Number

Fractions: Add/subtract: common denominator. Multiply: numerators × numerators, denominators × denominators. Divide: KFC — Keep, Flip, Change.

Conversions: Fraction → Decimal: divide. Decimal → %: ×100. % → Fraction: over 100, simplify.

Standard form: A × 10ⁿ where 1 ≤ A < 10. Example: 45,000 = 4.5 × 10⁴. 0.0032 = 3.2 × 10⁻³.

Indices rules: aᵐ × aⁿ = aᵐ⁺ⁿ. aᵐ ÷ aⁿ = aᵐ⁻ⁿ. (aᵐ)ⁿ = aᵐⁿ. a⁰ = 1. a⁻ⁿ = 1/aⁿ.

Surds: Simplify by finding square factors. √12 = 2√3. Rationalise denominator: 1/√5 = √5/5.

Deep dive: build the idea, then use it

Conceptual model

Number is a system for representing size, ratio, and scale. Fractions, decimals, and percentages are different names for the same proportion; choose the form that makes the operation visible. Standard form separates a number's significant digits from its scale. Index laws work because repeated multiplication is being compressed, while a negative index means reciprocal, not a negative value. A surd is an exact value: keep it exact until a question asks for a decimal approximation.

Worked example

Calculate (3/5 - 1/4) ÷ 7/10. First make the subtraction comparable: 3/5 = 12/20, 1/4 = 5/20, so it is 7/20. Dividing by 7/10 means multiplying by its reciprocal: 7/20 × 10/7 = 1/2. For scale, (6 × 10^5)(3 × 10^-2) = 18 × 10^3 = 1.8 × 10^4; adjust the coefficient so it is between 1 and 10.

Exam-method habit

Translate the question before calculating: mark whether it asks for an exact value, a percentage change, or a rounded estimate. Put brackets around negative powers and around the whole numerator/denominator. For percentage increase, find the multiplier 1 + r; for a decrease use 1 - r. State rounding only at the final line unless earlier rounding is explicitly requested.

Common errors to catch early

Adding denominators when adding fractions; treating a^m + a^n as a^(m+n); writing 0.35% = 0.35 instead of 0.0035; leaving 18 × 10^3 as standard form; and rounding an intermediate calculator display so much that the final answer shifts.

Retrieval drill — close the notes, phir try karo

Without notes: convert 7/16 to a decimal and percentage; simplify 2^5 × 2^-3; write 0.000709 in standard form; explain in one sentence why √50 = 5√2. Finally estimate whether 19.8 ÷ 0.49 should be nearer 4, 40, or 400.

A Pakistan-relevant use

A student comparing two prepaid data bundles can calculate cost per GB and then compare the percentage difference. The useful decision is not which logo is familiar; it is whether the allowance, validity period, and price make the unit rate genuinely better for their own usage.

Concise summary

Keep equivalent forms connected, preserve exact values where useful, and let place value control standard form. Number work is dependable when every operation has a visible reason.

A reliable self-check routine

Before accepting an answer, say what each quantity, symbol, particle, or graph feature means. Then check its unit, sign, direction, size, or conservation rule. In a calculation, write the relationship first, substitute with units, calculate, and decide whether the result is sensible. In an explanation, make a chain: cause → mechanism → observed result. This is not extra decoration; it is how a reader can follow your thinking and how you catch a copied digit or an attractive-but-wrong statement. If the question gives a new context, do not hunt for a memorised sentence. Identify the model underneath it and apply that model. Short, precise working beats a long paragraph that never answers the command word.

Practice plan

Try one straightforward question without notes, one mixed question where you choose the method, and one question where you explain why an answer is reasonable. Mark the exact first step that felt uncertain. Revisit that step the next day for two minutes instead of rereading everything. Small retrieval loops make the topic stick, yaar.

Connect and transfer

This topic becomes stronger when you deliberately meet it in an unfamiliar wrapper. A diagram may be rotated, a calculation may use an awkward unit, a practical may describe an everyday object, or a question may provide more information than you need. Pause and sort the information into three columns: given, wanted, and relationship. That small pause prevents the common rush of putting every number into the first formula remembered. If the answer is qualitative, decide whether the task is asking for a prediction, a description, or a mechanism; these need different sentences. If it is numerical, estimate its order of magnitude before the calculator. If it involves a graph or table, describe the relevant trend using the actual variables before explaining it.

Teach the idea out loud in sixty seconds as though a friend missed the lesson. Avoid specialist words you cannot unpack. Then add those words back with their exact meaning. This exposes the difference between recognition (“that looks familiar”) and recall (“I can construct the answer”). Keep an error log with a corrected example, not a list of scores. For the next attempt, cover the correction and reproduce the decision that led to it. The goal is calm, repeatable reasoning—not racing through a page. When your final answer differs from a friend’s, compare the model and assumptions before comparing calculators. Often the useful learning is in the first different step.