🔢 Quant

Fractions, Ratios & Percents

Ratio multipliers, percent change, successive discounts, profit, simple and compound interest, two-way tables.

~6h

to master

Fractions, ratios, percents — one idea, three costumes

All three express part-to-whole or part-to-part relationships. The GMAT switches costumes mid-question; your job is to translate freely: $\frac{3}{4} = 0.75 = 75\% = 3:1$ (part-to-part). Memorize the conversion anchors: $\frac{1}{8} = 12.5\%$ , $\frac{1}{6} \approx 16.7\%$ , $\frac{1}{3} \approx 33.3\%$ , $\frac{3}{8} = 37.5\%$ .

Ratio mechanics

A ratio $a:b$ fixes relative size only. Actual quantities are $ax$ and $bx$ for some multiplier $x$ — write the $x$ immediately.
Order matters: "disposed-to-recycled" is not "recycled-to-disposed".
Setting two ratios equal gives a proportion; clear it by cross-multiplying: $\frac{n}{12} = \frac{3}{4} \Rightarrow 4n = 36$ .
Ratios with a shared term chain: if $a:b = 2:3$ and $b:c = 4:5$ , scale to a common $b = 12$ : $a:b:c = 8:12:15$ .

✏️Worked Example

Bouquet logic. With 15 white and 85 red tulips and every bouquet identical, the number of bouquets must divide both 15 and 85 — the answer is a greatest-common-divisor in disguise: $\gcd(15, 85) = 5$ . Ratio constraints often hide integer constraints.

Percent change — the original is the denominator

📐Core Rule

$\%\ \text{change} = \dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%$ . From 24 to 30 is $+25\%$ ; from 30 back to 24 is $-20\%$ . The two directions are never equal, because the base changes. A decrease can't exceed 100%, but an increase can ("price is 300% of 2003" = up 200%).

Successive changes multiply, never add. A 20% discount then a 30% discount leaves

0.8 \times 0.7 = 0.56

— a 44% total discount, not 50%. Same machinery for markups, population growth, and .

⚡Shortcut

"Discounted by $p\%$ " = "pay $(100-p)\%$ ". Going backwards from a final price: divide by the multiplier. Paid $24 after 25% off → original $= 24 / 0.75 = \$ 32$. Never add the percent back.

Profit and interest

Profit = revenue − cost. "Profit equal to 50% of cost" with cost $30 → sell at $30 \times 1.5 = \$ 45$. Read carefully whether a percent is of cost or of selling price.
Simple interest grows linearly: $\text{interest} = P \cdot r \cdot t$ . Three months at 6% annual on $8,000: $8000 \times 0.06 \times \frac{3}{12} = \$ 120$.
Compound interest applies the rate to principal plus accumulated interest: value $= P(1 + \frac{r}{n})^{nt}$ for $n$ periods per year. 10% annual compounded half-yearly for a year = two rounds of 5%: $P(1.05)^2$ — slightly more than 10% simple.

Two-way tables beat Venn diagrams here

"40% of items are red; half are small; 10% are red and small; 40 items are green and large…" — draw a 2×2 grid (rows: small/large; columns: red/green), fill the forced cells, and the unknown total falls out of whichever cell gets a real count. This layout solves an entire question family mechanically — including .

⚠️GMAT Trap

Percent of what? Every percent in a multi-group problem has its own base: "40% of the men" ≠ "40% of all people". Before computing, annotate each percent with its base population. Mixed bases are the single most common percent error.

Checklist

Ratios → introduce the multiplier $x$ at once
Percent change over the original; direction asymmetry
Successive percents multiply
Reverse a discount by dividing
Multi-group percents → two-way table, bases annotated

Sample Questions

22 practice questions

Hard

A furniture dealer bought a cabinet for $360 and then set the selling price equal to the purchase price plus a markup that was 40 percent of the selling price. If the dealer sold the cabinet at the selling price, what was the amount of the dealer's gross profit from the purchase and the sale of the cabinet?

Hard

The present ratio of players to coaches at a sports academy is 20 to 1. If the number of players were to increase by 88 and the number of coaches were to increase by 10, the ratio of players to coaches would then be 12 to 1. What is the present number of coaches?

Create a free account to access all 22 practice questions on this topic.

Continue Your Prep

Flashcards →

Bite-size concept cards

Question Bank →

Every GMAT Focus question

Mock Test →

Full GMAT Focus simulation

Practice Fractions, Ratios & Percents