Applied Problems Subtest in Factors, Multiples, and Patterns

Complete applied problems subtest in factors, multiples, and patterns with detailed solutions and visual infographics.

Comprehensive Summary: Applied Problems Subtest

Key Definitions

Factor: A number that divides another number evenly without remainder.

Multiple: The product of a number and any whole number.

Pattern: A sequence that follows a predictable rule.

Applied Problem: A real-world situation that requires mathematical thinking to solve.

Core Rules & Laws

Factor rule: If a×b=n, then a and b are factors of n.

Multiple rule: Multiples of n are n, 2n, 3n, 4n, etc.

Pattern rule: Find the relationship between position and value.

Word problem rule: Identify keywords to determine the operation needed.

Methods & Procedures

Factor finding: Systematically divide by numbers 1 to √n.

LCM finding: List multiples until finding the first common one.

Pattern recognition: Look for consistent differences or ratios.

Problem solving: Read, plan, solve, check.

Tips & Tricks

Factor tip: Always include 1 and the number itself.

Multiple tip: Skip counting helps identify multiples.

Pattern tip: Make a table to visualize relationships.

Problem tip: Draw pictures or make lists for complex problems.

1
Read Carefully
2
Identify Keywords
3
Choose Method
4
Solve Systematically
5
Check Answer
KEY TAKEAWAY: Applied problems connect math concepts to real-world situations. Always identify what the problem is asking before choosing your approach!
Subtest: Basic Applied Problems
1 Basic Factors Easy
SUBTEST ITEM Example 1
FACTOR PROBLEM Tom has 20 marbles and wants to put them in equal groups for his friends. What are all the possible ways he can group the marbles? Which grouping gives him the most groups?
Definition:

Factors: Numbers that divide evenly into another number. For example, the factors of 20 are numbers that divide 20 without leaving a remainder.

Method to find factors:
  1. Start with 1 and the number itself
  2. Check each number up to the square root of the number
  3. If a number divides evenly, add both the divisor and the quotient
  4. List all unique factors
Problem
Group 20 marbles equally
Find factors of 20
1, 2, 4, 5, 10, 20
Step 1: Find all factors of 20

1×20=20, 2×10=20, 4×5=20

So factors are: 1, 2, 4, 5, 10, 20

Step 2: Interpret factors as possible groupings

Each factor represents a possible number of groups or marbles per group

1 group of 20 marbles, 2 groups of 10 marbles, 4 groups of 5 marbles, etc.

Step 3: List all possible groupings

6 different ways: 1×20, 2×10, 4×5, 5×4, 10×2, 20×1

Step 4: Identify grouping with most groups

The grouping with the largest number as groups gives the most groups

20 groups of 1 marble each gives the most groups (20 groups)

6 different groupings possible, 20 groups of 1 gives most groups
Final answer:

Tom can group his 20 marbles in 6 different ways:

1 group of 20, 2 groups of 10, 4 groups of 5, 5 groups of 4, 10 groups of 2, or 20 groups of 1

The grouping with the most groups is 20 groups of 1 marble each.

Applied rules:

• Factor identification: Find all numbers that divide 20 evenly

• Application: Each factor pair represents a possible grouping

• Optimization: The largest factor as number of groups gives the most groups

🔮
Factor Tree
20
4×5
2×2×5
📊
Possible Groupings
1×20
2×10
4×5
5×4
10×2
2 Basic Multiples Medium
SUBTEST ITEM Example 2
MULTIPLE PROBLEM A school bus comes every 10 minutes and a city bus comes every 15 minutes. If both buses arrive at the station at 9:00 AM, when will they next arrive together? How many times will each bus have come by that time?
Definition:

Least Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers.

Problem
School bus every 10 min, City bus every 15 min
Find LCM of 10 and 15
LCM(10, 15) = 30
Next arrival
9:00 AM + 30 min = 9:30 AM
Step 1: List multiples of each number

Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90...

Multiples of 15: 15, 30, 45, 60, 75, 90, 105...

Step 2: Identify the least common multiple

The smallest number that appears in both lists is 30

So LCM(10, 15) = 30

Step 3: Calculate the next simultaneous arrival

They will both arrive together again in 30 minutes

9:00 AM + 30 minutes = 9:30 AM

Step 4: Calculate number of arrivals by that time

School bus: 30 ÷ 10 = 3 times (but we started at 0, so 3 additional arrivals)

City bus: 30 ÷ 15 = 2 times (but we started at 0, so 2 additional arrivals)

Total: School bus arrives 4 times (including 9:00), City bus arrives 3 times (including 9:00)

Both will arrive together at 9:30 AM
Final answer:

The buses will next arrive together at 9:30 AM.

By that time, the school bus will have arrived 4 times (at 9:00, 9:10, 9:20, 9:30) and the city bus will have arrived 3 times (at 9:00, 9:15, 9:30).

Applied rules:

• LCM concept: Find the smallest common multiple of given numbers

• Time calculation: Add LCM minutes to the starting time

• Counting principle: Divide total time by interval to count occurrences

🚌
School Bus Schedule
9:00
9:10
9:20
9:30
9:40
🚍
City Bus Schedule
9:00
9:15
9:30
9:45
9:60
3 Basic Patterns Easy
SUBTEST ITEM Example 3
PATTERN PROBLEM A pattern of stars follows this sequence: 5, 10, 15, 20, 25, ... If the pattern continues, what will be the 15th number in the sequence? How many stars are needed for the first 6 figures?
Definition:

Arithmetic sequence: A sequence where each term increases by a constant amount called the common difference.

Pattern
5, 10, 15, 20, 25, ...
Rule
5n
15th term
5(15) = 75
Step 1: Identify the pattern rule

5 = 1×5, 10 = 2×5, 15 = 3×5, 20 = 4×5, 25 = 5×5

The nth term is 5n

Step 2: Find the 15th term

Using the rule: 5(15) = 75

Step 3: Find sum of first 6 terms

First 6 terms: 5, 10, 15, 20, 25, 30

Sum = 5 + 10 + 15 + 20 + 25 + 30 = 105

15th term: 75, Sum of first 6: 105
Final answer:

The 15th number in the sequence is 75.

The first 6 figures require 105 stars total.

Applied rules:

• Pattern recognition: Identify the rule that generates the sequence

• Formula application: Use the rule to find any term in the sequence

• Series summation: Add consecutive terms to find totals

⭐
Sequence Pattern
5
10
15
20
25

Rule: 5n

🎯
Term Calculation
nth term
5n
5(15) = 75
Subtest: Intermediate Applied Problems
Factor × Factor = Product
Factors Relationship
Factor Identification
1, 2, 4, 5, 10, 20
Factors of 20
Multiples
10, 20, 30, 40...
Multiples of 10
Pattern Rule
5n
nth term
Key definitions:

Factors: Numbers that divide evenly into another number

Multiples: Products of a number and any integer

Patterns: Sequences that follow a predictable rule

Problem-solving methodology:
  1. Read carefully: Understand what the problem is asking
  2. Identify key words: Look for terms like "arrange," "together," "sequence," "pattern"
  3. Choose method: Determine if it's a factor, multiple, or pattern problem
  4. Solve systematically: Apply the appropriate mathematical operation
  5. Check answer: Verify your solution makes sense
Tip 1: When finding factors, work systematically from 1 up to the square root.
Tip 2: For LCM problems, list multiples until you find a common one.
Tip 3: For patterns, look for the relationship between position and value.

Tip 4: Always verify your answer by substituting back into the original problem.
Subtest: Advanced Applied Problems
4 Greatest Common Factor Medium
SUBTEST ITEM Example 4
GCF PROBLEM A teacher has 30 books and 45 notebooks. She wants to make identical reading packets with the same number of books and notebooks in each packet, using all supplies. What is the greatest number of reading packets she can make? How many books and notebooks will be in each packet?
Definition:

Greatest Common Factor (GCF): The largest number that divides two or more numbers without leaving a remainder.

Given
30 books, 45 notebooks
Find GCF of 30, 45
GCF(30, 45) = 15
Reading packets
15 packets
Step 1: Find factors of 30 and 45

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Factors of 45: 1, 3, 5, 9, 15, 45

Step 2: Identify common factors

Common factors: 1, 3, 5, 15

Greatest common factor: 15

Step 3: Calculate items per reading packet

Books per packet: 30 ÷ 15 = 2

Notebooks per packet: 45 ÷ 15 = 3

Step 4: Verify the solution

15 packets × 2 books = 30 books ✓

15 packets × 3 notebooks = 45 notebooks ✓

15 packets with 2 books and 3 notebooks each
Final answer:

The teacher can make 15 identical reading packets.

Each packet will contain 2 books and 3 notebooks.

Applied rules:

• GCF application: Use GCF to find maximum number of identical groups

• Division: Divide total items by group count to get items per group

• Verification: Multiply back to ensure original quantities are preserved

📚
Book Distribution
30 books
÷15 packets
2 books per packet
📝
Notebook Distribution
45 notebooks
÷15 packets
3 notebooks per packet
5 Advanced Pattern Hard
SUBTEST ITEM Example 5
COMPLEX PATTERN A tile pattern starts with 7 tiles in the first figure, 13 tiles in the second figure, 19 tiles in the third figure. If this pattern continues, how many tiles will be in the 25th figure? Which figure will contain exactly 151 tiles? How many tiles are in the first 10 figures combined?
Definition:

Linear pattern: A pattern where the number of elements increases by a constant amount each time.

Pattern
7, 13, 19, ...
Rule
6n + 1
25th figure
6(25) + 1 = 151
Step 1: Identify the pattern rule

Figure 1: 7 tiles, Figure 2: 13 tiles, Figure 3: 19 tiles

Each figure adds 6 tiles: 7, 7+6, 7+6+6, ...

Pattern rule: 6n + 1 (where n is the figure number)

Step 2: Find tiles in 25th figure

Using the rule: 6(25) + 1 = 150 + 1 = 151 tiles

Step 3: Find which figure has 151 tiles

We already found that the 25th figure has 151 tiles

Let's verify: 6n + 1 = 151 → 6n = 150 → n = 25

Step 4: Find total tiles in first 10 figures

Using the rule for n=1 to n=10:

Sum = Σ(6n+1) from n=1 to 10 = 6×Σn + 10 = 6×55 + 10 = 330 + 10 = 340

Or: 7+13+19+25+31+37+43+49+55+61 = 340

Step 5: Verify the results

25th figure: 6(25) + 1 = 151 ✓

Figure with 151 tiles: n = 25 ✓

First 10 figures: 340 tiles ✓

25th figure: 151 tiles, 25th figure: 151 tiles, first 10: 340 tiles
Final answer:

The 25th figure will contain 151 tiles.

The 25th figure will contain exactly 151 tiles.

The first 10 figures contain 340 tiles combined.

Applied rules:

• Pattern recognition: Identify the constant difference between terms

• Formula creation: Develop a rule based on the pattern

• Series summation: Use arithmetic series formulas to find totals

🧩
Pattern Rule
Fig 1: 7
Fig 2: 13
Fig 3: 19
Fig 4: 25
Fig 5: 31

Rule: 6n + 1

🔍
Finding Nth Term
6n + 1
6(25) + 1
151 tiles
Subtest Review: Key Concepts and Strategies
GCF(a,b) × LCM(a,b) = a × b
Relationship Formula
Key definitions:

Factor: A number that divides another number evenly

Multiple: The product of a number and an integer

Prime Number: A number with exactly two factors (1 and itself)

Composite Number: A number with more than two factors

Problem-solving strategies:
  1. Understand: Read the problem carefully and identify what is being asked
  2. Plan: Decide which mathematical concept to apply (factors, multiples, or patterns)
  3. Solve: Apply the appropriate method systematically
  4. Check: Verify your answer makes sense in the context of the problem
Tip 1: To find factors, try dividing by small prime numbers first.
Tip 2: For LCM, multiply the highest powers of all prime factors.
Tip 3: For patterns, look at the differences between consecutive terms.
Tip 4: Always draw a picture or make a table for complex word problems.
Common errors: Forgetting 1 and the number itself as factors, misidentifying pattern rules, miscalculating LCM/GCF.
Key to success: Practice identifying keywords in word problems, master basic multiplication facts, and verify all answers.
Essential formulas and rules:

• Factor pairs: If a×b = n, then a and b are factors of n

• Multiples: Multiples of n are n, 2n, 3n, 4n, ...

• GCF: Greatest common factor is the largest shared factor

• LCM: Least common multiple is the smallest shared multiple

• Pattern rule: Find the relationship between position and value

Subtest Visualization: Patterns and Relationships
SUBTEST ITEM Example 6: Visualizing Patterns
Consider the following pattern sequences:
Sequence A: 3, 8, 13, 18, 23, ... (start at 3, add 5 each time)
Sequence B: 4, 12, 20, 28, 36, ... (start at 4, add 8 each time)
Sequence C: 1, 9, 25, 49, 81, ... (odd perfect squares)
A:3
A:8
A:13
A:18
A:23
B:4
B:12
B:20
B:28
B:36
C:1
C:9
C:25
C:49
C:81

Analysis: Different sequences follow different rules and grow at different rates.

  • Sequence A: Arithmetic sequence (rule: 5n-2)
  • Sequence B: Arithmetic sequence (rule: 8n-4)
  • Sequence C: Odd perfect squares (rule: (2n-1)²)

Subtest Questions & Answers

Question: I don't understand how to find all the factors of a number. Can you explain the systematic way to do it?

Answer: Great question! Here's the systematic approach to find all factors of a number:

  1. Start with 1 and the number itself (since 1 × number = the number)
  2. Check 2: Does 2 divide the number evenly? If yes, write down 2 and its partner
  3. Continue checking 3, 4, 5, etc., up to the square root of the number
  4. For each divisor found, write down both the divisor and its partner

Example with 20:

  • 1 × 20 = 20 → factors: 1, 20
  • 2 × 10 = 20 → factors: 2, 10
  • 4 × 5 = 20 → factors: 4, 5
  • 5 already listed, so we stop here

All factors of 20: 1, 2, 4, 5, 10, 20

Question: How do I know when a word problem is about factors versus multiples?

Answer: Look for these key words and situations:

Factor problems usually involve:

  • "Arranging" items into equal groups (like arranging marbles in groups)
  • "Dividing" items equally among people or containers
  • Finding "how many ways" something can be arranged
  • Keywords: arrange, divide, share, distribute equally

Multiple problems usually involve:

  • Events happening at regular intervals (buses, trains)
  • Items bought in packages of certain amounts
  • Finding when things happen "together" or "at the same time"
  • Keywords: every, together, simultaneously, schedule

Remember: Factors divide a number, multiples are created by multiplying a number.

Question: I have trouble seeing patterns in number sequences. Any tips?

Answer: Here are effective strategies for recognizing patterns:

  1. Look at the differences: Subtract consecutive terms to see if there's a consistent change
  2. Check for multiplication: See if each term is multiplied by the same number
  3. Look for geometric patterns: Shapes or visual representations often follow predictable growth
  4. Make a table: Organize position vs. value to see relationships

Example: 5, 10, 15, 20, 25...

  • Differences: 10-5=5, 15-10=5, 20-15=5, 25-20=5
  • Pattern: Add 5 each time
  • Rule: 5n (multiply position by 5)

Practice with simple patterns first, then move to more complex ones!