Ch9. Advanced Numerical Reasoning — Mastering Data Interpretation and Applied Calculation

Structure of Numerical Reasoning

Numerical reasoning competency consists of two core question types:

Basic Arithmetic: Four operations, fractions and decimals, ratios and percentages, equations
Data Interpretation: Reading and analyzing tables and graphs to draw conclusions

Data interpretation carries more weight in professional workforce assessments. This lecture focuses on advanced data interpretation and the applied calculation patterns that appear most often.

The 5-Step Data Interpretation Process

Step 1: Understand the Data Structure (30 seconds)

Identify the chart type (bar, line, pie, combo)
Check the units (thousands, millions, percent, index)
Note the time range and categories covered

Step 2: Read the Answer Choices (30 seconds)

Read the answer choices first so you know exactly which values need to be calculated.

Step 3: Extract Only the Necessary Data

Don’t read every number. Only locate the values required by each answer choice.

Step 4: Use Estimation to Narrow Down

Before calculating precisely, use rounded numbers to eliminate 2–3 answer choices.

Step 5: Verify (Only When Needed)

Recalculate only if your answer seems uncertain.

Key Calculation Type 1: Percent Change

Formula

Percent Change (%) = (Comparison Period − Base Period) ÷ Base Period × 100

Trap: Absolute Change vs. Percent Change

Item	2020	2021
Product A	1,000	1,100
Product B	500	700

Product A: absolute change = 100, percent change = 10%
Product B: absolute change = 200, percent change = 40%

“Product with the highest percent change” = B (40%) “Product with the greatest absolute increase” = B (200)

Both answers happen to be B here, but this won’t always be the case.

Compound Annual Growth Rate (CAGR)

CAGR = (End Value / Start Value)^(1/n) − 1

On assessments, you rarely need to calculate exact CAGR. It usually appears as a relative comparison question.

Market Share (%) = A ÷ Total × 100

Trap: If the total changes, every item’s share changes too.

Compound Ratio Problems

“Country A’s exports account for 30% of its GDP, and Product B makes up 25% of those exports. What share of GDP does Product B’s exports represent?”

30% × 25% = 7.5%

When multiplying percentages, always convert to decimals first: 0.30 × 0.25 = 0.075

Key Calculation Type 3: Index Numbers and Base Periods

Reading an Index

An index sets a reference period to 100 and expresses all other periods relative to it.

“If the Consumer Price Index for 2023 is 115 using 2020 as the base year of 100, what is the rate of inflation?” → 15% increase (115 − 100 = 15)

Base Year Trap

Two index series with different base years cannot be directly compared. Doing so produces a wrong answer.

Key Calculation Type 4: Weighted Average

Used when each value carries a different weight.

Mixing Problems (Concentration and Price)

200 g of 10% saltwater mixed with 300 g of 20% saltwater — what is the combined concentration?

Salt = 200 × 0.1 + 300 × 0.2 = 20 + 60 = 80 g
Total solution = 200 + 300 = 500 g
Concentration = 80 / 500 = 16%

Average Speed (Watch Out!)

A traveler goes 60 miles at 60 mph, then another 60 miles at 30 mph. What is the average speed for the whole trip?

Simple average: (60 + 30) / 2 = 45 mph ← Wrong! Actual average: Total distance / Total time = 120 / (1 + 2) = 120 / 3 = 40 mph

Key rule: When equal distances (not equal times) are traveled, use the harmonic mean for average speed.

Key Calculation Type 5: Work Rate Problems

Core Formula

Work Rate = 1 / Time to Complete (treating the whole job as 1)

A alone takes 6 days; B alone takes 4 days. Together?

A's rate: 1/6, B's rate: 1/4
Combined: 1/6 + 1/4 = 2/12 + 3/12 = 5/12
Time to finish: 12/5 = 2.4 days

Partial Completion, Then Collaborate

A works 3 days, then B joins to finish the rest:

A’s 3-day contribution: 3 × (1/6) = 1/2
Remaining work: 1 − 1/2 = 1/2
Combined rate of A+B: 5/12
Time for the rest: (1/2) ÷ (5/12) = 6/5 = 1.2 days

Probability Fundamentals

Addition Rule

For events A and B:

P(A or B) = P(A) + P(B) − P(A and B)

If A and B are mutually exclusive: P(A or B) = P(A) + P(B)

Multiplication Rule

For independent events:

P(A and B) = P(A) × P(B)

Conditional Probability

P(A | B) = P(A and B) / P(B)

“The probability that A occurs given that B has already occurred”

High-Frequency Type: Combinations

Choose 2 from 5 people — how many ways?

₅C₂ = 5! / (2! × 3!) = 10

Choose 1 team leader and 2 team members from 10 people:

₁₀C₁ × ₉C₂ = 10 × 36 = 360

Situation	Shortcut
×0.9	×(1−0.1) = 10% decrease
×1.1 × 1.1	≈ ×1.21 (not a full 20% increase)
Comparing fractions	Estimate using ratio of numerator to denominator
Division	Convert to multiplication by the reciprocal
Percent → fraction	25%=1/4, 33%≈1/3, 20%=1/5

Study Checklist

Can distinguish between absolute change and percent change
Can convert percentages to decimals when computing compound ratios
Can explain why weighted average differs from simple average
Can set up work rate problems by treating the whole job as 1
Can apply the addition and multiplication rules to calculate probabilities