Understanding Number Analogy Problems

Number analogy questions test your ability to find the relationship between a pair of numbers and apply that same relationship to another pair or group of numbers to find a missing element. You need to observe the given pairs carefully and identify the underlying rule or pattern.

Analyzing the Given Number Analogies

The problem presents the following analogy structure:

7 : 42 :: ? : 110 :: 9 : 72

This means the relationship between 7 and 42 is the same as the relationship between the missing number and 110, and also the same as the relationship between 9 and 72.

Discovering the Relationship Pattern

Let's examine the first pair: 7 : 42

What is the relationship between 7 and 42? We can look for simple mathematical operations:

• Adding: $7 + \text{something} = 42$ (something is 35)
• Multiplying: $7 \times \text{something} = 42$ (something is 6)
• Squaring: $7^2 = 49$. How is 42 related to 49? $49 - 7 = 42$. This can be written as $n^2 - n$.

Let's consider the pattern $n \times (n-1)$. For $n=7$, this would be $7 \times (7-1) = 7 \times 6 = 42$. This matches the first pair.

Now, let's examine the third pair: 9 : 72

Let's test the pattern $n \times (n-1)$ with $n=9$.

$9 \times (9-1) = 9 \times 8 = 72$. This also matches the third pair perfectly.

Therefore, the pattern is likely that the second number in each pair is obtained by multiplying the first number by one less than itself, i.e., $n \times (n-1)$ or $n^2 - n$.

Applying the Pattern to Find the Missing Number

Now we apply this pattern to the second part of the analogy: ? : 110

Let the missing number be $x$. According to the pattern, the relationship is $x : x \times (x-1)$.

So, we have the equation:

$x \times (x-1) = 110$

Step-by-Step Solution Calculation

We need to solve the equation $x(x-1) = 110$ for $x$.

Expand the left side:

$x^2 - x = 110$

Rearrange into a standard quadratic equation form $ax^2 + bx + c = 0$:

$x^2 - x - 110 = 0$

We can solve this quadratic equation. One way is by factoring, but we can also try to guess integer factors of 110 that have a difference of 1. The factors of 110 are (1, 110), (2, 55), (5, 22), (10, 11). The pair (10, 11) has a difference of 1.

So we can rewrite the equation as:

$x^2 - 11x + 10x - 110 = 0$

$x(x - 11) + 10(x - 11) = 0$

$(x - 11)(x + 10) = 0$

This gives two possible solutions for $x$:

• $x - 11 = 0 \Rightarrow x = 11$
• $x + 10 = 0 \Rightarrow x = -10$

Since the numbers in the given pairs (7 and 9) are positive, the missing number is likely positive. Therefore, we choose $x = 11$.

Alternatively, we could recognize that we are looking for a number $x$ such that $x$ multiplied by the previous integer ($x-1$) equals 110. We are looking for two consecutive integers whose product is 110. By checking simple multiplications, we find $10 \times 11 = 110$. Since we need $x \times (x-1)$, if $x=11$, then $x-1=10$, and $11 \times 10 = 110$. This matches the equation.

Verifying the Solution

Let's check if our missing number, 11, fits the pattern:

For the pair 11 : 110, applying the pattern $n \times (n-1)$:

$11 \times (11-1) = 11 \times 10 = 110$

This is correct and matches the given analogy ? : 110.

Thus, the missing number is 11.

Revision Table: Number Analogy Concepts

Additional Information on Number Series and Analogies

Number analogy questions are a type of reasoning puzzle often found in aptitude tests. They are closely related to number series questions. While analogies involve pairs or groups sharing a relationship, number series involve finding the pattern in a sequence of numbers.

Common patterns in number analogies and series include:

• Arithmetic operations (addition, subtraction, multiplication, division).
• Operations involving squares, cubes, square roots, cube roots.
• Relationships with consecutive numbers (like $n+1$, $n-1$, $n \times (n+1)$, $n \times (n-1)$).
• Prime numbers, composite numbers.
• Digit manipulation (sum of digits, product of digits).
• Combination of multiple operations.

To solve these problems effectively, it's helpful to practice recognizing common patterns and be comfortable with basic arithmetic and algebra.