Basic Mathematics — Solved One Paper Course
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Written for PPSC Intelligence Officer (BS-14) beginners. Read these original notes, practise recall, then open the MCQ dashboard for timed questions. Dates and dynamic details should be rechecked before the exam.
1. Number systems and BODMAS
Begin with integers, factors, multiples, and order of operations. Brackets are evaluated before multiplication/division and addition/subtraction. Work left to right for operations of equal precedence.
Key points
- Example: 5 + 3 × 4 = 17
- Example: (5 + 3) × 4 = 32
- An even integer is divisible by 2
- A prime has exactly two positive factors
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about number systems and bodmas?
Example: 5 + 3 × 4 = 17
Recall point 2: what did you learn about number systems and bodmas?
Example: (5 + 3) × 4 = 32
Recall point 3: what did you learn about number systems and bodmas?
An even integer is divisible by 2
Recall point 4: what did you learn about number systems and bodmas?
A prime has exactly two positive factors
2. Fractions and decimals
Fractions represent parts of a whole; decimal and percentage forms are alternate representations. For addition, obtain a common denominator; for multiplication, multiply numerators and denominators.
Key points
- 1/2 equals 0.5
- 3/4 equals 0.75
- 1/8 equals 0.125
- 1/4 + 1/2 = 3/4
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about fractions and decimals?
1/2 equals 0.5
Recall point 2: what did you learn about fractions and decimals?
3/4 equals 0.75
Recall point 3: what did you learn about fractions and decimals?
1/8 equals 0.125
Recall point 4: what did you learn about fractions and decimals?
1/4 + 1/2 = 3/4
3. Percentages
Percent means per hundred. To calculate 15% of 200, multiply 200 × 15/100 = 30. Percentage increase is change divided by original amount times 100.
Key points
- 20% of 150 is 30
- 10% of 450 is 45
- From 80 to 100 is 25% increase
- From 100 to 80 is 20% decrease
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about percentages?
20% of 150 is 30
Recall point 2: what did you learn about percentages?
10% of 450 is 45
Recall point 3: what did you learn about percentages?
From 80 to 100 is 25% increase
Recall point 4: what did you learn about percentages?
From 100 to 80 is 20% decrease
4. Ratio and proportion
A ratio compares parts. If 50 is divided in the ratio 2:3, the total parts are five; the shares are 20 and 30. A proportion equates two ratios.
Key points
- 2:3 means 2 shares versus 3 shares
- Divide 100 in 1:4 gives 20 and 80
- Ratios should use comparable units
- Cross multiplication solves simple proportions
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about ratio and proportion?
2:3 means 2 shares versus 3 shares
Recall point 2: what did you learn about ratio and proportion?
Divide 100 in 1:4 gives 20 and 80
Recall point 3: what did you learn about ratio and proportion?
Ratios should use comparable units
Recall point 4: what did you learn about ratio and proportion?
Cross multiplication solves simple proportions
5. Average, median and mode
Mean is the sum of values divided by count. Median is middle value after sorting; mode is the most frequent value. Exam questions often try to confuse all three.
Key points
- Mean of 4, 8, 12 is 8
- Median of 3, 7, 9 is 7
- Mode of 2, 2, 5 is 2
- An outlier can change the mean
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about average, median and mode?
Mean of 4, 8, 12 is 8
Recall point 2: what did you learn about average, median and mode?
Median of 3, 7, 9 is 7
Recall point 3: what did you learn about average, median and mode?
Mode of 2, 2, 5 is 2
Recall point 4: what did you learn about average, median and mode?
An outlier can change the mean
6. Profit and loss
Profit = selling price minus cost price. Profit percentage is normally calculated relative to cost price, not selling price. Discount percentage is normally based on marked price.
Key points
- If cost is 100 and sale is 120 profit is 20
- 20 profit on cost 100 means 20% profit
- If cost 100 and sale 90 loss is 10%
- Discount of 10% on 500 is 50
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about profit and loss?
If cost is 100 and sale is 120 profit is 20
Recall point 2: what did you learn about profit and loss?
20 profit on cost 100 means 20% profit
Recall point 3: what did you learn about profit and loss?
If cost 100 and sale 90 loss is 10%
Recall point 4: what did you learn about profit and loss?
Discount of 10% on 500 is 50
7. Simple and compound interest
Simple interest is P×R×T/100 when annual rate R is a percentage and time is years. Compound interest adds interest to principal in subsequent periods.
Key points
- P=1000 R=10 T=2 gives SI 200
- Principal is starting amount
- Compound interest can earn interest on earlier interest
- Always check time period and compounding frequency
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about simple and compound interest?
P=1000 R=10 T=2 gives SI 200
Recall point 2: what did you learn about simple and compound interest?
Principal is starting amount
Recall point 3: what did you learn about simple and compound interest?
Compound interest can earn interest on earlier interest
Recall point 4: what did you learn about simple and compound interest?
Always check time period and compounding frequency
8. Time speed distance
Speed = distance/time and distance = speed×time. Convert time to hours when speed is km/h. For equal distances at different speeds, average speed is generally not the arithmetic average.
Key points
- 60 km/h for 2h gives 120 km
- Half an hour equals 0.5 hours
- 120 km in 3 hours means 40 km/h
- For return trips use total distance divided by total time
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about time speed distance?
60 km/h for 2h gives 120 km
Recall point 2: what did you learn about time speed distance?
Half an hour equals 0.5 hours
Recall point 3: what did you learn about time speed distance?
120 km in 3 hours means 40 km/h
Recall point 4: what did you learn about time speed distance?
For return trips use total distance divided by total time
9. Work and time
If one person can finish in 6 days, their one-day work is 1/6. For two workers, add their rates, then invert to find total time. This assumes constant independent work rates.
Key points
- Worker A in 6 days does 1/6 per day
- Worker B in 3 days does 1/3 per day
- Combined rate 1/2 gives 2 days
- Work and time problems use rates, not sum of durations
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about work and time?
Worker A in 6 days does 1/6 per day
Recall point 2: what did you learn about work and time?
Worker B in 3 days does 1/3 per day
Recall point 3: what did you learn about work and time?
Combined rate 1/2 gives 2 days
Recall point 4: what did you learn about work and time?
Work and time problems use rates, not sum of durations
10. Algebra and equations
Solve equations by performing the same operation on both sides. In 3x + 4 = 19, subtract 4 then divide by 3 to get x=5. Check by substitution.
Key points
- 2x + 3 = 13 gives x=5
- 5x = 35 gives x=7
- x/4 = 3 gives x=12
- The equals sign means two expressions have equal value
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about algebra and equations?
2x + 3 = 13 gives x=5
Recall point 2: what did you learn about algebra and equations?
5x = 35 gives x=7
Recall point 3: what did you learn about algebra and equations?
x/4 = 3 gives x=12
Recall point 4: what did you learn about algebra and equations?
The equals sign means two expressions have equal value
11. Geometry fundamentals
Rectangle area is length×width; triangle area is half base×height. Circles use πr² for area and 2πr for circumference. Distinguish area (square units) from perimeter (linear units).
Key points
- Rectangle 4 by 5 has area 20
- Square of side 6 has perimeter 24
- Triangle base 8 height 5 has area 20
- Circle diameter is twice radius
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about geometry fundamentals?
Rectangle 4 by 5 has area 20
Recall point 2: what did you learn about geometry fundamentals?
Square of side 6 has perimeter 24
Recall point 3: what did you learn about geometry fundamentals?
Triangle base 8 height 5 has area 20
Recall point 4: what did you learn about geometry fundamentals?
Circle diameter is twice radius
12. Data interpretation and logical reasoning
Read graph labels and units before calculating. For percentages, identify the correct base. In logical patterns, test successive differences, ratios and repeated operations.
Key points
- Difference between 30 and 45 is 15
- A table may use thousands or millions as units
- The sequence 2 4 8 16 doubles each time
- To compute percentage change use original value as base
Active recall
Close the answers and explain these facts without looking at the options. Then practise the matching subject in the MCQ dashboard.
Recall point 1: what did you learn about data interpretation and logical reasoning?
Difference between 30 and 45 is 15
Recall point 2: what did you learn about data interpretation and logical reasoning?
A table may use thousands or millions as units
Recall point 3: what did you learn about data interpretation and logical reasoning?
The sequence 2 4 8 16 doubles each time
Recall point 4: what did you learn about data interpretation and logical reasoning?
To compute percentage change use original value as base
Study references
These are recommended sources for checking and extending the notes; they do not certify every individual item above.