7.2 Percentages, Reverse Percentages & Proportional Scaling

7.2 Percentages, Reverse Percentages & Proportional Scaling

Percentage calculations, ratio scaling, and fractional proportions form the core backbone of the UCAT Quantitative Reasoning subtest, appearing in over 40% of all test items. Test authors deliberately embed subtle structural traps—such as confusing percentage points with relative growth, compounding sequential changes additively, or applying discounts to the wrong base value.

Mastering the algebraic formulas and decimal multiplier mechanics outlined below ensures both speed and computational precision.

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Core Percentage Mechanics & Percentage Change

The fundamental formula for relative percentage change evaluates the shift in value relative to the original starting baseline:

$$\text{Percentage Change} = \left(\frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}}\right) \times 100\% = \left(\frac{\Delta V}{V_{\text{initial}}}\right) \times 100\%$$

Decimal Multipliers for Rapid Scaling

Rather than executing multi-step addition or subtraction, transform percentage shifts into direct decimal multipliers:

• Percentage Increase of $r\%$: Multiply by $\left(1 + \frac{r}{100}\right)$. For a $15\%$ increase: $\times 1.15$.
• Percentage Decrease of $r\%$: Multiply by $\left(1 - \frac{r}{100}\right)$. For a $35\%$ decrease: $\times 0.65$.

┌─────────────────────────────────────────────────────────────────────────┐
│                    DECIMAL MULTIPLIER QUICK REFERENCE                   │
├─────────────────────┬───────────────────┬───────────────────────────────┤
│ Percentage Change   │ Multiplier Format │ Example Calculation (£80 base)│
├─────────────────────┼───────────────────┼───────────────────────────────┤
│ +5% Increase        │ × 1.05            │ £80 × 1.05 = £84.00           │
│ +17.5% Increase     │ × 1.175           │ £80 × 1.175 = £94.00          │
│ +80% Increase       │ × 1.80            │ £80 × 1.80 = £144.00          │
│ +120% Increase      │ × 2.20            │ £80 × 2.20 = £176.00          │
│ -8% Decrease        │ × 0.92            │ £80 × 0.92 = £73.60           │
│ -25% Decrease       │ × 0.75            │ £80 × 0.75 = £60.00           │
│ -60% Decrease       │ × 0.40            │ £80 × 0.40 = £32.00           │
└─────────────────────┴───────────────────┴───────────────────────────────┘

The Base-Value Trap

Consider an emergency ward where patient admissions rose from 40 in January to 50 in February:

• Correct Calculation: $\frac{50 - 40}{40} \times 100\% = \frac{10}{40} \times 100\% = \mathbf{+25.0\%}$
• Distractor Trap: Calculating $\frac{10}{50} \times 100\% = \mathbf{20.0\%}$ (dividing by the new value instead of the original baseline). Test authors routinely include this exact distractor in QR answer options.

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Percentage Points vs. Relative Percentage Change

A critical conceptual distinction tested frequently in medical statistics and public health scenarios is the difference between an absolute percentage point shift and a relative percentage change.

  Baseline Rate: 20% ────────────────────────► New Rate: 25%
  
  1. Percentage Point Difference:  25% - 20% = +5 Percentage Points
  2. Relative Percentage Change:   ((25 - 20) / 20) × 100% = +25.0% Growth

Clinical Example: Treatment Efficacy

A clinical trial compares two oncology regimens for 5-year survival rates:

• Regimen A (Standard): $40\%$ survival rate
• Regimen B (Novel Immunotherapy): $50\%$ survival rate

$$\begin{aligned} \text{Absolute Difference} &= 50\% - 40\% = \mathbf{10\text{ percentage points}} \\ \text{Relative Improvement} &= \left(\frac{50 - 40}{40}\right) \times 100\% = \frac{10}{40} \times 100\% = \mathbf{25.0\%} \end{aligned}$$

If a UCAT question asks: "What was the relative percentage increase in survival for patients on Regimen B compared to Regimen A?", the answer is $25.0\%$, not $10\%$.

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Reverse Percentages: Recovering Original Baselines

Reverse percentage problems require you to determine the original pre-change value ($V_{\text{initial}}$) when given the final post-change value ($V_{\text{final}}$) and the percentage rate ($r$).

$$\mathbf{V_{\text{initial}} = \frac{V_{\text{final}}}{1 \pm \frac{r}{100}}}$$

Case 1: VAT / Sales Tax Removal

A hospital purchases a mobile X-ray unit for £96,000 inclusive of 20% VAT. What was the pre-tax cost of the machine?

$$\text{Pre-Tax Cost} = \frac{\text{Gross Price}}{1 + 0.20} = \frac{96,000}{1.20} = \mathbf{\text{£}80,000}$$

The Fatal Reduction Fallacy: Never deduct $20\%$ from the gross price ($96,000 \times 0.80 = \text{£}76,800$). Taking $20\%$ of £96,000 deducts £19,200, whereas VAT was levied as $20\%$ of the original £80,000 (£16,000). Deducting the percentage from the final figure always overshoots the correction.

Case 2: Post-Discount Price Recovery

A clinic buys boxes of surgical sutures on sale for £36.00 each after a 25% bulk discount. What was the normal retail price per box?

$$\text{Retail Price} = \frac{\text{Sale Price}}{1 - 0.25} = \frac{36.00}{0.75} = \mathbf{\text{£}48.00}$$

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Multi-Stage Percentage Compounding

When a quantity undergoes multiple consecutive percentage changes, the changes cannot be added arithmetically. They must be multiplied sequentially as combined multipliers.

$$\mathbf{\text{Total Combined Multiplier} = M_1 \times M_2 \times \dots \times M_n}$$ $$\mathbf{\text{Net Percentage Change} = (\text{Total Multiplier} - 1) \times 100\%}$$

Worked Scenario: Departmental Budget Compounding

A medical research laboratory's funding changes over three consecutive years:

• Year 1: Increased by $+20\%$ ($M_1 = 1.20$)
• Year 2: Decreased by $-15\%$ ($M_2 = 0.85$)
• Year 3: Increased by $+10\%$ ($M_3 = 1.10$)

$$\begin{aligned} \text{Combined Multiplier} &= 1.20 \times 0.85 \times 1.10 \\ &= 1.02 \times 1.10 = \mathbf{1.122} \\ \text{Net Change} &= (1.122 - 1) \times 100\% = \mathbf{+12.2\%} \end{aligned}$$

(Note: Adding the percentages directly yields $20 - 15 + 10 = +15.0\%$, which is completely incorrect).

Symmetric Percentage Shift Trap

If a share price increases by $25\%$ and then decreases by $25\%$:

$$\text{Combined Multiplier} = 1.25 \times 0.75 = 0.9375 \implies (0.9375 - 1) \times 100\% = \mathbf{-6.25\% \text{ net loss}}$$

An equal percentage rise and fall always results in a net overall loss.

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Ratios, Proportional Division & Dilution Chemistry

1. The Unitary Method for Multi-Part Ratios

To divide a total quantity $Q$ among entities in the ratio $a : b : c$:

1. Find total parts: $T = a + b + c$.
2. Calculate the value of 1 part: $\text{Unit Value} = \frac{Q}{T}$.
3. Multiply each ratio term by the unit value: $\text{Share}_a = a \times \text{Unit Value}$.

Worked Example: NHS Endowment Allocation

A £108,000 charitable trust endowment is split between Pediatrics, Cardiology, and Oncology in the ratio $4 : 3 : 2$.

$$\begin{aligned} \text{Total Parts} &= 4 + 3 + 2 = 9\text{ parts} \\ \text{Value of 1 Part} &= \frac{108,000}{9} = \text{£}12,000 \\ \text{Pediatrics } (4\text{ parts}) &= 4 \times 12,000 = \mathbf{\text{£}48,000} \\ \text{Cardiology } (3\text{ parts}) &= 3 \times 12,000 = \mathbf{\text{£}36,000} \\ \text{Oncology } (2\text{ parts}) &= 2 \times 12,000 = \mathbf{\text{£}24,000} \end{aligned}$$

2. Clinical Dilution Ratios: Solute-to-Solvent vs. Solution Fraction

In pharmacology and laboratory calculations, pay meticulous attention to the phrasing of dilution instructions:

┌─────────────────────────────────────────────────────────────────────────┐
│                     DILUTION NOTATION DISTINCTIONS                      │
├──────────────────────────┬───────────────────────┬──────────────────────┤
│ Terminology              │ Physical Composition  │ Active Concentration │
├──────────────────────────┼───────────────────────┼──────────────────────┤
│ **Ratio "1:4 Dilution"** │ 1 Part Solute +       │ 1 / (1 + 4) = 1/5    │
│ (Parts solute to solvent)│ 4 Parts Diluent/Water │ = **20.0%**          │
├──────────────────────────┼───────────────────────┼──────────────────────┤
│ **Fraction "1 in 4"**    │ 1 Part Solute in      │ 1 / 4                │
│ (Solute in total volume) │ 4 Parts Total Volume  │ = **25.0%**          │
└──────────────────────────┴───────────────────────┴──────────────────────┘

Practical Dilution Scenario:

A pharmacist needs to prepare 500 mL of a 1:4 dilution of chlorhexidine skin disinfectant from a pure stock solution:

• Total parts $= 1 + 4 = 5\text{ parts}$.
• Volume per part $= \frac{500\text{ mL}}{5} = 100\text{ mL}$.
• Formulation requires: 100 mL pure chlorhexidine stock + 400 mL sterile water.