A fraction is a way of expressing a part of a whole. Write it as numerator / denominator — the numerator counts how many parts you have, the denominator tells you how many equal parts the whole is divided into. So 3/8 means the whole has been cut into 8 equal pieces and you are holding 3 of them.
Here is the analogy that works in every Class 1-5 classroom: think of a pizza cut into equal slices. If you cut it into 4 slices and eat 1, you ate 1/4 (एक चौथाई). If your friend cuts the same-sized pizza into 8 slices and eats 2, they also ate 2/8 — the same amount. That is the core idea of equivalent fractions: different-looking numbers, same actual value.
Decimals are just fractions in disguise, using powers of 10 as the denominator. The decimal 0.75 is 75/100, which simplifies to 3/4. Every decimal you see in CTET can be converted to a fraction by reading the place value: one digit after the decimal point means tenths, two digits mean hundredths, three mean thousandths.
Why does CTET Paper I test this topic so heavily? Because you are not just solving for a number — you are expected to understand why fractions work the way they do, so you can teach them. The questions you will face mix straightforward computation with word problems designed to trip you on the two-step logic (Monday's sale affects Tuesday's base, not the original stock). That layered reasoning is where marks are lost.
The difficulty here is not the arithmetic itself — it is keeping track of what the denominator refers to at each step of a word problem. Lock that habit in and most CTET fraction questions become straightforward.
5/8.9/4.2 1/4. To convert: 2 1/4 = (2×4+1)/4 = 9/4.3/4 = 6/8 = 15/20.3/7 + 2/7 = 5/7.Addition and Subtraction — always work with a common denominator.
2/3 − 1/4: LCM of 3 and 4 is 12. Convert: 8/12 − 3/12 = 5/12.
The shortcut for two unlike fractions a/b − c/d: result = (ad − bc) / bd. If the resulting fraction is messy, simplify by HCF. But be careful — this cross-multiply shortcut only works cleanly when b and d are coprime. When they share factors, the LCM route is safer.
Multiplication — straightforward: multiply numerators together, multiply denominators together, then simplify.
14/5 × 5/3 = 70/15 = 14/3.
Look for cross-cancellation before multiplying — it saves simplification steps at the end. In 14/5 × 5/3, the 5 in the numerator and the 5 in the denominator cancel immediately, giving 14/3 in one step.
Division — flip the second fraction (take its reciprocal) and multiply.
(5/9) ÷ (10/27) = (5/9) × (27/10) = 135/90 = 3/2.
Again, cross-cancel first: 5 and 10 share factor 5 (giving 1 and 2), and 27 and 9 share factor 9 (giving 3 and 1). So (1/1) × (3/2) = 3/2 — done in a single glance.
Multiplication: multiply as integers, then count total decimal places across both numbers and insert the point.
4.05 × 1.5: treat as 405 × 15 = 6075, total decimal places = 2+1 = 3, so answer = 6.075.
Division: eliminate decimals by multiplying numerator and denominator by the appropriate power of 10.
2.025 ÷ 0.75: multiply both by 100 → 202.5 ÷ 75 = 2.7. Or multiply both by 1000 → 2025 ÷ 750 = 2.7.
3/4 = 75/100 = 0.75.35 ÷ 96 = 0.36458... → rounded to 3 places = 0.365.Memorise these benchmarks — they appear constantly:
| Fraction | Decimal | |----------|---------| | 1/2 | 0.5 | | 1/4 | 0.25 | | 3/4 | 0.75 | | 1/3 | 0.333... | | 2/3 | 0.667... | | 1/5 | 0.2 | | 1/8 | 0.125 | | 3/8 | 0.375 |
Convert mixed numbers to improper fractions immediately before doing any arithmetic. Students who leave 2 4/5 as a mixed number and try to multiply it get confused. The moment you see 2 4/5 × 1 2/3 ÷ 2 1/3, your first move is: 14/5 × 5/3 ÷ 7/3. Then cross-cancel, multiply, done.
The single most common CTET fraction mistake is applying Tuesday's fraction to the original stock rather than the remaining stock. When a problem says "sold 5/12 of the remaining stock on Tuesday", the base has changed. Monday's remainder becomes the new whole for Tuesday's calculation. Always label what "1 whole" means at each step.
When multiplying fractions, scan diagonally for common factors between any numerator and any denominator before multiplying. Example: (14/5) × (5/3) — the 5s cancel instantly → 14/3. Standard method: multiply to get 70/15, then find HCF(70,15) = 5, divide → 14/3. That is 3 steps vs 1. In a chain of 3 fractions, cross-cancellation typically saves 4-5 arithmetic steps.
Division (a/b) ÷ (c/d) = (a×d) / (b×c). Before multiplying, look for diagonal cancellations between a and c, and between b and d. For (5/9) ÷ (10/27): diagonal pairs are 5 vs 10 (cancel to 1 and 2) and 27 vs 9 (cancel to 3 and 1). Result reads off as (1×3)/(1×2) = 3/2. Standard path: 135/90 → HCF = 45 → 3/2. Cross-cancel: 2 steps vs 4 steps.
When you see "A exceeds B by N", write (larger fraction − smaller fraction) × x = N, take LCM of the denominators in a single step, solve for x. For 2x/3 − x/4 = 55: LCM(3,4) = 12, so (8x − 3x)/12 = 55 → 5x = 660 → x = 132. Do not convert to decimals — fraction form gives an exact integer answer faster. Decimal route introduces rounding errors and takes ~20s extra.
For "tank is A full, after adding V litres it is B full" problems: capacity = V ÷ (B − A). For the 3/5 → 9/10 problem: 9/10 − 3/5 = 9/10 − 6/10 = 3/10. So 3/10 × capacity = 18 → capacity = 60. This is a direct one-line formula. Setting up algebra from scratch takes ~40s; this pattern recognises the structure in ~10s.
For 4.05 × 1.5: count decimal places (2 + 1 = 3). Multiply integers: 405 × 15 = 6075. Place 3 decimal places from the right: 6.075. No long multiplication with decimals, no alignment anxiety. Compare to the standard column method with decimal alignment: ~50s vs ~15s for two-decimal × one-decimal products.
When a CTET fraction/decimal problem lands in front of you, run this decision tree:
Step 1 — Identify the operation. Is it a computation (add/subtract/multiply/divide) or a word problem? Word problems need an extra "label the whole" step first.
Step 2 — For word problems, underline what "1 whole" means at each stage. If the problem has two time-periods or two people, redraw the base at each stage before writing any equation.
Step 3 — For mixed numbers, convert to improper fractions immediately. No exceptions.
Step 4 — Before multiplying or dividing fractions, look for cross-cancellation diagonally. Do it before computing, not after.
Step 5 — For decimal operations, convert to integers (multiply by power of 10), operate, then reinsert the decimal point by counting places.
Step 6 — Sanity check: if the answer is supposed to be a fraction of the original quantity, it must be less than 1. If it is supposed to be a capacity or a number, it must be a positive integer or simple decimal. Odd-looking answers signal a base-change error at Step 2.
Total time target per question: 60-90 seconds.
Why this question: Tests whether you correctly rebase the denominator after Monday's sale — the exact trap described in the Deep Dive.
Solving path: Identify the two fractional states of the tank (3/5 and 9/10). The difference 9/10 − 3/5 = 3/10 of the total capacity equals 18 litres. So total capacity = 18 ÷ (3/10) = 18 × (10/3) = 60 litres. One subtraction, one division — 30 seconds.
Why this question: A clean "product known, one factor known, find the other" problem. Tests the division-of-fractions rule and cross-cancellation speed.
Solving path: Other fraction = (5/9) ÷ (10/27). Flip and multiply: (5/9) × (27/10). Cross-cancel: 5 and 10 → 1 and 2; 27 and 9 → 3 and 1. Answer = 3/2. No simplification step needed at the end because you cancelled first.
Why this question: Tests the "exceeds by" equation setup — extremely common in CTET. Students who set it up as 2x/3 + x/4 = 55 (adding instead of subtracting) lose the mark.
Solving path: 2x/3 − x/4 = 55. LCM(3,4) = 12. (8x − 3x)/12 = 55. 5x = 660. x = 132. Verify: 2(132)/3 = 88, 132/4 = 33, 88 − 33 = 55. Correct.
Why this question: Multi-step decimal arithmetic with BODMAS. The trap is doing subtraction before division.
Solving path: BODMAS — multiplication and division before subtraction. 4.05 × 1.5: multiply integers 405 × 15 = 6075, three decimal places → 6.075. 2.025 ÷ 0.75: multiply both by 1000 → 2025 ÷ 750 = 2.7. Final: 6.075 − 2.7 = 3.375.
Why this question: Mixed number chain — tests whether you can convert, cross-cancel, and reduce in one smooth sequence without losing track.
Solving path: Convert immediately: 2 4/5 = 14/5, 1 2/3 = 5/3, 2 1/3 = 7/3. Expression: (14/5 × 5/3) ÷ 7/3. Inner product: 5 and 5 cancel → 14/3. Division: (14/3) ÷ (7/3) = (14/3) × (3/7). Cross-cancel 14 and 7 → 2 and 1; 3 and 3 → 1 and 1. Answer = 2.
Applying Tuesday's fraction to the original stock: the phrase "of the remaining" changes the base. Always identify what 1 whole is at that step, not what it was at the start.
Adding fractions without finding LCM: 1/3 + 1/4 ≠ 2/7. The denominators cannot simply be added. Find LCM(3,4) = 12, then 4/12 + 3/12 = 7/12.
Forgetting to convert mixed numbers before multiplying: 2 1/2 × 1 1/3 is NOT 2 × 1 + 1/2 × 1/3. Convert first: 5/2 × 4/3 = 20/6 = 10/3.
BODMAS errors with decimals: in 4.05 × 1.5 − 2.025 ÷ 0.75, division and multiplication must be resolved before the subtraction. Students who subtract first get a wrong intermediate value and then multiply it.
Rounding too early: in multi-step problems, keep fractions exact throughout and only convert to decimal (or round) in the very last step. Rounding 2/3 to 0.67 early propagates error.
Reciprocal error in division: (5/9) ÷ (10/27) — you flip the second fraction, not the first. A common slip is computing (9/5) × (10/27), which gives 90/135 = 2/3 — wrong answer, wrong instinct.