A fraction represents a part of a whole. Write it as p/q where p is the numerator and q is the denominator. A decimal is simply a fraction whose denominator is a power of 10 — 0.7 is just 7/10, 0.37 is 37/100, and so on. They are two different notations for the same underlying quantity.
Here is the classification you need to hold in your head:
3/5. Its decimal value lies between 0 and 1.7/4. Its decimal value is ≥ 1.1¾. Convert to improper by multiplying the integer by the denominator and adding the numerator: 1¾ = (1×4 + 3)/4 = 7/4.0.125 = 125/1000.3/8 = 0.375. This happens when the denominator (in lowest terms) has only 2s and 5s as prime factors.1/3 = 0.333.... This happens when the denominator has prime factors other than 2 and 5.Think of the fraction bar as a division sign — 3/8 literally means 3 divided by 8. Every conversion question in AFCAT reduces to that one idea: long-divide the numerator by the denominator. The challenge is doing it quickly enough, and that is where the shortcuts in this page matter.
One analogy that works well: if a pizza is cut into 8 equal slices and you take 3, you hold 3/8 of the pizza. If someone asks "how much of the pizza is that as a decimal?", you divide the 3 slices by the 8-slice total: 3 ÷ 8 = 0.375, i.e., 37.5% of the pizza. The fraction, the decimal, and the percentage are all the same quantity in different dress.
Method 1 — Direct long division. Divide numerator by denominator. Place the decimal point when the numerator runs out of digits.
7 ÷ 20: 20 does not go into 7, so write 0., then divide 70 by 20 = 3 remainder 10, then divide 100 by 20 = 5. Result: 0.35.
Method 2 — Make denominator a power of 10. If the denominator factors into only 2s and 5s, multiply top and bottom to reach 10, 100, or 1000.
7/20: multiply both by 5 → 35/100 = 0.35. This is faster than long division when the denominator cooperates.
3/8: 8 = 2³. Multiply both by 125 (because 8 × 125 = 1000) → 375/1000 = 0.375.
Write the decimal digits as the numerator. The denominator is 10^(number of decimal places). Then simplify by dividing both by their HCF.
0.6 = 6/10. HCF(6, 10) = 2. Simplified: 3/5.
0.125 = 125/1000. HCF(125, 1000) = 125. Simplified: 1/8.
0.35 = 35/100. HCF(35, 100) = 5. Simplified: 7/20.
The cleanest approach in an exam hall: convert the fraction to decimal, then compare digit by digit from left to right.
Compare 0.7 and 5/8: 5/8 = 0.625. Now compare 0.700 and 0.625 — the first digit after the decimal point is already larger in 0.7 (7 > 6), so 0.7 wins.
Do not cross-multiply with a decimal — it is error-prone under time pressure. Convert the fraction, compare as decimals.
You will see questions like 1.5 - 3/4 or 0.4 + 1/5. The cleanest approach: convert everything to decimal, then operate.
1.5 - 3/4: convert 3/4 = 0.75. Then 1.5 - 0.75 = 0.75.
0.4 + 1/5: convert 1/5 = 0.2. Then 0.4 + 0.2 = 0.6.
Alternatively, convert everything to fractions and find a common denominator. Both methods work — choose decimal conversion when the fractions have denominators that are powers of 2 and 5 (they terminate cleanly), and choose fraction addition when the decimals are something like 0.333... (where keeping it as 1/3 is cleaner).
A fraction p/q is in its simplest form (lowest terms) when HCF(p, q) = 1. To simplify:
HCF(p, q) — by prime factorisation or Euclid's algorithm.p and q by that HCF.125/1000: factor 125 = 5³, factor 1000 = 2³ × 5³. HCF = 5³ = 125. Divide: 1/8.
35/100: factor 35 = 5 × 7, factor 100 = 2² × 5². HCF = 5. Divide: 7/20.
These come up directly in AFCAT-style questions. Knowing them saves 20–30 seconds per question:
| Fraction | Decimal |
|----------|---------|
| 1/2 | 0.5 |
| 1/4 | 0.25 |
| 3/4 | 0.75 |
| 1/5 | 0.2 |
| 2/5 | 0.4 |
| 3/5 | 0.6 |
| 4/5 | 0.8 |
| 1/8 | 0.125 |
| 3/8 | 0.375 |
| 5/8 | 0.625 |
| 7/8 | 0.875 |
| 1/20 | 0.05 |
| 7/20 | 0.35 |
If you walk into the exam with this table memorised cold, every question in this topic becomes a lookup, not a calculation.
When the denominator of a fraction has only 2s and 5s as prime factors, multiply top and bottom to reach the nearest power of 10. Denominator 4 → multiply by 25 to reach 100. Denominator 8 → multiply by 125 to reach 1000. Denominator 20 → multiply by 5 to reach 100.
Example: 7/20 × 5/5 = 35/100 = 0.35. No long division needed.
Standard long division path: 4 steps, ~25 seconds. Scaling path: 2 steps, ~8 seconds.
To convert a decimal to a fraction in one motion: count the digits after the decimal point (call it n), write the digits as the numerator, write 10^n as the denominator, simplify.
0.875 → 3 decimal places → 875/1000 → divide by HCF 125 → 7/8. Done.
Standard method (writing out 0.8 = 8/10, 0.07 = 7/100 and adding): 5 steps. This method: 3 steps.
The eighths family (1/8, 2/8, 3/8...) follows a clean pattern: each step adds 0.125. So 1/8 = 0.125, 2/8 = 0.25, 3/8 = 0.375, 4/8 = 0.5, 5/8 = 0.625, 6/8 = 0.75, 7/8 = 0.875. Memorise 0.125 as the base unit and count up by 0.125 for any eighth.
If you see 5/8 in a comparison question, count 5 steps of 0.125: 0.125 → 0.25 → 0.375 → 0.5 → 0.625. Takes about 5 seconds vs. long division (~20 seconds).
Never cross-multiply a decimal against a fraction — the arithmetic is messy. Convert the fraction to decimal first, then compare column by column left to right.
Example: compare 0.7 vs 5/8. Convert 5/8 = 0.625. Now line up: 0.700 vs 0.625. First column after decimal: 7 > 6. Stop. 0.7 wins.
Cross-multiply approach: 0.7 × 8 = 5.6 vs 5 × 1 = 5. Correct but requires fraction rearrangement. Decimal comparison: 3 seconds.
For expressions mixing decimals and fractions, identify which form is simpler for the specific numbers involved. If denominators are 4, 5, 8, 10, 20 — convert to decimal (they terminate cleanly). If decimals are recurring (like 0.333...) — keep as fraction.
0.75 + 1/8: both terminate → 0.75 + 0.125 = 0.875. One addition, done in ~5 seconds. Fraction route: 3/4 + 1/8 = 6/8 + 1/8 = 7/8 = 0.875 — also valid but requires an LCM step (~10 seconds more).
In the exam hall, run this decision tree the moment you read a decimals-and-fractions question:
Step 1 — Identify the task. Is it conversion (fraction → decimal or decimal → fraction), comparison, or arithmetic (add/subtract/multiply)?
Step 2 — Conversion question? Check if denominator has only 2s and 5s. If yes, use power-of-10 scaling (8 seconds). Otherwise, do short long division.
Step 3 — Comparison question? Convert the fraction to decimal using your memorised table or Step 2 method. Compare digit by digit left to right. Stop at the first differing digit.
Step 4 — Arithmetic question (mixed decimal + fraction)? Convert all terms to decimal if denominators are from {2, 4, 5, 8, 10, 20, 25}. Then operate. If any denominator produces a recurring decimal (3, 6, 7, 9...), convert to fraction instead and find LCM.
Step 5 — Simplification question? Find HCF of numerator and denominator, divide both. Cross-check: can you cancel a common factor by inspection before computing HCF formally?
Always verify your answer against the options. If your computed decimal does not match any option exactly, check your simplification step first — HCF error is the most common failure point.
Why this question: Tests whether you recognise a basic decimal-to-fraction conversion and can simplify correctly — the most common AFCAT question type in this topic.
Solving path: 0.6 = 6/10. Count common factors: 6 = 2 × 3, 10 = 2 × 5. HCF = 2. Divide: 3/5. Scan options — 3/5 is option B. Done in under 10 seconds.
Why this question: 0.125 is a classic benchmark. The question checks whether you know the eighths family or can apply power-of-10 denominator simplification.
Solving path: 0.125 = 125/1000. HCF(125, 1000): 125 = 5³, 1000 = 8 × 125. HCF = 125. Divide: 1/8. Or simply recall 1/8 = 0.125 from your table — 2 seconds.
Why this question: Mixed operation (decimal minus fraction) — tests whether you can unify the two forms before operating.
Solving path: Convert 3/4 to decimal. 3/4 × 25/25 = 75/100 = 0.75. Subtract: 1.5 - 0.75 = 0.75. Option A.
Why this question: Fraction-to-decimal conversion using scaling — the power-of-10 shortcut is directly applicable.
Solving path: 7/20. Multiply by 5/5: 35/100 = 0.35. Option A. No long division needed.
Why this question: Tests mixed addition and whether you handle the 1/5 = 0.2 benchmark cleanly.
Solving path: 1/5 = 0.2. Add: 0.4 + 0.2 = 0.6. Option A. Five seconds.
Why this question: Adds a three-decimal benchmark (1/8 = 0.125) to a two-decimal number. Tests precision with decimal place alignment.
Solving path: 1/8 = 0.125. Add: 0.750 + 0.125 = 0.875. Keep trailing zeros for alignment — this is where candidates make errors. Option A.
Why this question: Comparison of a decimal and a fraction — tests whether you convert first rather than trying to compare directly.
Solving path: Convert 5/8: count 5 steps of 0.125 → 0.625. Compare 0.700 vs 0.625. First digit after decimal: 7 > 6. So 0.7 is greater. Option A.
Why this question: Direct benchmark recall (3/8 = 0.375) versus options that are close but wrong (0.325, 0.385, 0.350). The question is designed to trap those who approximate rather than compute.
Solving path: 3/8. Multiply by 125/125: 375/1000 = 0.375. Or recall from the eighths table: 3 steps of 0.125 = 0.125 + 0.125 + 0.125 = 0.375. Option A.
Forgetting to simplify after conversion. Writing 6/10 as the final answer instead of 3/5 is wrong even if the arithmetic was correct. Always check HCF(numerator, denominator) = 1 before selecting an option.
Misaligning decimal places during addition. When adding 0.75 + 0.125, writing 0.875 is correct, but many candidates add 0.75 + 0.125 as 0.875 after writing 75 + 125 = 875 — which accidentally works here, but the method fails for 0.7 + 0.125. Always align on the decimal point, not the last digit.
Comparing fractions and decimals without converting. Looking at 0.7 and 5/8 and guessing 5/8 is larger because "5 is a bigger number than 7" is a classic trap. Convert first, always.
Wrong power-of-10 multiplier. For 3/8, some candidates multiply by 12 (trying to reach a round number) instead of 125 (to reach 1000). The rule is strict: you must reach exactly 10, 100, 1000, etc. Compute 10^n ÷ denominator to find the right multiplier.
Treating 0.125 as 1/125 instead of 1/8. The number of decimal places determines the denominator: 3 decimal places → denominator is 1000, not 125. 0.125 = 125/1000, not 1/125.
Mixed fraction conversion errors. When converting 1¾ to improper: 1¾ = (1×4 + 3)/4 = 7/4, not (1+3)/4 = 4/4. You multiply the integer by the denominator first, then add the numerator. This is the step that gets rushed under time pressure.