems almost always swap these two — using the new value as the base or reporting the multiplier instead of the change.
Summary
Translate words into multipliers and equations first, identify the base every time, and remember that 'integer' is a stricter constraint than it looks.
Practice arithmetic: integers, fractions, percentages adaptively
Reading the rule is the start. Working GRE-format questions on this sub-topic with adaptive selection, watching your mastery score climb in real time, and seeing the items you missed return on a spaced-repetition schedule — that's where score lift actually happens. Free for seven days. No credit card required.
Start your free 7-day trialFrequently asked questions
What is arithmetic: integers, fractions, percentages on the GRE?
GRE arithmetic problems test whether you can move fluently between integer properties, fraction operations, and percentage relationships under time pressure. The right answer almost always rewards translating the words into a clean equation or ratio first, then computing — never the reverse. What students miss is that these problems are engineered around predictable slips: confusing percent OF with percent CHANGE, treating consecutive percent changes as additive, or forgetting that 'integer' rules out fractions and decimals (and includes negatives and zero).
How do I practice arithmetic: integers, fractions, percentages questions?
The fastest way to improve on arithmetic: integers, fractions, 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 GRE; 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, percentages?
'Percent of' gives you a value; 'percent change' gives you a ratio of difference to the original. The wrong answers in percent problems almost always swap these two — using the new value as the base or reporting the multiplier instead of the change.
Is there a memory aid for arithmetic: integers, fractions, percentages questions?
MBC: Multiplier, Base, Check. Convert percents to multipliers, underline the base, and check whether the answer is a percent, a percent change, or a raw value.
What is "Additive-percent trap" in arithmetic: integers, fractions, percentages questions?
treating $+25\%$ then $-20\%$ as $+5\%$ instead of multiplying.
What is "Wrong-base trap" in arithmetic: integers, fractions, percentages questions?
computing percent change off the new value instead of the original.
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