Decimals & Scientific Notation - Cheat Sheet
Your last-minute revision companion for mastering decimals, percentages, and scientific notation — essential for all Philippine college entrance exams
Decimal Operations
Formulas
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
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
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
Decimal
Example
0.75 = 75/100 = 3/4
A number expressing fractions in base 10 using decimal point
Terminating Decimal
Example
0.25, 0.125, 0.5
Decimal that ends (finite number of decimal places)
Repeating Decimal
Example
0.333... = 1/3
Decimal with digits that repeat infinitely
Diagrams to know
- Place value chart for decimals
- Decimal number line
Percentage, Base, and Rate (PBR)
Formulas
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)
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'
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%
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
Percentage
Example
25 is 50% of 50 (25 = percentage)
The result when a number is multiplied by a percent (associated with 'is')
Base
Example
25% of 200 (200 = base)
The whole amount from which percent is taken (associated with 'of')
Rate
Example
25% of 200 = 50 (25% = rate)
The percent itself, ratio of percentage to base
Diagrams to know
- PBR triangle diagram
- Percentage-decimal-fraction conversion chart
Ratio and Proportion
Formulas
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
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
Ratio
Example
3:5 or 3/5 or '3 is to 5'
Comparison of two quantities of same unit
Rate
Example
60 km/hr, ₱5 per orange
Comparison of two quantities with different units
Proportion
Example
1:2 = 5:10
Statement that two ratios are equal
Scientific Notation
Formulas
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
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
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
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
Scientific Notation
Example
3.2 × 10⁵ = 320,000
Method of writing numbers as a × 10ⁿ
Coefficient
Example
In 2.5 × 10³, coefficient is 2.5
The decimal number part (must be 1 ≤ a < 10)
Exponent
Example
In 2.5 × 10³, exponent is 3
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
Decimal vs Percentage vs Fraction
| Decimal | Percentage | Fraction | |
|---|---|---|---|
| Half | 0.5 | 50% | 1/2 |
| Quarter | 0.25 | 25% | 1/4 |
| One-tenth | 0.1 | 10% | 1/10 |
| Three-quarters | 0.75 | 75% | 3/4 |
Direct vs Inverse Proportion
| Direct Proportion | Inverse Proportion | |
|---|---|---|
| Relationship | Both increase/decrease together | One increases, other decreases |
| Formula | y = kx (k constant) | xy = k (k constant) |
| Example | Distance vs time (constant speed) | Speed vs time (constant distance) |
Diagrams
Decimals and Scientific Notation Overview
Complete overview of all major topics showing relationships between decimal operations, percentages, ratios, and scientific notation
PBR Problem Solving Process
Step-by-step flowchart for solving any PBR problem systematically
Scientific Notation Conversion Process
Decision tree for converting any number to proper scientific notation format