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UAT-UK SPEC: M3MODULE: MATHEMATICS 1TOPIC ID: ARCH-M1-04

Non-Linear Proportionality (y ∝ xⁿ) & Reverse Percentage Compounding

Master Non-Linear Proportionality (y ∝ xⁿ) & Reverse Percentage Compounding on the ESAT Mathematics 1 paper. Traditional classroom derivations waste valuable seconds. Using the 2-step Sub-90 invariant elimination method compresses solve time to 35s, saving +85s per question.

Standard Classroom Time
120s
Textbook multi-line derivations bleed exam clock
Sub-90 Target Pace
35s
Comfortably below the 88.8s per-question limit
Time Saved for Review
+85s
Banks clock buffer across 27 questions

2-Step Sub-90 Shortcut Formula

How to eliminate intermediate algebra and solve under 45 seconds.

Launch Interactive Lesson
STEP 1 • SETUP INVARIANT (LINE 1)10–15s
Line 1 (Ratio Scaling): y2=y1(x2x1)3y_2 = y_1 \left(\frac{x_2}{x_1}\right)^3.

Establish the structural invariant and eliminate impossible options immediately.

STEP 2 • FAST SOLVE (LINE 2)15–25s
Line 2 (Evaluate): 24×(32)3=24×278=3×27=8124 \times \left(\frac{3}{2}\right)^3 = 24 \times \frac{27}{8} = 3 \times 27 = 81 (20s).

Evaluate the simplified expression using integer factoring or unit elimination.

Distractor Trap Forensics

The predictable cognitive traps engineered by examiners into the wrong options.

TRAP #1sign_or_direction_flip

Calculating pre-sale price after 20% discount as 1.20 * P(sale) instead of P(sale) / 0.80

Applying percentage increase to the discounted price instead of dividing by the multiplier (1 - r).

Sub-90 Fix: Test invariants before expanding equations.
TRAP #2factor_of_two_or_stoichiometry_trap

Multiplying percentage change by power n (e.g. 2 * 20% = 40%) instead of (1.20)² - 1 = 44%

Using linear approximation n*r instead of exact multiplier (1 + r)ⁿ.

Sub-90 Fix: Test invariants before expanding equations.
TRAP #3fence_post_or_domain_trap

Using y₂ = y₁ * (x₂/x₁) instead of (x₂/x₁)³ for inverse cube proportionality

Omitting the exponent or inversion on the scaling ratio.

Sub-90 Fix: Test invariants before expanding equations.

Try 1 Timed Question: Non-Linear Proportionality (y ∝ xⁿ) & Reverse Percentage Compounding

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Full Lesson All 15 Questions (ARCH-M1-04)
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ARCH-M1-04: Non-Linear Proportionality (y ∝ xⁿ) & Reverse Percentage Compounding (Sample)Mathematics 1
00:00.0
SUB-90 ZONE (<30S)
0.0s30s Sub-9090s ESAT Budget150s Classroom
Question 1 of 1ARCH-M1-04 (6-Card Lesson)Format: Standard 5-Option
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Question 1 of 1 • 90s Target

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Mathematics 1•ARCH-M1-04•Standard 5-Option
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