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REET Level 1 & 2 · Maths

Fractions and Decimals: REET Maths Notes, Shortcuts and Solved Questions

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This page brings together types of fractions, comparing and ordering fractions, operations on fractions, terminating and recurring decimals, and how to teach all of this in class. REET 2024 had 8 questions from this one topic, which makes it one of the highest-scoring areas in the Maths section.

  • 8 questions in REET 2024
  • Class 3–8 level
  • 9 min read
Verified 17 September 2026 · Source: NCERT Textbooks (Maths, Class 6–9)

How REET asks this topic

Across REET 2024's three shifts, this topic carried 8 questions — 4 in Level 1 (Shift 1) and 4 across Level 2 (Shifts 2 and 3). The difficulty sits around Class 6–8, but the real test is speed: comparing fractions and arranging them in ascending order (3 questions), converting recurring decimals to p/q form (2 questions), a telescoping sum of fractions (1/20 + 1/30 + …), a 'remaining part' word problem (a pole/pillar), and converting a unit into its decimal part (seconds/hour). Expect fewer long calculations and more shortcuts plus conceptual clarity.

Types of Fractions

TypeHow to identify itExample
Proper fractionNumerator < denominator; value less than 13/7, 5/9
Improper fractionNumerator ≥ denominator; value 1 or more14/11, 9/4
Mixed fractionWhole number + proper fraction1 3/11 = 14/11
Like fractionsAll have the same denominator2/9, 5/9, 7/9
Unlike fractionsDifferent denominators1/4, 2/9
Unit fractionNumerator is 11/8, 1/20
Equivalent fractionsNumerator and denominator multiplied/divided by the same number3/5 = 6/10 = 9/15
Lowest termsHCF of numerator and denominator = 127/99 → 3/11

Improper ↔ mixed fraction: 14 ÷ 11 = 1 remainder 3, so 14/11 = 1 3/11.[Source 1]

Comparing and ordering fractions — REET shortcuts

Standard method: take the LCM of all the denominators, convert to a common denominator, and then compare numerators. For just two fractions a/b and c/d, cross-multiplication is fastest — if a×d > b×c, then a/b is larger. Example: for 3/7 and 5/11, 3×11 = 33 and 7×5 = 35, so 5/11 > 3/7.

Rule 1 — if the numerators are equal, the fraction with the smaller denominator is larger (1/4 > 1/5). Rule 2 — if the denominators are equal, the fraction with the larger numerator is larger. Rule 3 — if the gap between denominator and numerator is the same across all the fractions (e.g. 13/15, 15/17, 17/19, 19/21 all have a gap of 2), then among proper fractions the one with the larger numerator-denominator pair is the largest, because the 'missing' part (like 2/21) is the smallest there. For improper fractions (like 3/2, 5/4) this rule flips.

Converting to decimals to compare: 2/9 ≈ 0.222, 1/4 = 0.25, 3/7 ≈ 0.429, 5/11 ≈ 0.455 — so the ascending order is 2/9, 1/4, 3/7, 5/11. Often just checking the first and last fraction among the options is enough to spot the answer.

Decimals: terminating, non-terminating recurring, and converting to p/q

CaseRuleExample
Terminating decimalIn lowest terms, the denominator's only prime factors are 2 and/or 53/8 = 0.375, 1/20 = 0.05
Non-terminating recurringThe denominator has a prime factor other than 2 or 51/3 = 0.3̅, 3/11 = 0.2̅7̅
Purely recurring → p/qDivide the repeating digits by as many 9s0.8̅ = 8/9, 0.2̅7̅ = 27/99 = 3/11
Mixed recurring → p/q(whole number − non-repeating part) ÷ (as many 9s as repeating digits + as many 0s as non-repeating digits)0.12̅ = (12 − 1)/90 = 11/90
With an integer partAdd the whole-number part separately1.2̅7̅ = 1 + 3/11 = 14/11
Terminating decimal → p/qRemove the decimal point and divide by 10, 100, 10000.125 = 125/1000 = 1/8

Worth memorising: 1/8 = 0.125, 1/16 = 0.0625, 1/7 = 0.1̅4̅2̅8̅5̅7̅, 1/9 = 0.1̅, 1/11 = 0.0̅9̅.[Source 1]

Operations, telescoping sums, and word problems

Addition/subtraction: take the LCM of the denominators to get a common denominator. Multiplication: numerator × numerator over denominator × denominator (cancel first where possible). Division: multiply by the reciprocal of the second fraction — 3/4 ÷ 3/8 = 3/4 × 8/3 = 2. Finding a fraction of a number is just multiplication: 3/5 of 60 = 36.

Telescoping sums: 1/(n(n+1)) = 1/n − 1/(n+1). So 1/20 + 1/30 + 1/42 + 1/56 = (1/4 − 1/5) + (1/5 − 1/6) + (1/6 − 1/7) + (1/7 − 1/8) = 1/4 − 1/8 = 1/8 = 0.125. Spot this pattern whenever the denominators are the product of two consecutive numbers (20 = 4×5, 30 = 5×6).

'Remaining part' word problems: add up all the given fractional parts, subtract from 1; the leftover fraction equals the given quantity, and you scale up to find the whole. Pole example: 3/10 + 3/5 = 9/10, remaining 1/10 = 2 metres, so the whole pole = 20 metres. Salary, tank-filling and distance problems follow the same pattern.

Decimal operations: line up the decimal points for addition/subtraction. For multiplication, add up the total decimal places in both numbers (0.2 × 0.03 = 0.006). Multiplying by 10, 100, 1000 shifts the decimal point right; dividing shifts it left. Unit conversion: 1 second = 1/3600 hour ≈ 0.000278 hour; 250 grams = 0.25 kg.

Quick facts

Between two fractions
There are infinitely many fractions between any two given fractions; (a/b + c/d) ÷ 2 always lies between them
LCM of fractions
LCM of numerators ÷ HCF of denominators
HCF of fractions
HCF of numerators ÷ LCM of denominators
Value of 0.9̅
9/9 = 1
Reciprocal
The reciprocal of a/b is b/a; the reciprocal of 0 is not defined
2(0.8̅)
2 × 8/9 = 16/9 = 1.7̅

How to teach children

Activity (Class 3–5)

Fold a strip of paper or a picture of a roti into equal parts to show 1/2, 1/4, 3/4. Stress 'equal parts' — unequal pieces are not fractions.

Teaching aids

Fraction strips, fraction circles, a number line, and for decimals a 10×10 grid (100 boxes = 1; 1 box = 0.01), plus familiar contexts like rupees-and-paise or metres-and-centimetres.

Common misconception 1

'1/8 is bigger than 1/4 because 8 > 4' — children apply whole-number rules to fractions. Fix: cut two identical strips into 4 and 8 parts and let them compare directly.

Common misconception 2

'0.25 > 0.3 because 25 > 3', and 1/2 + 1/3 = 2/5 (adding numerators and denominators separately). Fix: shade 0.25 and 0.3 on a 10×10 grid to compare, and use fraction strips to show that 1/2 + 1/3 cannot be smaller than 1/2 (i.e. it cannot be 2/5).

Assessment idea

Ask everyday questions like 'who got the bigger share?' and have children explain their reasoning, not just state the answer. Asking them to mark 3/4 and 0.75 at the same point on a number line is a great diagnostic question.

Memory trick / exam tip

When comparing fractions, check the options first: if 'denominator − numerator' is the same for every proper fraction, the one with the largest digits is the largest — no calculation needed. And for recurring decimals, the denominator gets as many 9s as there are recurring digits, and as many 0s as there are non-recurring digits.

Questions asked in REET 2024 (with solutions)

Question 93

In the form of P/Q, 1.2̅7̅ (27 recurring) will be equal to

  1. A17/11
  2. B11/14
  3. C14/11
  4. D11/12

Explanation

1.2̅7̅ में 27 आवर्ती है, अतः 0.2̅7̅ = 27/99 = 3/11। इसलिए 1.2̅7̅ = 1 + 3/11 = 14/11। विकल्प (B) 11/14 इसका व्युत्क्रम है, जो 1 से छोटा है, अतः गलत है।

Question 105

The value of 1/20 + 1/30 + 1/42 + 1/56 is

  1. A0.152
  2. B1.125
  3. C0.125
  4. D0.120

Explanation

20 = 4×5, 30 = 5×6, 42 = 6×7, 56 = 7×8 तथा 1/(n(n+1)) = 1/n − 1/(n+1)। अतः योग = (1/4 − 1/5) + (1/5 − 1/6) + (1/6 − 1/7) + (1/7 − 1/8) = 1/4 − 1/8 = 1/8 = 0.125। विकल्प (B) 1.125 में दशमलव गलत है — चार छोटी भिन्नों का योग 1 से अधिक नहीं हो सकता।

Question 113

Which fraction among 13/15, 15/17, 17/19 and 19/21 is the largest fraction ?

  1. A19/21
  2. B13/15
  3. C17/19
  4. D15/17

Explanation

प्रत्येक भिन्न में हर, अंश से 2 अधिक है: 13/15 = 1 − 2/15, 15/17 = 1 − 2/17, 17/19 = 1 − 2/19, 19/21 = 1 − 2/21। जिसका हर सबसे बड़ा है उसमें 1 से घटने वाला भाग (2/21) सबसे छोटा है, अतः 19/21 सबसे बड़ी भिन्न है।

Question 118

If 3/10 th part of a pole is in mud, 3/5 th part is in water and the remaining 2 meters is above the water surface, then the length of a pole will be

  1. A10 मीटर
  2. B15 मीटर
  3. C30 मीटर
  4. D20 मीटर

Explanation

कीचड़ और पानी में कुल भाग = 3/10 + 3/5 = 3/10 + 6/10 = 9/10। शेष भाग = 1 − 9/10 = 1/10, जो 2 मीटर है। अतः खम्भे की लम्बाई = 2 × 10 = 20 मीटर।

Question 102

How many decimal parts of an hour is one second ?

  1. A0.002561
  2. B0.000277
  3. C0.000123
  4. D0.002077

Explanation

1 घंटा = 60 × 60 = 3600 सेकण्ड। अतः 1 सेकण्ड = 1/3600 घंटा = 0.000277... घंटा (लगभग)। इसलिए विकल्प (B) 0.000277 सही है।

Question 113

The sum of the largest and smallest fractions among the fractions 5/9, 3/7, 11/13, 6/11 and 7/8 is :

  1. A1 5/8
  2. B1 17/53
  3. C1 17/56
  4. D1 13/99

Explanation

दशमलव में बदलने पर: 5/9 ≈ 0.556, 3/7 ≈ 0.429, 11/13 ≈ 0.846, 6/11 ≈ 0.545, 7/8 = 0.875। सबसे बड़ा भिन्न 7/8 और सबसे छोटा 3/7 है। योग = 7/8 + 3/7 = (49 + 24)/56 = 73/56 = 1 17/56। इसलिए (C) सही है।

Question 103

Arrangement in ascending order of 3/7, 2/9, 5/11, 1/4 is

  1. A5/11, 3/7, 1/4, 2/9
  2. B2/9, 1/4, 3/7, 5/11
  3. C3/7, 5/11, 1/4, 2/9
  4. D2/9, 1/4, 5/11, 3/7

Explanation

दशमलव रूप में: 2/9 ≈ 0.222, 1/4 = 0.25, 3/7 ≈ 0.429, 5/11 ≈ 0.455। अतः आरोही (छोटे से बड़े) क्रम है 2/9, 1/4, 3/7, 5/11। विकल्प (D) में 5/11 और 3/7 का क्रम उलटा है (3/7 < 5/11 क्योंकि 3×11 = 33 < 5×7 = 35)।

Question 119

Solution of 2(0.8̅) is

  1. A0.1̅6̅
  2. B0.1̅7̅
  3. C1.6̅
  4. D1.7̅

Explanation

0.8̅ (8 आवर्ती) = 8/9। अतः 2 × 8/9 = 16/9 = 1.777… = 1.7̅ (7 आवर्ती)। 2 × 0.8 = 1.6 सीधे गुणा करने की गलती है, क्योंकि आवर्ती दशमलव में 0.888… × 2 = 1.777… होता है।

Practice questions

Question 1

Which is the smallest among 7/12, 5/8, 2/3 and 3/4?

  1. A7/12
  2. B5/8
  3. C2/3
  4. D3/4

Explanation

Taking the LCM as 24, the fractions become 14/24, 15/24, 16/24 and 18/24. The smallest numerator is 14, so 7/12 is the smallest fraction.

Question 2

0.3̅6̅ (36 recurring) in simplest p/q form is

  1. A36/100
  2. B4/11
  3. C9/25
  4. D6/17

Explanation

For a purely recurring decimal, divide the repeating digits by as many 9s: 36/99 = 4/11. Note that 36/100 = 0.36 is a terminating decimal, not a recurring one.

Question 3

0.23̅ (only 3 recurring) in p/q form is

  1. A23/99
  2. B23/90
  3. C7/30
  4. D2/9

Explanation

This is a mixed recurring decimal: (23 − 2)/90 = 21/90 = 7/30. Getting 23/90 means forgetting to subtract the non-repeating digit 2, and 23/99 is actually the value of the purely recurring 0.2̅3̅.

Question 4

Which fraction has a terminating decimal expansion?

  1. A7/12
  2. B11/30
  3. C9/40
  4. D5/21

Explanation

40 = 2³ × 5, which has only 2 and 5 as prime factors, so 9/40 = 0.225 terminates. 12, 30 and 21 each have a factor of 3 or 7, so their decimal expansions recur.

Question 5

The value of 1/6 + 1/12 + 1/20 + 1/30 is

  1. A1/3
  2. B2/5
  3. C1/2
  4. D4/15

Explanation

6 = 2×3, 12 = 3×4, 20 = 4×5, 30 = 5×6; so the sum telescopes: (1/2 − 1/3) + (1/3 − 1/4) + (1/4 − 1/5) + (1/5 − 1/6) = 1/2 − 1/6 = 1/3.

Question 6

2/5 of a tank was emptied in the morning and 1/3 in the afternoon. The remaining water is 120 litres. What is the full capacity of the tank?

  1. A360 लीटर
  2. B540 लीटर
  3. C400 लीटर
  4. D450 लीटर

Explanation

Emptied fraction = 2/5 + 1/3 = 11/15, so the remaining fraction = 4/15. Since 4/15 of the tank equals 120 litres, the full tank = 120 × 15/4 = 450 litres.

Question 7

15 minutes is what decimal part of 2 hours?

  1. A0.25
  2. B0.075
  3. C0.15
  4. D0.125

Explanation

2 hours = 120 minutes, so 15/120 = 1/8 = 0.125. 0.25 would be the decimal fraction of one hour, not two.

Question 8

The value of 3/4 ÷ 9/16 is

  1. A27/64
  2. B4/3
  3. C3/4
  4. D2/3

Explanation

Division means multiplying by the reciprocal of the second fraction: 3/4 × 16/9 = 48/36 = 4/3. 27/64 would come from multiplying instead of dividing, and 3/4 comes from mistakenly taking the reciprocal of the first fraction instead of the second.

Question 9

A Class 5 student writes 0.35 > 0.5 because 35 > 5. Which is the most suitable way to remove this misconception?

  1. Aदशमलव तुलना का नियम पाँच बार लिखवाना
  2. B10×10 ग्रिड पर दोनों संख्याएँ रंगवाकर तुलना कराना
  3. Cउत्तर गलत बताकर अगला प्रश्न देना
  4. Dकेवल अधिक अभ्यास प्रश्न देना

Explanation

Shading 0.35 as 35 boxes and 0.5 as 50 boxes on the grid lets the child see for themselves that 0.5 is larger. Concrete representation builds the concept, while rote correction or extra practice alone does not address the root of the misconception.

Question 10

Which is the most suitable starting point to introduce the concept of fraction in Class 3?

  1. Aभिन्न की परिभाषा याद करवाना
  2. Bभिन्नों के जोड़ का सूत्र बताना
  3. Cरोटी या कागज़ को बराबर भागों में बाँटने की गतिविधि
  4. Dसंख्या रेखा पर 7/9 अंकित करवाना

Explanation

At the primary stage, maths is taught by moving from the concrete to the abstract. A familiar activity like dividing a roti or paper into equal parts builds the meaning of 'half' and 'quarter'; definitions, formulas and the number line come at a later stage.

Frequently asked questions

How many questions come from Fractions and Decimals in REET?+

REET 2024 had 8 questions from this topic in total — 4 in one Level 1 shift and 4 across two Level 2 shifts. The exact number can vary by shift, but it's a regular part of the Maths section every year.

How do you convert a recurring decimal to a fraction (p/q)?+

For a purely recurring decimal, divide the repeating digits by as many 9s (0.2̅7̅ = 27/99). For a mixed recurring decimal, subtract the non-repeating part from the whole number, and divide by as many 9s as there are repeating digits followed by as many 0s as there are non-repeating digits (0.12̅ = 11/90).

Which fractions give a terminating decimal?+

In its lowest terms, if a fraction's denominator has only 2 and/or 5 as prime factors, its decimal expansion terminates — for example 7/20 = 0.35.

How do you arrange fractions in ascending order?+

The fastest way is to convert the fractions to approximate decimals, or use cross-multiplication to compare them pairwise. Ascending means smallest to largest, descending means largest to smallest.

Read next

Whole Numbers and Integers

Previous topic: the foundations of the number system

LCM and HCF

Next topic: also needed for LCM-HCF of fractions

REET Syllabus

The complete official syllabus for Maths and Pedagogy

REET 2024 Solved Papers

Questions from every shift with official answer key and explanations

Official Sources

  1. NCERT Textbooks (Maths, Class 6–9)
  2. NCERT — Position Paper: National Focus Group on Teaching of Mathematics (NCF 2005)
  3. BSER — REET 2024 official question papers and final answer keys (original PDFs; no longer hosted on the board portal)

Change Log

  • 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked