Basic Statistics & Consecutive Numbers - Revision Notes
This chapter covers fundamental statistical concepts and consecutive number problems commonly found in Philippine civil service examinations and college entrance tests. You'll learn to calculate averages, solve missing term problems, work with weighted averages, and tackle consecutive number sequences. These skills are essential for success in UPCAT, CSE, LET, NLE, NMAT, ACET, and USTET examinations.
Basic Statistical Measures
Formulas
Average = Sum of all terms ÷ Number of terms
Example
Quiz scores: 16, 14, 22, 19, 17. Average = (16+14+22+19+17) ÷ 5 = 88 ÷ 5 = 17.6
Variables
Average = mean value, Sum = total of all values, Number = count of items
Application
Finding the central tendency of a dataset
Missing term = Sum of original terms - Sum of remaining terms
Example
Average height of 5 people is 150cm. Four heights are 153, 150, 151, 152. Fifth height = (150×5) - (153+150+151+152) = 750 - 606 = 144cm
Variables
Missing term = unknown value, Sum of original = total before removal, Sum of remaining = total after removal
Application
Finding unknown values in a dataset when average is known
Sum of all terms = Average × Number of terms
Example
If average score of 5 games is 147, total score = 147 × 5 = 735
Variables
Sum = total of all values, Average = mean value, Number = count of items
Application
Finding total when average and count are known
Weighted Average = Sum of weighted terms ÷ Total number of terms
Example
26 boys (40kg avg) + 14 girls (35kg avg): Weighted avg = (26×40 + 14×35) ÷ (26+14) = 1530 ÷ 40 = 38.25kg
Variables
Weighted terms = each value multiplied by its weight/frequency, Total number = sum of all weights
Application
When different values have different importance or frequency
Exam tips
- Always write down what you're looking for before starting calculations
- Use the sum formula (Sum = Average × Number) as your main tool
- Check if your answer makes logical sense compared to given values
- Practice mental math for common calculations to save time
Key points
- Average (mean) is the sum of all values divided by the number of values
- Use the sum formula to find missing terms or total values
- Weighted average considers the importance or frequency of each value
- Always identify what the problem is asking for before solving
Definitions
Average (Mean)
The sum of all values in a dataset divided by the number of values
Importance
Most commonly tested statistical measure in Philippine exams
Weighted Average
An average that takes into account the relative importance or frequency of each value
Importance
Essential for solving complex average problems with different group sizes
Sum Formula
The relationship between average, sum, and number of terms used to find missing values
Importance
Key technique for solving missing term problems in examinations
Common mistakes
- Forgetting to convert mixed numbers to improper fractions before calculating
- Not identifying whether the problem asks for a missing term or a new average
- Confusing simple average with weighted average when groups have different sizes
- Making arithmetic errors when adding or subtracting large numbers
Unit Conversion
Formulas
New Value = Original Value × Conversion Factor
Example
644 days to weeks: 644 × (1 week/7 days) = 644/7 = 92 weeks
Variables
New Value = result in desired units, Original Value = given measurement, Conversion Factor = ratio between units
Application
Converting between different units of measurement
Exam tips
- Memorize key conversion factors: 12 inches = 1 foot, 3 feet = 1 yard, 7 days = 1 week
- Always set up conversion as a fraction to ensure proper unit cancellation
- Double-check that your final answer has the correct units
- Practice with both exact and approximate conversions
Key points
- Master common conversion factors used in Philippine exams
- Use dimensional analysis to ensure correct unit conversions
- Pay attention to whether the answer should be a fraction or decimal
- Practice converting between metric and imperial systems
Definitions
Conversion Factor
A ratio that expresses how many of one unit equal another unit
Importance
Essential for accurate unit conversions in science and math problems
Dimensional Analysis
A method using conversion factors to change from one unit to another
Importance
Prevents errors and ensures correct unit cancellation in complex conversions
Common mistakes
- Using incorrect conversion factors (e.g., 1 foot = 10 inches instead of 12)
- Forgetting to cancel units properly in dimensional analysis
- Not converting mixed numbers to improper fractions before multiplying
- Rounding too early in multi-step conversions
Fraction Word Problems
Formulas
Whole = Part ÷ Fraction
Example
Tank is 3/8 full, needs 60L more. Capacity = 60 ÷ (1 - 3/8) = 60 ÷ (5/8) = 96L
Variables
Whole = complete amount, Part = known portion, Fraction = decimal/fraction representing the part
Application
Finding total capacity when a fractional part is known
Part = Whole × Fraction
Example
Inherited 6/7 of estate, sold 2/3 of share. Sold = (6/7) × (2/3) = 4/7 of total estate
Variables
Part = portion being calculated, Whole = total amount, Fraction = decimal/fraction representing the portion
Application
Finding a specific portion of a known total
Exam tips
- Draw diagrams or use visual representations for complex fraction problems
- Always check if fractions need to add up to 1 (whole)
- Practice finding common denominators quickly
- Verify answers by substituting back into the original problem
Key points
- Identify what fraction represents the whole in each problem
- Use addition and subtraction to find missing parts
- Convert mixed numbers to improper fractions for easier calculation
- Set up equations when dealing with relationships between parts
Definitions
Fractional Part
A portion of a whole expressed as a fraction
Importance
Foundation for solving complex word problems involving parts and wholes
Complementary Fraction
The remaining fraction that adds with given fractions to make a whole (1)
Importance
Used to find missing portions in word problems
Common mistakes
- Adding fractions without finding common denominators
- Forgetting that fractions in word problems often don't add up to 1
- Misinterpreting 'of' as addition instead of multiplication
- Not converting mixed numbers before performing operations
Consecutive Numbers
Formulas
Sum of n consecutive integers = n × (first term + last term) ÷ 2
Example
Sum of 1, 2, 3, 4, 5 = 5 × (1 + 5) ÷ 2 = 5 × 6 ÷ 2 = 15
Variables
n = number of terms, first term = smallest number, last term = largest number
Application
Finding sum of consecutive number sequences
For consecutive integers: Let first = n, then numbers are n, n+1, n+2, ...
Example
Three consecutive integers sum to 24: n + (n+1) + (n+2) = 24, so 3n + 3 = 24, n = 7
Variables
n = first integer in sequence
Application
Setting up equations for consecutive integer problems
Exam tips
- Always define your variable clearly (what does n represent?)
- Check your answer by substituting back into the original conditions
- Practice recognizing consecutive number patterns in word problems
- Use algebraic methods rather than guess-and-check for efficiency
Key points
- Consecutive integers differ by 1: n, n+1, n+2, etc.
- Consecutive even numbers differ by 2: n, n+2, n+4, etc.
- Consecutive odd numbers also differ by 2: n, n+2, n+4, etc.
- Use algebraic expressions to represent consecutive numbers
Definitions
Consecutive Numbers
Numbers that follow each other in order without gaps
Importance
Common in algebra problems and number theory questions
Arithmetic Sequence
A sequence where consecutive terms have a constant difference
Importance
Consecutive numbers are the simplest form of arithmetic sequences
Common mistakes
- Using wrong increments (using +1 for even numbers instead of +2)
- Setting up incorrect equations for word problems
- Forgetting to check if solutions are actually consecutive
- Confusing consecutive integers with consecutive even/odd numbers
Connections
- Statistics connects to data analysis and probability in advanced math courses
- Unit conversion is essential for physics and chemistry calculations
- Fraction problems appear in percentage, ratio, and proportion topics
- Consecutive number problems relate to arithmetic sequences and series
- Average calculations are used in grade computation and performance analysis
- These concepts appear across multiple sections of standardized tests
Quick review questions
Find the average of 16, 14, 22, 19, and 17.
Answer
17.6
Sum = 16+14+22+19+17 = 88. Average = 88÷5 = 17.6
A tank is 7/9 full. After removing 27L, it's 4/9 full. What's the tank capacity?
Answer
81 liters
Let x = capacity. (7/9)x - 27 = (4/9)x. Solving: (3/9)x = 27, so x = 81L
Convert 644 days to weeks.
Answer
92 weeks
644 days × (1 week/7 days) = 644/7 = 92 weeks
If 1/3 take public transport and 2/5 drive cars, what fraction walk?
Answer
4/15
Transport + Drive = 5/15 + 6/15 = 11/15. Walk = 1 - 11/15 = 4/15
Three consecutive integers sum to 36. Find the middle number.
Answer
12
Let n, n+1, n+2 be the numbers. 3n+3=36, so 3n=33, n=11. Middle = n+1 = 12
Diagrams
Basic Statistics & Consecutive Numbers Overview
Comprehensive overview showing the four main topics and their key components, helping students understand how all concepts relate to each other.
Average Problem Solving Process
Step-by-step flowchart showing the decision-making process for different types of average problems, helping students identify the correct approach.
Unit Conversion Process
Systematic approach to unit conversion problems, emphasizing the importance of proper unit cancellation and verification.
Fraction Word Problem Strategy
Decision tree for approaching fraction word problems, helping students determine whether they need to find a part, whole, or missing fraction.
Consecutive Number Problem Steps
Sequential process showing the interaction between student and problem-solving steps for consecutive number problems.
Exam strategy
Focus on pattern recognition and formula memorization. Practice identifying problem types quickly - is it asking for an average, missing term, or weighted average? Set up equations systematically for word problems. Master basic arithmetic operations with fractions to avoid calculation errors. Time management is crucial: solve easier problems first, then return to complex multi-step problems. Always verify your answers make logical sense in the context of the problem.