Topic 1: Quantitative Aptitude Basics for NABE

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Core Formulas — The NABE Must-Know:

Topic	Formula
Ratio	a:b = a/b
Proportion	a:b = c:d → ad = bc
Percentage	X% = X/100
Average	Sum of values / Number of values
Simple Interest	SI = P × R × T / 100
Compound Interest	CI = P(1 + R/100)^T - P
Profit/Loss %	(SP - CP)/CP × 100

Quick Conversions — NABE High-Yield:

• 1/2 = 50% | 1/3 = 33.33% | 1/4 = 25% | 1/5 = 20%
• 1/8 = 12.5% | 1/10 = 10% | 1/20 = 5% | 1/25 = 4%

Worked Example (Ratio): If the ratio of boys to girls in a class is 3:2 and there are 30 boys, how many girls?

• 3/2 = 30/x → x = 20 girls

Worked Example (Percentage): What is 25% of 240? → 25/100 × 240 = 60

⚡ Exam tip: NABE quantitative questions are designed to be solved without calculators. Master the common fractions-to-percentages conversions above. They appear in nearly every exam.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Ratio and Proportion

Ratio

A ratio is a comparison between two quantities of the same kind, expressed in the same units. It tells us how many times one quantity is of the other.

Notation: a : b (read as “a is to b”)

• a is called the first term or antecedent
• b is called the second term or consequent

Key Properties of Ratios:

1. The order matters — 3:2 is NOT the same as 2:3
2. Both quantities must be of the same kind
3. Ratio has no unit
4. A ratio can be simplified by dividing both terms by their HCF (Highest Common Factor)
5. Multiplying or dividing both terms by the same non-zero number does not change the ratio

Examples:

• Simplify 15:25 → divide by 5 → 3:5
• Simplify 48:36 → divide by 12 → 4:3
• If a:b = 3:4 and b:c = 4:5, then a:b:c = 3:4:5

Comparing Ratios

To compare two ratios, convert them to equivalent fractions and compare:

Example: Which is greater — 3:5 or 4:7?

• 3:5 = 3/5 = 0.6
• 4:7 = 4/7 ≈ 0.571
• 3:5 is greater

Types of Ratios

1. Duplicate Ratio: a²:b² — e.g., 3²:4² = 9:16
2. Sub-duplicate Ratio: √a:√b — e.g., √3:√12 = √3:2√3 = 1:2
3. Compound Ratio: (a×c):(b×d) — e.g., 2:3 and 4:5 → compound = 8:15
4. Inverse Ratio: b:a is the inverse of a:b

Proportion

Proportion states that two ratios are equal.

Notation: a : b :: c : d (read as “a is to b as c is to d”)

• Extremes: a and d
• Means: b and c
• Property: Product of extremes = Product of means → a × d = b × c

Example: 3:4 :: 9:12 → 3 × 12 = 4 × 9 → 36 = 36 ✓

Continued Proportion

Three quantities a, b, c are in continued proportion if a:b = b:c

• This means b² = a × c (b is the mean proportional between a and c)

Example: 4, 8, 16 → 4:8 :: 8:16 → 8² = 4 × 16 → 64 = 64 ✓

Percentages

Definition and Conversion

Percent (%) means “per hundred” — a fraction with denominator 100.

Formula:

Percentage = (Part / Whole) × 100
Part = (Percentage / 100) × Whole

Key Conversions to Memorize

Fraction	Decimal	Percentage
1/1	1.0	100%
1/2	0.5	50%
1/3	0.333…	33.33%
2/3	0.666…	66.67%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
2/5	0.4	40%
3/5	0.6	60%
4/5	0.8	80%
1/8	0.125	12.5%
3/8	0.375	37.5%
5/8	0.625	62.5%
7/8	0.875	87.5%
1/10	0.1	10%
1/20	0.05	5%
1/25	0.04	4%
1/50	0.02	2%

Percentage Increase and Decrease

Increase:

New Value = Original × (1 + Increase%)

Decrease:

New Value = Original × (1 - Decrease%)

Reverse Percentage (Finding Original): If a price is increased by 25% to become Rs. 500, what was the original?

• Original = New / (1 + 25/100) = 500 / 1.25 = Rs. 400

Worked Example — Percentage Increase: The population of a town is 50,000 and increases by 10% per year. What is the population after 1 year?

• Increase = 50,000 × 10/100 = 5,000
• New population = 50,000 + 5,000 = 55,000

Worked Example — Percentage Decrease: A car worth Rs. 2,000,000 depreciates by 15% per year. What is its value after 1 year?

• Decrease = 2,000,000 × 15/100 = 300,000
• New value = 2,000,000 - 300,000 = Rs. 1,700,000

Percentage in Profit and Loss Context

NABE frequently asks these:

• If a shopkeeper sells at a 20% profit, his selling price = 120% of cost price
• If a shopkeeper sells at a 20% loss, his selling price = 80% of cost price

Averages

Definition

Average (Arithmetic Mean) is the sum of all values divided by the number of values.

Formula:

Average = (Sum of all values) / (Number of values)

Also written as:

x̄ = Σx / n

Where Σx = sum of all observations, n = number of observations

Properties of Averages

1. Sum = Average × Number of items (very useful!)
2. Adding the same value to every item increases the average by that same value
3. Removing or replacing some items changes the average accordingly
4. The average always lies between the minimum and maximum values

Worked Examples:

Example 1 — Basic Average: Find the average of 12, 15, 18, 21, 24.

• Sum = 12 + 15 + 18 + 21 + 24 = 90
• Average = 90 / 5 = 18

Example 2 — Using the Sum formula: The average of 10 numbers is 40. Their sum is 400.

• Sum = Average × n = 40 × 10 = 400 ✓

Example 3 — Weighted Average: In a class, 30 students have an average mark of 60, and 20 students have an average mark of 80. Find the combined average.

• Total marks = (30 × 60) + (20 × 80) = 1,800 + 1,600 = 3,400
• Total students = 30 + 20 = 50
• Combined average = 3,400 / 50 = 68

Average Speed

When distance is the same for two journeys:

Average Speed = (2 × Speed₁ × Speed₂) / (Speed₁ + Speed₂)

Example: A car travels from A to B at 60 km/h and returns from B to A at 40 km/h. Find the average speed for the round trip.

• Average speed = (2 × 60 × 40) / (60 + 40) = 4,800 / 100 = 48 km/h

Note: This is NOT the arithmetic mean of 60 and 40 (which would be 50). The average speed for equal distances is always closer to the slower speed.

Simple Interest and Compound Interest

Simple Interest (SI)

Simple Interest is calculated only on the original principal amount throughout the entire period.

Formula:

SI = (P × R × T) / 100

Where:
P = Principal (the initial amount borrowed or invested)
R = Rate of interest per year (in %)
T = Time period (in years)

Total Amount (A):

A = P + SI = P + (P × R × T) / 100 = P(1 + RT/100)

Worked Example — Simple Interest: Rs. 5,000 is invested at 8% per annum for 3 years. Find the simple interest.

• SI = (5,000 × 8 × 3) / 100 = (5,000 × 24) / 100 = 120,000 / 100 = Rs. 1,200
• Total amount = 5,000 + 1,200 = Rs. 6,200

Compound Interest (CI)

Compound Interest is calculated on the principal plus accumulated interest from previous periods — “interest on interest.”

Formula:

CI = P(1 + R/100)^T - P

Total Amount: A = P(1 + R/100)^T

Key difference from Simple Interest:

• SI: Interest is always calculated on the original principal P
• CI: Interest is calculated on (P + accumulated interest) — the base grows each period

Compound Interest — Annual Compounding (Most Common)

Formula:

A = P × (1 + R/100)^T
CI = A - P

Worked Example — Compound Interest: Rs. 10,000 is invested at 10% per annum for 3 years. Find the compound interest.

• Year 1: 10,000 × 10/100 = 1,000 → Amount = 11,000
• Year 2: 11,000 × 10/100 = 1,100 → Amount = 12,100
• Year 3: 12,100 × 10/100 = 1,210 → Amount = 13,310
• CI = 13,310 - 10,000 = Rs. 3,310

Using the formula: A = 10,000 × (1 + 10/100)³ = 10,000 × (1.1)³ = 10,000 × 1.331 = Rs. 13,310 CI = 13,310 - 10,000 = Rs. 3,310

SI vs. CI Comparison

Year	Simple Interest (P=10000, R=10%)	Compound Interest (P=10000, R=10%)
1	1,000	1,000
2	1,000	1,100
3	1,000	1,210
Total Interest	3,000	3,310

Pattern: For compound interest, the interest amount increases each year because the base grows. For simple interest, the interest amount stays constant.

Compound Interest — Half-Yearly Compounding

When interest is compounded half-yearly (twice a year):

• New rate = R/2 per half-year
• New time = 2T periods

Formula:

A = P × (1 + R/200)^(2T)

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Data Interpretation Basics

Data Interpretation (DI) is a key NABE quantitative section. It involves reading, analyzing, and drawing conclusions from tables, graphs, and charts.

Types of Data Presentation

1. Tables: Raw data in rows and columns
2. Bar Graphs: Vertical or horizontal bars comparing quantities
3. Pie Charts: Data shown as slices of a circle (parts of a whole = 100%)
4. Line Graphs: Data plotted as points connected by lines (show trends over time)
5. Mixed Graphs: Combination of two or more types

Reading Pie Charts

Key principle: The whole circle = 100% = 360°

• 1% = 3.6° of the circle

Example: A pie chart shows a country’s exports:

• Textiles: 60% → 60% × 360° = 216°
• Rice: 20% → 20% × 360° = 72°
• Leather: 10% → 10% × 360° = 36°
• Others: 10% → 10% × 360° = 36°
• Total: 360° ✓

Reading Bar Graphs

• Read the scale carefully — the height/length of bars corresponds to values
• Watch for the scale starting at 0 or at another value (truncated axis can exaggerate differences)

Practice DI — Worked Example

Question: The following table shows the marks obtained by five students in three subjects:

Student	Math	Science	English
Ali	70	80	65
Sara	85	75	90
Ahmed	60	70	75
Fatima	90	85	80
Hasan	75	60	70

Q1: What is Sara’s average marks?

• Average = (85 + 75 + 90) / 3 = 250 / 3 = 83.33

Q2: Which student has the highest total?

• Ali: 215, Sara: 250, Ahmed: 205, Fatima: 255, Hasan: 205
• Fatima (255 marks)

Q3: What is the class average in Science?

• Total Science marks = 80 + 75 + 70 + 85 + 60 = 370
• Average = 370 / 5 = 74

Time and Work

Basic Formula

Work Done = Rate × Time

If A can complete a job in 'a' days:
A's work rate = 1/a of the job per day

If A and B work together:
Combined rate = 1/a + 1/b
Time taken = 1 / (Combined rate)

Worked Example: A can complete a job in 10 days and B can complete the same job in 20 days. How long will they take working together?

• A’s rate = 1/10 per day
• B’s rate = 1/20 per day
• Combined rate = 1/10 + 1/20 = 3/20 per day
• Time = 1 / (3/20) = 20/3 = 6.67 days (approximately 6 days and 16 hours)

Time, Speed, and Distance

Basic Formula

Distance = Speed × Time
Speed = Distance / Time
Time = Distance / Speed

Important conversions for NABE:

• If speed is in km/h and time is in hours → distance is in km
• To convert km/h to m/s: multiply by 5/18
• To convert m/s to km/h: multiply by 18/5

Example: 72 km/h = 72 × 5/18 = 20 m/s

Profit and Loss

Key Formulas

Selling Price (SP) = Cost Price (CP) + Profit
OR
Selling Price (SP) = Cost Price (CP) - Loss

Profit % = (Profit / CP) × 100 = ((SP - CP) / CP) × 100
Loss % = (Loss / CP) × 100 = ((CP - SP) / CP) × 100

SP = CP × (1 + Profit%/100)
SP = CP × (1 - Loss%/100)

Worked Example — Discount and Profit: A shopkeeper buys an article for Rs. 400 and sells it at a 20% discount on the marked price. If he still makes a profit of 20%, what is the marked price?

• Profit = 20% of 400 = 80
• SP required = 400 + 80 = 480
• Let M be marked price
• SP after 20% discount = 80% of M = 480
• M = 480 / 0.8 = Rs. 600

All Formulas — Quick Reference Card

RATIO & PROPORTION
  a:b = a/b
  a:b = c:d → ad = bc
  Mean proportional: b² = ac

PERCENTAGE
  X% = X/100
  Percentage = (Part/Whole) × 100
  Increase%: (New - Old)/Old × 100
  Decrease%: (Old - New)/Old × 100

AVERAGES
  Average = Sum/n
  Sum = Average × n
  Average Speed (equal distance): 2v₁v₂/(v₁+v₂)

SIMPLE INTEREST
  SI = P×R×T/100
  Amount = P(1 + RT/100)

COMPOUND INTEREST
  Amount = P(1 + R/100)^T
  CI = P[(1 + R/100)^T - 1]

PROFIT & LOSS
  Profit% = (SP-CP)/CP × 100
  SP = CP × (1 + Profit%/100)

TIME & WORK
  Work = Rate × Time
  A's time = a days → Rate = 1/a
  Combined: 1/a + 1/b = 1/T

TIME, SPEED, DISTANCE
  Distance = Speed × Time
  km/h → m/s: × 5/18
  m/s → km/h: × 18/5

⚡ Exam Pattern Insight: In the NABE quantitative section, the majority of marks come from Percentages (15-20 questions), Averages (5-10 questions), and Simple/Compound Interest (5-10 questions). Ratio and Proportion questions often require cross-multiplication (ad = bc). Always check whether the question asks for Simple Interest or Compound Interest — they give different answers for the same principal, rate, and time.