A fraction is a way of expressing a part of a whole. Write it as p/q where p is the numerator (how many parts you have) and q is the denominator (how many equal parts the whole is divided into). The denominator can never be zero — dividing into zero parts is undefined.
Think of a pizza cut into 8 equal slices. If you eat 3 slices, you've consumed 3/8 of the pizza. The 8 tells you how the pizza was divided; the 3 tells you what you took.
Types you must know for SSC MTS:
3/7, 5/9. These fractions live between 0 and 1 on the number line.9/4, 7/7. The pizza analogy breaks — you'd need more than one pizza.2(3/4), which means 2 + 3/4. This is just an improper fraction written in a friendlier form. 2(3/4) = 11/4.1/3, 1/9. These are the building blocks of ancient fraction arithmetic.2/4 = 3/6 = 1/2. You get equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number.The practical skill SSC MTS tests is not just knowing these definitions — it's being fast at comparing fractions and computing with mixed fractions. Both types appear in almost every MTS paper.
To convert a mixed fraction a(b/c) to improper form:
a(b/c) = (a × c + b) / c
Example: 5(3/4) = (5 × 4 + 3)/4 = 23/4
To convert back from improper to mixed: Divide numerator by denominator. Quotient is the whole number, remainder is the new numerator.
Example: 45/12 — divide 45 by 12: quotient 3, remainder 9. So 45/12 = 3(9/12) = 3(3/4).
Addition and Subtraction — you need a common denominator (LCM of all denominators):
a/p + b/q = (a × q + b × p) / (p × q) when p and q share no common factor.
For mixed fractions, handle the whole-number parts and fractional parts separately, then combine. This is faster than converting everything to improper fractions when the whole numbers are large.
Example: (5 + 2/3) - (3 + 1/6)
Step 1: Whole number part: 5 - 3 = 2
Step 2: Fraction part: 2/3 - 1/6 = 4/6 - 1/6 = 3/6 = 1/2
Step 3: Combine: 2 + 1/2 = 2(1/2) — done.
Multiplication: Straight across. (a/b) × (c/d) = ac/bd. Cancel common factors before multiplying to keep numbers small.
Division: Flip the second fraction and multiply. (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc.
Method 1: Common Denominator Convert all fractions to the same denominator (the LCM). Then compare numerators directly.
Example: Compare 7/8, 6/7, 3/4.
LCM of 8, 7, 4 = 56.
7/8 = 49/56, 6/7 = 48/56, 3/4 = 42/56.
Order: 49 > 48 > 42, so 7/8 > 6/7 > 3/4. Clean.
Method 2: Cross-Multiplication (for two fractions)
To compare a/b and c/d, compute a×d vs b×c.
If a×d > b×c, then a/b > c/d.
Example: Compare 6/7 and 7/8.
6×8 = 48 vs 7×7 = 49. Since 48 < 49, we have 6/7 < 7/8. Done in 3 seconds.
Method 3: Decimal Conversion Divide numerator by denominator mentally. Best when fractions have denominators 4, 5, 8, 10, 20, 25.
4/5 = 0.80, 9/11 ≈ 0.818, 7/9 ≈ 0.778, 11/13 ≈ 0.846. Largest is 11/13.
Method 4: Complement Method (for fractions close to 1)
Look at what each fraction is missing from 1. 1 - 7/8 = 1/8. 1 - 6/7 = 1/7. 1 - 3/4 = 1/4.
The fraction with the smallest "deficit" is the largest. 1/8 < 1/7 < 1/4, so 7/8 > 6/7 > 3/4.
This is the fastest method when fractions are all close to 1 — which happens often in SSC MTS.
These appear as word problems. The logic: if 1/5 of girls and 1/8 of boys participated, and you need the fraction of total students, you can't just add 1/5 + 1/8. The denominators represent different group sizes.
The trick is to pick concrete numbers that satisfy the given ratios, compute the participants, then find the fraction of the total. This is covered in the PYQ section below.
When all fractions have numerators just one less than their denominators (like 3/4, 6/7, 7/8, 9/11), compute the "gap" from 1 for each: 1/4, 1/7, 1/8, 1/11 respectively. The fraction with the smallest gap is the largest. You compare unit fractions (bigger denominator = smaller unit fraction) instead of doing LCM. Standard method (LCM of 8,7,4 = 56, convert 3 fractions): ~40 seconds. Complement method: ~10 seconds.
When adding or subtracting mixed fractions like 10(2/3) + 3(1/6) - 5(5/6) - 2(1/3), group the whole numbers separately from the fractions. Whole part: 10 + 3 - 5 - 2 = 6. Fraction part: 2/3 + 1/6 - 5/6 - 1/3 = 4/6 + 1/6 - 5/6 - 2/6 = -2/6 = -1/3. Final answer: 6 - 1/3 = 5(2/3). Converting everything to improper fractions first requires handling numbers like 32/3 and 35/6, slowing you down by 3-4 steps.
When the problem gives fractions of two sub-groups (boys and girls), pick concrete numbers equal to the LCM of the denominators or use the ratio directly. If 2/5 of boys and 3/7 of girls participate, and boys:girls = 5:7, let boys = 5, girls = 7. Participants = 2 + 3 = 5. Total = 12. Fraction = 5/12. This avoids setting up algebraic equations entirely. Standard algebraic approach: ~60 seconds. Number-picking: ~20 seconds.
To decide which of two fractions is larger, cross-multiply and compare. For a/b vs c/d: compute a×d and b×c. The fraction on the side of the larger product wins. Example: Is 70/80 > 84/108? Cross: 70×108 = 7560 vs 80×84 = 6720. Since 7560 > 6720, 70/80 > 84/108. No LCM needed. Standard method (find LCM of 80 and 108 = 2160): ~45 seconds. Cross-multiply: ~15 seconds.
Always reduce fractions to lowest terms before comparing or computing. 84/108 — divide both by 12: 7/9. 42/60 — divide both by 6: 7/10. Now comparing 7/9 vs 7/10 is trivial (same numerator, larger denominator = smaller fraction). This eliminates 2-3 steps of working with large numbers. Works whenever both numerator and denominator share a visible common factor (look for multiples of 2, 3, 5, 7 first).
In the exam hall, classify the question type in 3 seconds:
Comparison question (which is largest/smallest/greater than X): → Are all fractions close to 1? Use complement method (fastest). → Only two fractions to compare? Cross-multiply. → Three or more with no obvious pattern? Simplify first, then convert to decimals or find LCM.
Arithmetic question (compute a sum/difference of mixed fractions):
→ Separate whole numbers from fractions. Add/subtract whole parts. Add/subtract fractional parts using LCM. Combine.
→ If the fractional part subtraction gives a negative result (e.g., 1/6 - 5/6 = -4/6), borrow 1 from the whole number part.
Word problem with fraction-of-a-group: → Identify the sub-groups and their fractions. → Pick concrete numbers (use the denominator values or given ratio directly). → Count participants, count total, divide.
Always reduce your final answer to lowest terms. SSC MTS options are always simplified — an unsimplified answer won't match any option even if your arithmetic is correct.
Why this question: The most fundamental comparison question — tests whether you know a method beyond "convert everything to decimals."
Solving path: The complement method is cleanest here. Deficits from 1: 3/4 is missing 1/4, 6/7 is missing 1/7, 7/8 is missing 1/8. Since 1/8 < 1/7 < 1/4, the order from largest to smallest is 7/8 > 6/7 > 3/4. Match with option C.
Why this question: Classic trap — students instinctively add 1/5 + 1/8 and get 13/40, which is option A (a planted distractor).
Solving path: You cannot add 1/5 and 1/8 directly because they apply to different-sized groups. Use concrete numbers: imagine 5 girls and 8 boys (13 students total). Girls participating: 1. Boys participating: 1. Total participating: 2 out of 13. Fraction = 2/13. If you chose 13/40, you fell for the trap.
Why this question: Tests the separate-parts method on a longer expression with four mixed fractions.
Solving path: Whole parts: 10 + 3 - 5 - 2 = 6. Fraction parts: 2/3 + 1/6 - 5/6 - 1/3. Convert to sixths: 4/6 + 1/6 - 5/6 - 2/6 = (4+1-5-2)/6 = -2/6 = -1/3. Combine: 6 - 1/3 = 5(2/3). Match option A.
Why this question: Tests whether you can identify which option exceeds a target fraction — requires simplifying the target first.
Solving path: Simplify 84/108: divide both by 12 to get 7/9 ≈ 0.778. Now scan options. 42/60 = 7/10 = 0.70 — less. 14/25 = 0.56 — less. 21/40 = 0.525 — less. 70/80 = 7/8 = 0.875 — greater. Answer: option A. The simplification step is what makes this fast.
Why this question: Four fractions without the near-1 pattern, so you need a reliable decimal or LCM method.
Solving path: Convert each to decimal. 4/5 = 0.800. 7/9 = 0.777.... 9/11 = 0.818.... 11/13 = 0.846.... The largest is 11/13. Note the complement method also works: deficits are 1/5=0.2, 2/9≈0.222, 2/11≈0.182, 2/13≈0.154. Smallest deficit is 2/13, confirming 11/13 is the largest.
Why this question: Combines ratio and fraction-of-a-group — a slightly harder word problem that often trips up students who set up algebraic equations.
Solving path: Boys:Girls = 5:7. Let boys = 5, girls = 7 (total = 12). Boys participating: 2/5 × 5 = 2. Girls participating: 3/7 × 7 = 3. Total participating: 2 + 3 = 5. Fraction = 5/12. The concrete-number approach avoids any algebraic setup.
Adding fractions of different groups directly. 1/5 of girls + 1/8 of boys ≠ (1/5 + 1/8) of all students. The groups have different sizes. Always convert to actual counts or use concrete numbers first.
Comparing fractions without simplifying. Trying to find LCM of 84 and 108 wastes time. Reduce 84/108 = 7/9 in 5 seconds and the comparison becomes trivial.
Forgetting to borrow when subtracting mixed fractions. If the fractional part of the second number is larger than the first (e.g., 5(1/6) - 3(5/6)), you need to borrow 1 from the whole number before subtracting. 5(1/6) = 4(7/6), then 4(7/6) - 3(5/6) = 1(2/6) = 1(1/3).
Leaving an improper fraction as the final answer. If you get 45/12 as a result, convert it to 3(3/4) and simplify. SSC options are always in simplest mixed or proper form.
Applying the complement method to fractions that are not all close to 1. Complement method works beautifully for fractions like 3/4, 6/7, 7/8, but gives you the wrong intuition for fractions like 2/9 and 7/8 in the same list. Check that all fractions in your comparison are "near 1" before using it.
Confusing the cross-multiplication direction. When comparing a/b vs c/d, you compute a×d (top-left times bottom-right) against b×c (bottom-left times top-right). The larger product is on the same side as the larger fraction. Getting the diagonal direction wrong flips your answer.