Select the option in which the numbers are related in the same way as are the numbers in the given set.
(15, 75, 135)
Solving Number Analogy Questions: Finding the Relationship
This question asks us to identify the option where the numbers share the same relationship as the numbers in the given set: (13, 65, 117).
To solve this type of number analogy problem, we first need to find the pattern or rule that connects the numbers in the provided set. Let's look at the numbers: 13, 65, and 117.
Analyzing the Given Set (13, 65, 117)
Let's try to find a relationship between these numbers using basic arithmetic operations like multiplication or division relative to the first number, 13.
- How is 65 related to 13? We can see that $13 \times 5 = 65$. So, the second number is 5 times the first number.
- How is 117 related to 13? Let's try multiplying 13 by another number. We know $13 \times 10 = 130$, so it's less than 10 times. Let's try 9. $13 \times 9 = (10 + 3) \times 9 = 90 + 27 = 117$. So, the third number is 9 times the first number.
The relationship in the given set (13, 65, 117) appears to be:
Second Number = First Number $\times 5$
Third Number = First Number $\times 9$
Applying the Relationship to Options
Now, we will test this relationship with the given options to find the set that follows the same rule.
Option 1: (14, 70, 127)
- First number = 14
- Second number check: $14 \times 5 = 70$. This matches the second number in the option.
- Third number check: $14 \times 9 = 126$. This does not match the third number (127) in the option.
So, Option 1 does not follow the rule.
Option 2: (15, 75, 135)
- First number = 15
- Second number check: $15 \times 5 = 75$. This matches the second number in the option.
- Third number check: $15 \times 9 = 135$. This matches the third number in the option.
So, Option 2 follows the rule.
Option 3: (17, 85, 163)
- First number = 17
- Second number check: $17 \times 5 = 85$. This matches the second number in the option.
- Third number check: $17 \times 9 = 153$. This does not match the third number (163) in the option.
So, Option 3 does not follow the rule.
Option 4: (12, 55, 109)
- First number = 12
- Second number check: $12 \times 5 = 60$. This does not match the second number (55) in the option. (No need to check the third number, as the second number didn't match).
So, Option 4 does not follow the rule.
Summary of Option Checks
| Option Set | Rule: $N_2 = N_1 \times 5$ | Rule: $N_3 = N_1 \times 9$ | Follows Rule? |
|---|---|---|---|
| (14, 70, 127) | $14 \times 5 = 70$ (Match) | $14 \times 9 = 126$ (No Match 127) | No |
| (15, 75, 135) | $15 \times 5 = 75$ (Match) | $15 \times 9 = 135$ (Match) | Yes |
| (17, 85, 163) | $17 \times 5 = 85$ (Match) | $17 \times 9 = 153$ (No Match 163) | No |
| (12, 55, 109) | $12 \times 5 = 60$ (No Match 55) | - | No |
Based on the analysis, only the numbers in Option 2 (15, 75, 135) are related in the same way as the numbers in the given set (13, 65, 117), following the pattern $N_2 = N_1 \times 5$ and $N_3 = N_1 \times 9$.
Revision Table: Key Learnings on Number Analogy
| Concept | Description | Application in this Problem |
|---|---|---|
| Identify the Pattern | Look for mathematical relationships (addition, subtraction, multiplication, division, squares, cubes, etc.) between the numbers in the given set. | Found that $65 = 13 \times 5$ and $117 = 13 \times 9$. |
| Formulate the Rule | Express the identified pattern as a clear rule connecting the numbers, usually based on the first number. | Rule: Second number = First number $\times 5$; Third number = First number $\times 9$. |
| Test the Options | Apply the formulated rule to each option set provided. | Applied the $\times 5$ and $\times 9$ rule to all four options. |
| Verify and Select | Check which option set consistently follows the rule derived from the given set. | Option (15, 75, 135) was the only set where $75 = 15 \times 5$ and $135 = 15 \times 9$. |
Additional Information: Types of Number Analogy Patterns
Number analogy questions can have various types of patterns. Understanding these can help solve problems faster.
- Arithmetic Progression: Numbers increase or decrease by a constant difference (e.g., 3, 6, 9, 12).
- Geometric Progression: Numbers are multiplied or divided by a constant ratio (e.g., 2, 4, 8, 16).
- Multiplication/Division Relationship: Numbers are direct multiples or divisions of each other, often related to the first number, as seen in this problem.
- Addition/Subtraction Relationship: Numbers are related by specific additions or subtractions, possibly involving the first number or a constant.
- Square or Cube Relationships: Numbers might be squares, cubes, or related to the squares/cubes of the first number or its position in the sequence.
- Combination of Operations: Patterns might involve a mix of operations, like multiply and then add, or square and then subtract.
Always start by checking simple relationships before moving to more complex ones. Multiplication and division relative to the first term are common patterns in analogy questions.
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