🔢 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

After a price was reduced by 25 percent and the reduced price was then lowered by an additional 10 percent, the final price was $81. What was the original price?

Hard

At a company, 60 percent of the employees are engineers and the rest are designers. If 30 percent of the engineers and 65 percent of the designers work remotely, what percent of all the employees work remotely?

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