Understanding the Number Analogy Pattern

The question asks us to find a missing term in a number analogy. We are given three pairs of numbers, where the relationship between the numbers in the first pair (10 : 45) and the third pair (11 : 55) follows a specific pattern. We need to identify this pattern and apply it to the second pair (? : 28) to find the missing term.

The given analogy structure is:

10 : 45 :: ? : 28 :: 11 : 55

This means that the relationship between 10 and 45 is the same as the relationship between the missing term and 28, and also the same as the relationship between 11 and 55.

Analyzing Number Relationship in Analogy Pairs

Let's carefully examine the two complete pairs provided: 10 : 45 and 11 : 55.

Analyzing the Pair 10 : 45 Analogy

How can we get 45 from 10? Let's think about common mathematical operations:

• Addition: $10 + 35 = 45$. If this were the rule, the difference would be constant.
• Multiplication: $10 \times 4.5 = 45$. This involves multiplication by 4.5.
• Squaring: $10^2 = 100$, which is not 45.
• Combining operations: Maybe $10 \times 5 - 5 = 45$? Or $10 \times 4 + 5 = 45$?

Analyzing the Pair 11 : 55 Analogy

Now let's look at the pair 11 : 55:

• Addition: $11 + 44 = 55$. The difference is 44, which is different from the previous pair's difference (35). So, constant addition is not the rule.
• Multiplication: $11 \times 5 = 55$. This involves multiplication by 5.

We have multipliers 4.5 for the first pair (10:45) and 5 for the third pair (11:55). These multipliers seem to be related to the first term of each pair (10 and 11).

Identifying the Number Analogy Pattern

Let the first term of a pair be represented by $n$ and the second term by $m$. We saw that for $n=10$, the multiplier was 4.5, and for $n=11$, the multiplier was 5.

Let's see if there's a relationship between $n$ and the multiplier. If the multiplier is $k$, then $m = n \times k$.

When $n=10$, $k=4.5$. Notice that $4.5 = 10/2 - 0.5$ or $4.5 = (10-1)/2 + ?$ or $4.5 = (10-1)/2 \times 1$? $(10-1)/2 = 9/2 = 4.5$.

Let's test the pattern: Multiplier = $(n-1)/2$. So, the second term $m = n \times \frac{n-1}{2}$.

• Check for 10 : 45: $n=10$. Second term $= 10 \times \frac{10-1}{2} = 10 \times \frac{9}{2} = 10 \times 4.5 = 45$. This matches the given pair.
• Check for 11 : 55: $n=11$. Second term $= 11 \times \frac{11-1}{2} = 11 \times \frac{10}{2} = 11 \times 5 = 55$. This matches the given pair.

The pattern is confirmed: The second term in each pair is obtained by multiplying the first term by (the first term minus 1) divided by 2.

Finding the Missing Term in the Analogy

Now we apply this pattern to the second pair: ? : 28. Let the missing term (the first term) be $x$. The second term is 28.

According to the pattern, we have:

\begin{equation*} x \times \frac{x-1}{2} = 28 \end{equation*}

To solve for $x$, multiply both sides of the equation by 2:

\begin{equation*} x(x-1) = 56 \end{equation*}

Expand the left side:

\begin{equation*} x^2 - x = 56 \end{equation*}

Rearrange the terms to form a quadratic equation:

\begin{equation*} x^2 - x - 56 = 0 \end{equation*}

We need to factor this quadratic equation. We look for two numbers that multiply to -56 and add up to -1 (the coefficient of the $x$ term). These numbers are -8 and +7.

So, we can factor the equation as:

\begin{equation*} (x-8)(x+7) = 0 \end{equation*}

This equation gives two possible solutions for $x$: $x-8 = 0$, which means $x=8$, or $x+7 = 0$, which means $x=-7$.

Looking at the numbers in the other pairs (10 and 11), they are positive integers. Therefore, the missing term is most likely the positive integer solution.

The missing term is 8.

Verification of the Missing Term

Let's check if the pair 8 : 28 fits the identified pattern using $n=8$:

According to the rule $m = n \times \frac{n-1}{2}$:

\begin{equation*} 8 \times \frac{8-1}{2} = 8 \times \frac{7}{2} = 8 \times 3.5 = 28 \end{equation*}

This calculation results in 28, which is the second term in the pair ? : 28. This confirms that our missing term, 8, is correct according to the pattern observed in the other pairs.

Let's summarize the pattern application in a table:

The missing term is 8.

Conclusion on Number Analogy Solution

By analyzing the relationship between the first and second terms in the given pairs (10:45 and 11:55), we discovered a consistent pattern: the second term is the product of the first term and (the first term minus 1) divided by 2. Applying this pattern to the pair ?:28 allowed us to set up and solve an equation, finding the missing term to be 8. This fits perfectly with the observed pattern and completes the number analogy.

Revision Table: Common Number Analogy Patterns

Solving number analogy questions effectively requires recognizing various types of patterns. Here are some common ones encountered in reasoning tests:

• Basic Arithmetic: Adding/subtracting a constant, multiplying/dividing by a constant.
• Sequential Operations: Adding/subtracting or multiplying/dividing by a number that follows a sequence (e.g., add 2, then 3, then 4).
• Squares and Cubes: The second term is the square or cube of the first term, or related to it ($n^2 \pm k$, $n^3 \pm k$).
• Operations on Digits: Sum of digits, product of digits, reversing digits.
• Specific Series: Fibonacci series, prime numbers, triangular numbers, etc.
• Combined Operations: A combination of multiplication/division and addition/subtraction (like the pattern in this question).

Additional Information: Strategies for Solving Reasoning Questions

Improving your skill in solving number analogy and other logical reasoning problems involves practice and systematic thinking. Here are some tips:

• Start Simple: Always check for the most basic arithmetic relationships first (addition, subtraction, multiplication, division).
• Consider Squares and Cubes: These are very common patterns, so check if the numbers are perfect squares or cubes, or close to them.
• Look at the Differences/Ratios: Calculate the differences or ratios between terms in the known pairs. See if there is a pattern in these differences or ratios.
• Break Down the Numbers: For larger numbers, consider operations on their individual digits.
• Be Flexible: If one pattern doesn't fit all given pairs, try another. Be prepared to combine different types of operations.
• Work Backwards: Sometimes, looking at how the second term relates to the first can reveal the pattern.
• Practice Regularly: The more types of patterns you encounter, the quicker you will become at identifying them during a test.