Understanding Number Analogy and Finding the Pattern

This question asks us to identify the relationship between the numbers in the given sets and apply that same relationship to the options provided. We are given two example sets: (15, 5, 375) and (14, 3, 210). The note specifies that we should treat the numbers as whole numbers and not break them down into individual digits.

Analyzing the Given Number Sets

Let's look at the first set: (15, 5, 375). We need to find a mathematical operation or combination of operations involving 15 and 5 that results in 375.

• If we multiply the first two numbers: $15 \times 5 = 75$. How is 75 related to 375? $375 \div 75 = 5$. So, $75 \times 5 = 375$.
• This suggests a possible rule: (First number $\times$ Second number) $\times$ Second number = Third number. Or, written differently, First number $\times$ (Second number)$^2$ = Third number.

Let's test this potential rule with the second set: (14, 3, 210).

• First number is 14, Second number is 3.
• According to the rule: (First number $\times$ Second number) $\times$ Second number = $(14 \times 3) \times 3 = 42 \times 3 = 126$.

The result 126 is not 210. So, this rule (First number $\times$ Second number) $\times$ Second number is incorrect.

Let's reconsider the relationship from the first set: $15 \times 5 = 75$, and $75 \times 5 = 375$. We saw that multiplying the product of the first two by the second number worked for the first set. For the second set: $14 \times 3 = 42$. To get 210 from 42, we need to multiply by $210 \div 42 = 5$. So, $42 \times 5 = 210$.

Notice that in the first set, we multiplied $15 \times 5$ by 5 (the second number). In the second set, we multiplied $14 \times 3$ by 5 (which is NOT the second number, 3). This '5' appears consistently as the final multiplier.

This leads to a new potential rule: Third number = First number $\times$ Second number $\times$ 5.

Verifying the Pattern Rule

Let's verify this rule with both given sets:

The rule "Third number = First number $\times$ Second number $\times$ 5" holds true for both the given sets.

Applying the Rule to the Options

Now, we will apply this rule to each of the given options to find the set that follows the same pattern.

Option 1: (10, 12, 320)

• First number = 10, Second number = 12.
• Calculated third number = $10 \times 12 \times 5 = 120 \times 5 = 600$.
• The given third number is 320. $600 \neq 320$. This option does not follow the rule.

Option 2: (17, 19, 920)

• First number = 17, Second number = 19.
• Calculated third number = $17 \times 19 \times 5$.
• $17 \times 19 = 323$.
• Calculated third number = $323 \times 5 = 1615$.
• The given third number is 920. $1615 \neq 920$. This option does not follow the rule.

Option 3: (14, 15, 780)

• First number = 14, Second number = 15.
• Calculated third number = $14 \times 15 \times 5$.
• $14 \times 15 = 210$.
• Calculated third number = $210 \times 5 = 1050$.
• The given third number is 780. $1050 \neq 780$. This option does not follow the rule.

Option 4: (11, 16, 880)

• First number = 11, Second number = 16.
• Calculated third number = $11 \times 16 \times 5$.
• $11 \times 16 = 176$.
• Calculated third number = $176 \times 5 = 880$.
• The given third number is 880. $880 = 880$. This option follows the rule.

Therefore, the set (11, 16, 880) is related in the same way as the numbers in the given sets.

Revision Table: Number Analogy Pattern

Additional Information on Number Reasoning Patterns

Number analogy or number series problems are common in logical and quantitative reasoning tests. They require you to find the hidden mathematical rule or pattern that connects a set of numbers. Common patterns include:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Operations on consecutive numbers (addition, subtraction, multiplication, division, squares, cubes)
• Combinations of operations
• Rules involving a constant factor or term
• Rules based on the position of the number in the sequence

Solving these problems involves careful observation, trying different basic arithmetic operations, and testing potential rules against all given examples before applying them to the options. Remember the constraint about using whole numbers and not breaking them down into digits, as specified in this question.