Understanding Numerical Analogies

Numerical analogies are problems that require you to find a relationship between a pair of numbers and then apply that same relationship to another number to find a missing term. The key is to identify the pattern or rule that connects the first two numbers.

Analyzing the Pattern in 4 : 80

We are given the relationship between 4 and 80. Let's look for a pattern. We can try different mathematical operations involving 4 to get 80.

• Multiplication: $4 \times 20 = 80$. If we apply this to 5, $5 \times 20 = 100$. This is not among the options.
• Powers: $4^2 = 16$, $4^3 = 64$. How can we get 80 using powers of 4? We can try combining them.
• Consider $4^3 + 4^2$. $4^3 = 4 \times 4 \times 4 = 64$. $4^2 = 4 \times 4 = 16$. Summing these gives $64 + 16 = 80$. This matches the second number.

So, the pattern identified is that the second number is obtained by adding the cube and the square of the first number. Let's express this pattern for a number $n$ as $n^3 + n^2$. This can also be written as $n^2 \times (n+1)$. For $n=4$, this is $4^2 \times (4+1) = 16 \times 5 = 80$.

Applying the Pattern to Find the Missing Number

Now we apply the same rule to the third number, which is 5. According to the pattern, the missing number will be $5^3 + 5^2$.

Calculation Details

Let's calculate the value:

• Calculate the cube of 5: $5^3 = 5 \times 5 \times 5 = 125$.
• Calculate the square of 5: $5^2 = 5 \times 5 = 25$.
• Add the results: $125 + 25 = 150$.

Alternatively, using the $n^2 \times (n+1)$ form:

• Calculate the square of 5: $5^2 = 25$.
• Calculate $(5+1)$: $5+1 = 6$.
• Multiply the results: $25 \times 6 = 150$.

Both methods yield 150.

Result

Following the same pattern $n^3 + n^2$ or $n^2 \times (n+1)$, when the first number is 5, the second number is 150.

Revision Table: Key Concepts

Additional Information: Common Numerical Patterns

Numerical analogy questions can use various patterns. Some common ones include:

• Basic arithmetic operations (addition, subtraction, multiplication, division).
• Powers and roots (squares, cubes, square roots, cube roots).
• Combinations of operations and powers (like $n^2+n$, $n^3-1$, etc.).
• Sequences (arithmetic progression, geometric progression).
• Digit-based operations (sum of digits, product of digits).
• Prime numbers, composite numbers, odd/even numbers.

Solving these questions requires careful observation and trying different potential relationships between the numbers.