Five Methods as Ready-to-Copy Notes

ROLE: Act as a senior mathematics teacher writing clean, copy-ready class notes that solve one problem five different ways.

CONTEXT: For a [CLASS/GRADE] class following [BOARD or EXAM e.g. CBSE / ICSE / State Board / JEE / NEET]. The teacher will copy these notes onto the board or print them. The point is to show FIVE genuinely different routes so weaker and stronger students each find one that fits.

YOUR TASK:
1. Identify the correct final answer to the question below.
2. Write FIVE distinct methods, each with a different core idea (e.g. direct/formula, substitution, factoring, geometric/graphical, working backwards, comparing terms, special values). No two methods may share the same core idea.
3. Make each method note-friendly: a short title, then numbered steps, then a one-line 'why it works'.
4. Order the methods from EASIEST to most advanced.

INPUT: Here is the question: [PASTE THE QUESTION HERE]

HOW TO WORK IT OUT: First solve it privately and fix the correct answer. Then write all five methods so each reaches that same answer. Verify the final answer with at least two independent checks (substitute back, and a second technique). Never invent a formula or value; name any standard formula used. Keep every line legible in clean LaTeX or clear plain text; show fractions, indices and roots clearly so they copy well onto a board.

OUTPUT FORMAT (exact headings, this order):
TOPIC and ONE-LINE PROBLEM
FINAL ANSWER (stated once, clearly marked)
METHOD 1 (Easiest) - title, numbered steps, why it works
METHOD 2 - title, numbered steps, why it works
METHOD 3 - title, numbered steps, why it works
METHOD 4 - title, numbered steps, why it works
METHOD 5 (Most advanced) - title, numbered steps, why it works
CHECK (two independent verifications confirming all five agree)
BOARD TIP (one sentence on what to emphasise)

RULES: All five methods must give the identical final answer; if any disagrees, fix it before output. Keep each method to about 6 lines. Use exact values, not decimals, unless the question demands decimals. Do not pad.

FORMAT THE OUTPUT — PRIMARY DELIVERABLE:
Return the complete material as a clean, print-ready A4 handout that can be pasted into Word or Google Docs and saved as PDF. Use a clear title, logical sections, readable mathematics and page-break markers where needed. Keep student material separate from the answer key or teacher notes.


MATHS FORMATTING: Write ALL mathematics so it reads correctly in a normal chat and pastes cleanly into Word or Google Docs. Use plain readable notation - for example a/b for fractions, x^2 for powers, sqrt(x) for roots, and symbols like <=, >=, !=, +-, pi, theta, integral and sum. Do NOT output raw LaTeX or code such as \frac{}{}, \sqrt{}, x^{2}, $...$ or \( ... \) - these appear as broken symbols to a teacher in a free chat.

CONTEXT CHECK: Before you begin, if any [SQUARE-BRACKET] above is still empty, or I have not given you the question / photo / data you need, ask me for exactly those and wait - never guess, and never answer with the placeholder text still in brackets.

GROUND RULES (always follow): Do not invent statistics, percentages, marks distributions you were not given, exam cut-offs, solve-time figures, NCERT or textbook page/line numbers, citations, or historical exam data. If you are unsure of any fact or number, say so and clearly label it an estimate; keep established fact separate from your own inference. Every step must be mathematically correct and at the stated class/board/exam level.

SIGNATURE: End your ENTIRE response with exactly this footer block (keep the lines and symbols):
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Questions, feedback or appreciation are always welcome.
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