Integers, Fractions and Decimals (P2)

Fractions — Types, Comparison and Operations

A fraction p/q represents a part of a whole, where p is the numerator (number of parts taken) and q is the denominator (total equal parts). At the upper-primary level, children work with all three types: proper fractions (p < q), improper fractions (p ≥ q), and mixed numbers (whole part + proper fraction).

Converting between forms: Mixed number a(b/c) = (ac+b)/c. Operations on mixed numbers are always cleaner after converting to improper fractions first.

Comparing fractions: The most reliable method for any comparison is a common denominator. Find LCM of all denominators, convert each fraction, then compare numerators. Cross-multiplication works for pairs: p/q > r/s iff ps > qr.

Comparing several fractions — worked example: Compare 6/7, 2(3/8), 3/14, 2(11/14). Convert mixed numbers: 2(3/8) = 19/8; 2(11/14) = 39/14. Now compare 6/7, 19/8, 3/14, 39/14. Common denominator = 56. Values: 48/56, 133/56, 12/56, 156/56. Largest: 156/56 = 39/14; smallest: 12/56 = 3/14. Sum = 39/14 + 3/14 = 42/14 = 3.

Operations on fractions:

• Addition/subtraction: Common denominator required; then add/subtract numerators only.
• Multiplication: Multiply numerator × numerator, denominator × denominator. Simplify first (cancel common factors across numerator and denominator).
• Division: Multiply by the reciprocal of the divisor (KFC: Keep, Flip, Change).

Decimals and their connection to fractions: Every terminating decimal is a rational number. 0.75 = 75/100 = 3/4. Every recurring decimal is rational: 0.333… = 1/3. Non-recurring non-terminating decimals are irrational. Decimal arithmetic follows the same rules as fraction arithmetic, with the denominator being a power of ten.

Percentage — Conversions and Applications

A percentage is a fraction with denominator 100. 'Per cent' means 'per hundred.' The three most important skills for CTET Paper 2 are: converting between fraction, decimal, and percentage; finding percentage of a quantity; and finding the original quantity given a percentage result.

Key conversions:

• Fraction to %: multiply by 100. (3/4 × 100 = 75%)
• % to fraction: divide by 100. (75% = 75/100 = 3/4)
• % to decimal: divide by 100. (75% = 0.75)
• Decimal to %: multiply by 100. (0.75 = 75%)

Benchmark percentages to memorise: 1/8 = 12.5%, 1/6 ≈ 16.67%, 1/5 = 20%, 1/4 = 25%, 1/3 ≈ 33.33%, 3/8 = 37.5%, 1/2 = 50%, 2/3 ≈ 66.67%, 3/4 = 75%.

Finding percentage of a quantity: x% of N = (x/100) × N. Example: 15% of 240 = 36.

Reverse percentage (finding original): If after an increase of x%, the new value is N, original = N × 100/(100+x). If after a decrease of x%, new value is N, original = N × 100/(100−x).

Percentage increase/decrease: % change = (change/original) × 100. Note: a 25% increase followed by a 20% decrease does NOT return to the original — the base changes at each step.

Profit and Loss — Formulas and Market Price

Profit and loss problems are real-world applications of percentage and are tested heavily in CTET Paper 2.

Core formulas:

• Profit = Selling Price (SP) − Cost Price (CP) [when SP > CP]
• Loss = CP − SP [when CP > SP]
• Profit% = (Profit/CP) × 100
• Loss% = (Loss/CP) × 100
• SP = CP × (100 + Profit%)/100
• SP = CP × (100 − Loss%)/100
• CP = SP × 100/(100 + Profit%)
• CP = SP × 100/(100 − Loss%)

Marked Price (MP) and Discount: MP (also called List Price or Market Price) is the price before any discount. Discount = MP − SP. Discount% = (Discount/MP) × 100. SP after discount = MP × (100 − Discount%)/100.

Profit despite discount: It is possible to give a discount and still make a profit if the article was marked above CP. Example: 12 fans bought at ₹24,000 (CP = ₹2000 each). Mark-up: 25% above CP → MP = ₹2500. Discount: 12% on MP → SP = ₹2500 × 88/100 = ₹2200. Profit per fan = ₹200. Total profit on 12 fans = ₹2400.

Selling some at profit, some at loss: Calculate total cost, total revenue, and compare.

Important trap: Profit% and loss% are always calculated on CP, not on SP. Many candidates confuse this and lose marks.

Simple Interest and Compound Interest

Interest calculations are tested in almost every CTET Paper 2 cycle. The key is knowing when to apply Simple Interest (SI) and when Compound Interest (CI) is intended, and being able to distinguish the two from a word problem.

Simple Interest (SI):
SI = P × R × T / 100
Amount = P + SI = P(1 + RT/100)

Where P = Principal, R = Rate per annum (%), T = Time in years.

Finding principal from compound interest: If a sum invested at 10% p.a. compounded annually becomes ₹3267 in 2 years:
A = P(1 + 10/100)² = P(1.1)² = 1.21P = 3267
P = 3267/1.21 = ₹2700.

Compound Interest (CI):
A = P(1 + R/100)^T
CI = A − P = P[(1 + R/100)^T − 1]

Half-yearly compounding: Rate becomes R/2 per half-year; time becomes 2T half-years.
A = P(1 + R/200)^(2T)

Finding rate from SI: If ₹12,000 amounts to ₹15,972 in 3 years under SI:
SI = 15972 − 12000 = 3972
R = (SI × 100)/(P × T) = (3972 × 100)/(12000 × 3) = 11.03% ≈ 11%.

Difference between SI and CI: For the same principal, rate, and time, CI > SI (because CI earns interest on previously earned interest). The difference for 2 years = P(R/100)². This is a frequently tested formula.

Integers — Operations, Properties and Misconceptions

Integers extend the number line in both directions: …, −3, −2, −1, 0, 1, 2, 3, … The introduction of negative numbers at Class 6 is one of the major conceptual shifts in school mathematics and one of the richest sources of CTET questions on both content and pedagogy.

Addition and subtraction on the integer number line: Adding a positive integer means moving right; adding a negative integer means moving left. Subtracting a positive means moving left; subtracting a negative means moving right (two negatives 'cancel'). 5 − (−3) = 5 + 3 = 8.

Multiplication and division rules:

• Positive × positive = positive
• Positive × negative = negative
• Negative × positive = negative
• Negative × negative = positive

The 'negative × negative = positive' rule is counter-intuitive for children. The most effective teaching strategy is the pattern extension approach: (−2)(3) = −6, (−2)(2) = −4, (−2)(1) = −2, (−2)(0) = 0, so (−2)(−1) = +2, (−2)(−2) = +4 — the pattern continues.

Properties of integers: Closure under +, −, × (not ÷); commutativity under + and × (not −); associativity under +, × (not −); distributivity of × over +; additive identity = 0; multiplicative identity = 1; additive inverse of a is −a.

Common misconceptions at upper-primary level:

• −7 > −3 (incorrect: on the number line, −7 is further left, so −7 < −3).
• The 'minus' sign in −5 means the operation subtraction (incorrect: it means the negative integer five).
• |−a| = −a (incorrect for a > 0: absolute value is always non-negative).

Decimals, Scientific Notation and Surds

Decimal numbers and their properties form a natural bridge between fractions and the real number system. At the upper-primary level, key skills include: converting between fractions and decimals, adding and subtracting decimals, multiplying and dividing by powers of 10, comparing decimal numbers, and expressing numbers in standard form.

Adding/subtracting decimals: Align decimal points vertically before adding or subtracting — this aligns place values correctly. A common error is to align from the right (which works for whole numbers but fails for decimals).

Comparing decimals: Compare digit by digit from left to right after aligning decimal points. 0.9 > 0.189 because 9 > 1 in the tenths place. Children often believe 0.189 > 0.9 because 189 > 9 — this is the 'longer is larger' misconception.

Scientific notation (standard form): A number is in standard form when written as a × 10ⁿ where 1 ≤ a < 10. Moving the decimal point left increases n; moving right decreases n. Example: combining 12.34 × 10¹⁰ and 5.67 × 10⁸ — convert both to the same power: 1234 × 10⁸ − 5.67 × 10⁸ = 1228.33 × 10⁸ = 1.22833 × 10¹¹.

Surds and decimal approximation: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236. For comparison: √198 × √550 = √(198×550) = √108900 = 330 exactly. These products simplify when the numbers have a neat product.

Finding what to add to restore a decimal: If x = 1.011 + 10.11 − 12.101 + 0.1011 = −0.8789, then what must be added to x to get the nearest whole number 0? Answer: 0.8789. Always add vertically, aligning decimal points.

Teaching Rational Numbers — Strategies and Misconceptions

Rational number instruction is one of the most researched topics in mathematics education. The same concepts that seem obvious to an adult specialist create profound difficulties for children aged 11–14. A CTET Paper 2 candidate must understand both the mathematical content and the research-backed teaching strategies.

The five hardest conceptual shifts in rational number learning (NCERT 2012):

1. Part-whole to quotient: A fraction 3/4 means not only '3 out of 4 equal parts of a whole' but also '3 ÷ 4'. Many children never make this link and cannot see that 3/4 = 0.75 is the result of dividing 3 by 4.
2. Operator conception: 3/4 of 12 = 9 — the fraction as an operator that scales a quantity. This requires understanding multiplication of fractions.
3. Ratio conception: 3/4 as a ratio (3:4) comparing two quantities. This is essential for proportion problems.
4. Measure conception: 3/4 as a point on the number line, between 0 and 1. Many children struggle to locate fractions on a number line.
5. Equivalence: 3/4 = 6/8 = 9/12 — the same number has infinitely many representations. Children often treat these as different numbers because they look different.

Classroom strategies endorsed by NCF 2005:

• Use fraction strips, pattern blocks, and area models before introducing symbolic notation.
• Use the number line as a central representation — it unifies integers, fractions, and decimals.
• Pose estimation tasks: 'Is 7/8 + 12/13 closer to 1, 2, or 3?' (Answer: 2.) This prevents mindless algorithm application.
• Discuss and value multiple strategies — not just the standard algorithm.

CTET Exam Focus

Fractions, decimals, and commercial arithmetic questions appear in every CTET Paper 2 Mathematics section. The following patterns cover the key question types.

Pattern 1 — Compound interest to find principal. A sum at R% p.a. grows to Amount A after T years. Use A = P(1+R/100)^T. Example: ₹3267 at 10% for 2 years → P = 3267/1.21 = ₹2700.

Pattern 2 — Simple interest to find rate. Given P, T, and Amount: SI = Amount − P. R = (SI × 100)/(P × T). Example: ₹12000 grows to ₹15972 in 3 years → SI = 3972 → R = 11.03% ≈ 11%.

Pattern 3 — Profit with marked price and discount. CP of each item → add mark-up % to get MP → subtract discount % from MP to get SP → SP − CP = Profit. Profit on all items = profit per item × number of items.

Pattern 4 — Fraction comparison and sums. Convert all to improper fractions, find common denominator, identify largest and smallest, then compute sum/difference. Common error: forgetting to convert mixed numbers first.

Pattern 5 — Decimal addition/subtraction with restoration. Compute a multi-term decimal expression; find what must be added to reach a target (usually the nearest integer). Always align decimal points.

Common traps: (i) Confusing SI and CI formulas — SI is linear, CI is exponential. (ii) Computing profit% on SP instead of CP. (iii) When comparing fractions, assuming the fraction with the larger numerator is always larger (without considering denominators). (iv) In percentage problems, applying the percentage to the wrong base (new vs original).