ARCH-M1-01: Compound Units, Alloy Density Mixing & Area/Volume Prefix Scaling

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-M1-01: Why the 135s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~135s (Blows 90s Clock)

Let each leg have length

d

. The outbound time is

t_1 = \frac{d}{20}

and the return time is

t_2 = \frac{d}{30}

. Summing over a common denominator gives

t_{\text{total}} = \frac{3d + 2d}{60} = \frac{5d}{60} = \frac{d}{12}

. Dividing total distance

2d

\frac{d}{12}

gives

24\text{ m s}^{-1}

(~120s).

The Fast ShortcutTarget: 40s (Saves 95s)

Line 1 (Harmonic Mean Invariant): For equal distances,

v_{\text{avg}} = \frac{2v_1 v_2}{v_1 + v_2}

.
Line 2 (Cancel Tens):

\frac{2 \times 20 \times 30}{20 + 30} = \frac{1200}{50} = 24\text{ m s}^{-1}

(30s).

Where Students Lose Time: In 17 past UAT-UK questions (7.56% of Mathematics 1), candidates lose 95s by applying linear 1d conversion factor (100) to 2d area or 3d volume units. Instead, apply M1 with Maths 1 skill M1.2.

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