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Numerical Ability - CIVIL SERVICE Study Notes

01

MDAS / PEMDAS Rule (Order of Operations)

Order of Operations is the foundation of all numerical problems. In the Philippine Civil Service Exam, Strict PEMDAS is followed: multiplication and division are equal, addition and subtraction are equal.

Priority	Operation	Rule
1st	Parenthesis ()	Solve inside brackets first.
2nd	Exponents x² / Roots	Resolve powers and square roots.
3rd	Multiplication / Division	Left to Right Rule!
4th	Addition / Subtraction	Left to Right Rule!

Solve: 12 \div 3 \times 4 + (10 - 2²)

1
Resolve Parenthesis: 10 - 4 = 6.
2
Multiplication & Division: 12 ÷ 3 = 4. Then 4 × 4 = 16.
3
Final Addition: 16 + 6 = 22.

⚠️ THE PEMDAS TRAP

"A" does not always come before "S". Whichever is on the LEFT side goes first. Always solve tied ranks from left to right.

02

Integers Mastery (Signed Numbers)

Mastering positive and negative numbers is critical. Think of them as Pera (Money) and Utang (Debt).

Addition Rules

Same Sign: Add values, keep sign.
-5 + -3 = -8
Different Sign: Subtract smaller from bigger, copy bigger sign.
-10 + 3 = -7

Multiplication Rules

Same Sign: Always POSITIVE.
Different Sign: Always NEGATIVE.

🚀 PR Hack: Love Logic

Friend (+) of my Friend (+) is my Friend (+). Enemy (-) of my Enemy (-) is my Friend (+). Friend (+) of my Enemy (-) is my Enemy (-).

03

Fractions: The Butterfly Method

Solve addition and subtraction of fractions in seconds without the long LCD process.

Solve: 25 + 13

1
Cross-Multiply: 2 × 3 = 6. 5 × 1 = 5.
2
Top Sum: 6 + 5 = 11.
3
Bottom Product: 5 × 3 = 15.
4
Answer: 11/15.

🚀 PR Hack: K.C.F. Division

For Division: Keep (1st), Change (÷ to ×), Flip (2nd). Then multiply straight!

04

Percentage Triangle (P-B-R)

Master the relationship between Percentage (Part), Base (Whole), and Rate (%).

P = B × R

Percentage

B = P / R

Base

R = P / B

Rate (%)

20% of 1,500 = 300

🚀 PR Hack: The "IS" over "OF" Rule

To find the Rate (%), use IS / OF. 5 IS what percent OF 20? → 5/20 = 1/4 = 25%. The number after "OF" is usually the Base.

05

Ratio & Proportion Mastery

There are three major types of proportions used in word problems.

1. Direct Proportion

When one goes up, the other goes up.
Ex: 3 shirts cost ₱600. How much for 7 shirts?

3 : 600 = 7 : x \to 3x = 4200 \to x = ₱1,400.

2. Inverse Proportion

When one goes up, the other goes down.
Ex: 5 workers finish in 4 days. How many days for 10 workers?

(5 \times 4) = (10 \times x) \to 20 = 10x \to x = 2 days.

3. Partitive Proportion

Dividing a whole into specific ratio parts.
Ex: Divide ₱1,200 in ratio 1:2:3.

Total units = 6. 1200 / 6 = 200 per unit. Parts are 200, 400, 600.

06

Work Problems (The Speed Hack)

Don't use complex fractions for two-person work problems. Use the Product/Sum shortcut.

Combined Time = (Time A \times Time B) / (Time A + Time B)

Problem: A paints a wall in 2 hours. B paints it in 3 hours. How long together?

Logic: (2 × 3) / (2 + 3) = 6 / 5 = 1.2 hours.

⚠️ THE LEAK TRAP

If a pipe is leaking (emptying), use MINUS in the reciprocal equation: 1/Total = 1/Fill - 1/Leak.

07

Motion: Distance, Speed, Time

Laging tandaan ang DST Triangle. Distance is always on Top.

Direction	Speed Logic	Combined Speed
Towards each other	ADD speeds	Fast closing gap
Same direction (Chasing)	SUBTRACT speeds	Slowly closing gap

The Average Speed Trap: A car travels at 40 km/h and returns at 60 km/h. Average speed is NOT 50!

Hack: [2 × 40 × 60] / [40 + 60] = 48 km/h.

08

Age Problems: The Table Strategy

Solve age problems by setting up a table for Past (-), Present (x), and Future (+).

Person	Past (-5)	Present	Future (+10)
Maria	x - 5	x	x + 10
Father	3x - 5	3x	3x + 10

🚀 PR Hack: Fixed Age Gap

The difference in ages NEVER changes. If you are 5 years younger than your brother, you will always be 5 years younger forever.

09

Mixture Problems (The Box Method)

Mixing concentrations? Use the Box Method to visualize the parts and the whole.

(V1 \times C1) + (V2 \times C2) = (Total Vol \times Final Conc)

⚠️ CONCENTRATION CLUES

Pure Water = 0%. Pure Alcohol/Acid = 100% (1.0). Always convert percentages to decimals before solving.

10

Simple Interest & Loan Problems

I = P \times r \times t

!
Principal (P): The original amount borrowed or invested.
!
Rate (r): The annual interest rate (e.g., 5% → 0.05).
!
Time (t): Must be in YEARS.

🚀 PR Hack: Discount Series

20% off plus 10% off is not 30%. Multiply the paying parts: 0.8 × 0.9 = 0.72. You pay 72%, so the discount is 28%.

11

Probability (AND vs OR Rules)

P = (Desired) / (Total Outcomes)

The "AND" Rule

Multiple events together? MULTIPLY the probabilities.

The "OR" Rule

Either of two events? ADD the probabilities.

12

Consecutive Integers (3-Second Shortcut)

Example: The sum of 3 consecutive integers is 33. Find the numbers.

33 \div 3 = 11

The answer is ALWAYS the Middle Number.

10, 11, 12

13

Number Series Patterns

Pattern	Logic	Example
Arithmetic	Addition/Sub	5, 10, 15, 20 (+5)
Geometric	Mult/Div	3, 6, 12, 24 (×2)
Fibonacci	Sum of prev 2	1, 1, 2, 3, 5, 8, 13
Alternating	Double skip	10, 2, 12, 4, 14 (Skip-1!)

14

Clock Hand Angle (30-5.5 Hack)

Quickly calculate the angle between hands without a drawing.

Angle = |30H - 5.5M|

Example: At 2:30, what is the angle?

Calc: |30(2) - 5.5(30)| = |60 - 165| = 105°.

15

Data Interpretation Strategy

Reading charts and graphs under time pressure. Focus on these 3 keys:

1
Scan the Labels: Read the Title, Axis, and Legend before looking at data.
2
Slope Logic: The steepest line means the fastest change (growth or drop).
3
Estimate: Accuracy is secondary to speed. Round numbers to solve faster.

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