Fraction Fundamentals
A fraction $\frac{a}{b}$ represents $a$ parts out of $b$ equal parts. On the GMAT, fractions appear in three key contexts: arithmetic, ratios, and algebraic relationships.
Fraction Operations
Ratio Architecture
A ratio $a:b$ means for every $a$ units of the first, there are $b$ units of the second. Ratios scale — multiply both parts by any positive number to get equivalent ratios.
Key GMAT Insight: The ratio tells you the proportion, not the actual quantities. You always need at least one more piece of information (total, difference, or one actual quantity) to find the absolute values.
Proportions & Scaling
A proportion is an equation of two ratios: $\frac{a}{b} = \frac{c}{d}$. Solve by cross-multiplying: $ad = bc$.
10 Fraction & Ratio Traps
1. Adding fractions without LCD
$\frac{1}{3} + \frac{1}{4} \neq \frac{2}{7}$. Always find LCD first.
2. Ratio vs actual quantity
A ratio of 3:5 does NOT mean there are 3 and 5 of something — it could be 30 and 50.
3. Division inversion
$\frac{a}{b} \div \frac{c}{d}$ is NOT $\frac{a}{b} \div \frac{d}{c}$. Invert the divisor, not the dividend.
4. Part-to-part vs part-to-whole
The ratio 3:5 means parts are $\frac{3}{8}$ and $\frac{5}{8}$ of the whole — not $\frac{3}{5}$ and $\frac{5}{3}$.
5. Combining ratios directly
If A:B = 2:3 and B:C = 4:5, you cannot combine to A:C = 2:5 directly — you must normalize B.
6. Fraction × fraction direction
Multiplying a fraction by a proper fraction makes it smaller. $\frac{3}{4} \times \frac{1}{2} = \frac{3}{8} < \frac{3}{4}$.
7. Cross-multiplying inequalities with negatives
If $\frac{a}{b} > \frac{c}{d}$ and $b$ or $d$ is negative, cross-multiplying flips the inequality.
8. "Of" creates multiplication
25% of 80 = 0.25 × 80 = 20. The word "of" always means multiply.
9. Simplification before multiplying
Simplify diagonally across a multiplication before computing: $\frac{4}{9} \times \frac{3}{8} = \frac{1}{6}$.
10. Mixed number division
Always convert mixed numbers to improper fractions before any operation.
10 GMAT Practice Questions
What is the value of $\dfrac{3}{4} + \dfrac{5}{6} - \dfrac{1}{3}$?
(A) $\frac{4}{3}$. LCD = 12. $\frac{9}{12} + \frac{10}{12} - \frac{4}{12} = \frac{15}{12} = \frac{5}{4}$. Wait — $9+10-4=15$ and $\frac{15}{12}=\frac{5}{4}$. Answer is (C) $\frac{5}{4}$. (C) is correct.
In a class, the ratio of boys to girls is 3:4. If there are 28 students in total, how many girls are there?
(D) 16. Boys = 3k, Girls = 4k. Total = 7k = 28 → k = 4. Girls = 4(4) = 16.
What is $\dfrac{5}{6} \div \dfrac{10}{3}$?
(A) $\frac{1}{4}$. Dividing by $\frac{10}{3}$ means multiplying by $\frac{3}{10}$: $\frac{5}{6} \times \frac{3}{10} = \frac{15}{60} = \frac{1}{4}$.
If the ratio of A to B is 2:3 and the ratio of B to C is 9:4, what is the ratio of A to C?
(B) 3:2. Normalize B: A:B = 2:3 = 6:9. B:C = 9:4. So A:B:C = 6:9:4. A:C = 6:4 = 3:2.
A recipe requires $2\frac{1}{2}$ cups of flour for every $\frac{3}{4}$ cup of sugar. How many cups of flour are needed if 3 cups of sugar are used?
(C) 10. Ratio: $\frac{5/2}{3/4} = \frac{5}{2} \times \frac{4}{3} = \frac{10}{3}$ cups flour per cup sugar. For 3 cups sugar: $\frac{10}{3} \times 3 = 10$.
What fraction of $\dfrac{3}{5}$ is $\dfrac{9}{20}$?
(B) $\frac{3}{4}$. "What fraction of $\frac{3}{5}$" means divide: $\frac{9/20}{3/5} = \frac{9}{20} \times \frac{5}{3} = \frac{45}{60} = \frac{3}{4}$.
Is $\dfrac{x}{y}$ greater than 1?
(1) $x > y$
(2) $y > 0$
(C) BOTH together sufficient. From (1) alone: $x>y$ but if $y=-3, x=-1$, then $x/y = 1/3 < 1$. NOT sufficient. From (2) alone: $y>0$ tells us nothing about $x$. NOT sufficient. Together: $x>y$ and $y>0$, so $x>y>0$, meaning $x/y > 1$. Sufficient.
In a mixture, red balls and blue balls are in a ratio of 5:3. If 10 red balls are removed, the ratio becomes 1:1. How many blue balls are there?
(B) 15. Initially: red = 5k, blue = 3k. After removing 10: $\frac{5k-10}{3k} = 1$ → $5k-10 = 3k$ → $2k = 10$ → $k = 5$. Blue = 3(5) = 15.
If $\dfrac{2x}{3} = \dfrac{8}{y}$, what is $xy$?
(C) 12. Cross-multiply: $2xy = 24$ → $xy = 12$.
A bag contains red and green marbles in a ratio of 7:3. The total number of marbles is between 40 and 60. What is one possible total number of marbles?
(C) 50. Total must be a multiple of 10 (since 7+3=10). Between 40 and 60: possibilities are 50. Verify: 7+3 = 10, so total = 10k. Only 50 (k=5) is strictly between 40 and 60.
Lesson Summary — Key Takeaways
Multiplier model for ratios
Write a:b as ak and bk. You need one more piece of info to find k and the actual quantities.
LCD before adding/subtracting
Never add fractions with different denominators. Find LCD and convert first.
Divide = multiply by reciprocal
Flip the second fraction and multiply. Never flip the first fraction.
Part-to-whole vs part-to-part
A ratio of 3:5 means the parts are 3/8 and 5/8 of the whole.