Civil Service Exam (Subprofessional) Numerical Ability — Decimals & Scientific NotationCheat Sheet
Decimals & Scientific Notation cheat sheet for Civil Service Exam (Subprofessional) aspirants. If you could only take one sheet of paper into your review session, this is what it would look like. Civil Service Commission (CSC)'s most-tested concepts, all in one place.
Exam context
The Career Service Examination — Subprofessional Level is conducted by Civil Service Commission (CSC) and is scheduled for Bi-annual — March and August 2026. The Numerical Ability subtest is marked as "~25% weightage" in the official pattern, and Decimals & Scientific Notation appears in position 3rd of 9 in the Civil Service Exam (Subprofessional) Numerical Ability review rotation. Passing mark: 80%. Recent Civil Service Exam (Subprofessional) 2026 papers have drawn roughly 17 questions from this subject.
Decimals & Scientific Notation - Cheat sheet
Your last-minute revision companion for mastering decimals, percentages, and scientific notation — essential for all Philippine college entrance exams
Sections
Formulas
Formula
When multiplying decimals: Count total decimal places in factors = decimal places in product
Meaning
0.25 × 0.5 = 0.125 (2 + 1 = 3 decimal places)
Watch Out
Students forget to count decimal places and place decimal point incorrectly
When To Use
Any decimal multiplication problem
Formula
When dividing decimals: Move decimal point right in both dividend and divisor equally
Meaning
0.56 ÷ 0.07 becomes 56 ÷ 7 = 8
Watch Out
Moving decimal point different number of places in dividend vs divisor
When To Use
When divisor is a decimal
Common Values
Value
0.5
Symbol
50%
Quantity
1/2
Value
0.25
Symbol
25%
Quantity
1/4
Value
0.75
Symbol
75%
Quantity
3/4
Section Title
Decimal Operations
Important Facts
- To add/subtract decimals: Align decimal points vertically
- Add zeros to the right if needed for alignment
- Decimal point in answer goes directly below aligned decimal points
- To convert fraction to decimal: Divide numerator by denominator
- To convert decimal to percent: Move decimal point 2 places right, add %
Key Definitions
Term
Decimal
Example
0.75 = 75/100 = 3/4
Definition
A number expressing fractions in base 10 using decimal point
Term
Terminating Decimal
Example
0.25, 0.125, 0.5
Definition
Decimal that ends (finite number of decimal places)
Term
Repeating Decimal
Example
0.333... = 1/3
Definition
Decimal with digits that repeat infinitely
Diagrams To Know
- Place value chart for decimals
- Decimal number line
Formulas
Formula
Percentage = Rate × Base
Meaning
P = R × B (Triangle method: cover what you're solving for)
Watch Out
Convert percentage to decimal first: 25% = 0.25
When To Use
When given rate (%) and base (whole amount)
Formula
Rate = Percentage ÷ Base × 100%
Meaning
R = (P ÷ B) × 100% to get percentage form
Watch Out
Forgetting to multiply by 100% for percentage form
When To Use
When asked 'what percent' or 'what part'
Formula
Base = Percentage ÷ Rate
Meaning
B = P ÷ R (convert rate to decimal first)
Watch Out
Using percentage form instead of decimal form of rate
When To Use
When asked 'of what number' or finding the whole
Common Values
Value
0.1
Symbol
1/10
Quantity
10%
Value
0.2
Symbol
1/5
Quantity
20%
Value
0.5
Symbol
1/2
Quantity
50%
Section Title
Percentage, Base, and Rate (PBR)
Important Facts
- When rate < 100%, percentage < base
- When rate = 100%, percentage = base
- When rate > 100%, percentage > base
- Percentage is associated with word 'is'
- Base is associated with word 'of'
- Rate is the number with % sign or word 'percent'
Key Definitions
Term
Percentage
Example
25 is 50% of 50 (25 = percentage)
Definition
The result when a number is multiplied by a percent (associated with 'is')
Term
Base
Example
25% of 200 (200 = base)
Definition
The whole amount from which percent is taken (associated with 'of')
Term
Rate
Example
25% of 200 = 50 (25% = rate)
Definition
The percent itself, ratio of percentage to base
Diagrams To Know
- PBR triangle diagram
- Percentage-decimal-fraction conversion chart
Formulas
Formula
Cross multiplication: a:b = x:y → ay = bx
Meaning
Product of extremes = Product of means
Watch Out
Confusing which terms are extremes vs means
When To Use
Solving proportion problems with missing term
Section Title
Ratio and Proportion
Important Facts
- Ratios can be written three ways: word form, colon form, fraction form
- In proportion a:b = x:y, a and y are extremes, b and x are means
- Direct proportion: as one increases, other increases proportionally
- Inverse proportion: as one increases, other decreases proportionally
- Partitive proportion: dividing whole into more than two parts
Key Definitions
Term
Ratio
Example
3:5 or 3/5 or '3 is to 5'
Definition
Comparison of two quantities of same unit
Term
Rate
Example
60 km/hr, ₱5 per orange
Definition
Comparison of two quantities with different units
Term
Proportion
Example
1:2 = 5:10
Definition
Statement that two ratios are equal
Formulas
Formula
Scientific notation: a × 10ⁿ where 1 ≤ a < 10
Meaning
a = coefficient (1-9.999...), n = exponent (power of 10)
Watch Out
Coefficient must be between 1 and 10 (not 0.5 or 15)
When To Use
Expressing very large or very small numbers
Formula
Large numbers: Move decimal left, positive exponent
Meaning
45,000 = 4.5 × 10⁴ (moved 4 places left)
Watch Out
Counting decimal places incorrectly
When To Use
Numbers ≥ 10
Formula
Small numbers: Move decimal right, negative exponent
Meaning
0.0045 = 4.5 × 10⁻³ (moved 3 places right)
Watch Out
Using positive exponent for small numbers
When To Use
Numbers < 1
Common Values
Value
3 × 10⁸ m/s
Symbol
c
Quantity
Speed of light
Value
6.022 × 10²³
Symbol
Nₐ
Quantity
Avogadro's number
Section Title
Scientific Notation
Important Facts
- Used for very large numbers (astronomy, physics)
- Used for very small numbers (molecular, atomic scale)
- Makes calculations with extreme numbers manageable
- Standard form in scientific calculators
- Essential for physics and chemistry problems
Key Definitions
Term
Scientific Notation
Example
3.2 × 10⁵ = 320,000
Definition
Method of writing numbers as a × 10ⁿ
Term
Coefficient
Example
In 2.5 × 10³, coefficient is 2.5
Definition
The decimal number part (must be 1 ≤ a < 10)
Term
Exponent
Example
In 2.5 × 10³, exponent is 3
Definition
The power of 10 (positive for large, negative for small)
Diagrams To Know
- Powers of 10 scale
- Scientific notation conversion steps
Must Remember
- PBR Triangle: P = R × B, R = P/B × 100%, B = P/R
- Convert percentage to decimal by moving decimal 2 places left
- Align decimal points when adding/subtracting decimals
- Count decimal places in factors to place decimal in product
- Scientific notation: coefficient between 1 and 10, then × 10ⁿ
- Large numbers → positive exponent, small numbers → negative exponent
- Cross multiply in proportions: a:b = x:y → ay = bx
- Percentage associated with 'is', base associated with 'of'
- 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%
- When rate < 100%, percentage < base; when rate > 100%, percentage > base
Last Minute Tips
- In PBR problems, identify keywords: 'is' for percentage, 'of' for base, '%' for rate
- For scientific notation, count zeros to determine exponent: 45,000 has 3 zeros after 45
- In decimal division, make divisor whole number first, then divide normally
- Cross multiplication shortcut: multiply diagonally across the equal sign
- Convert mixed numbers to improper fractions before percentage calculations
Comparison Tables
Rows
Values
- 0.5
- 50%
- 1/2
Property
Half
Values
- 0.25
- 25%
- 1/4
Property
Quarter
Values
- 0.1
- 10%
- 1/10
Property
One-tenth
Values
- 0.75
- 75%
- 3/4
Property
Three-quarters
Columns
- Decimal
- Percentage
- Fraction
Table Title
Decimal vs Percentage vs Fraction
Rows
Values
- Both increase/decrease together
- One increases, other decreases
Property
Relationship
Values
- y = kx (k constant)
- xy = k (k constant)
Property
Formula
Values
- Distance vs time (constant speed)
- Speed vs time (constant distance)
Property
Example
Columns
- Direct Proportion
- Inverse Proportion
Table Title
Direct vs Inverse Proportion
Previous chapter
Fractions — Operations, Conversion & Comparison
Next chapter
Ratio, Proportion & Percentage
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