75-Archetype SyllabusMathematics 1 • ARCH-M1-04 • Spec M3

Full 6-Card Lesson 15-Question Timed Drill

UAT-UK SPEC CODE: M3 (Ratio and proportion)ASSUMED MATHS 1 PREREQUISITE: M3.4FREQUENCY: VERY HIGH

Non-Linear Proportionality (y ∝ x^n) & Reverse Percentage Compounding

Deconstructed from 29 past UAT-UK questions (12.89% of all historical Mathematics 1 papers). Students using standard A-Level classroom methods spend an average of 120s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 35s (saving 85s per question).

Historical Frequency

29 Past Qs

12.89% of Mathematics 1

Classroom Trap Time

120s

+30s over 90s budget

Sub-90 Target Time

35s

Banks +85s on the clock

Dominant UI Format

5-Option A–E

Maths 1 Skill: M3.4

Card 1 & Card 2 Preview: The Classroom Time Trap vs. Sub-90 2-Line Rule

Excerpted from our interactive 6-Step Topic Lesson for ARCH-M1-04.

Launch All 6 Active-Recall Cards

CARD 1 OF 6 • THE CLASSROOM TIME TRAP (120s)

Why Students Bleed Time:

Write

y = kx^3

. Substituting

x = 2, y = 24

gives

24 = k(2^3) = 8k \implies k = 3

. When

x = 3

y = 3(3^3) = 3 \times 27 = 81

(~95s).

In 29 past UAT-UK questions (12.89% of Mathematics 1), candidates lose 85s by applying percentage increase to the discounted price instead of dividing by the multiplier (1 - r). Instead, apply M3 with Maths 1 skill M3.4.

CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<35s)

Line 1 (Ratio Scaling):

y_2 = y_1 \left(\frac{x_2}{x_1}\right)^3

Line 2 (Evaluate):

24 \times \left(\frac{3}{2}\right)^3 = 24 \times \frac{27}{8} = 3 \times 27 = 81

(20s).

Sanity & Limit Check: Verify units/signs against sign_or_direction_flip (Calculating pre-sale price after 20% discount as 1.20 * P_sale instead of P_sale / 0.80) and factor_of_two_or_stoichiometry_trap (Multiplying percentage change by power n (e.g. 2 * 20% = 40%) instead of (1.20)² - 1 = 44%).

Examiner Distractor Blueprint for ARCH-M1-04

FOIL #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).

Open Sign / Direction Flip Clinic

FOIL #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.

Open Factor-of-2 / Stoichiometry Clinic

FOIL #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.

Open Fence-Post / Domain Trap Clinic

FOIL #4intermediate_stop_trap

Outputting the new multiplier 1.44 (144%) instead of the percentage increase 44%

Failing to subtract 100% from the final scale factor.

Open Intermediate Stop Trap Clinic

Try 1 Live Sample Question Under the 90-Second Stopwatch

Press W to open the 2-line dry-wipe scratchpad, select A–E, and lock in to reveal the Dual-View solution.

Full 6-Card Lesson Launch All 15 Questions (ARCH-M1-04)

ARCH-M1-04: Non-Linear Proportionality (y ∝ x^n) & Reverse Percentage Compounding (Sample)Mathematics 1

00:00.0

SUB-90 ZONE (<35S)

0.0s35s Sub-9090s ESAT Budget150s Classroom

Question 1 of 1ARCH-M1-04 (6-Card Lesson)Format: Standard 5-Option

Right-click any option to cross out • Keys A–E + Enter

Question 1 of 1 • 90s Target

Ready to Start Question 1?

The question stem, diagram, and answer options are hidden behind this pre-start gate. When you click below, the question will reveal and your 90-second split stopwatch will immediately begin ticking.

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

or press Enter to start