Select the option in which the numbers share the same relationship as that shared by the given pair of numbers.
40 ∶ 300
27 ∶ 196
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13-Operations on 13 such as adding/ subtracting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed)

Question

Select the option in which the numbers share the same relationship as that shared by the given pair of numbers.

40 ∶ 300

27 ∶ 196

(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13-Operations on 13 such as adding/ subtracting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed)

Prepp Team · Accepted Answer

Finding the Number Relationship in Analogies

This type of question, known as a number analogy or number relationship problem, requires us to identify the mathematical rule or pattern that connects the two numbers in the given pairs. We then need to find an option pair that follows the exact same rule.

Analyzing the Given Number Pairs

We are given two pairs of numbers:

40 ∶ 300
27 ∶ 196

We need to find a relationship, let's say $y = f(x)$, where $x$ is the first number and $y$ is the second number in each pair. The rule must work for both given pairs.

Let's try common mathematical operations:

Addition/Subtraction: $300 - 40 = 260$. $196 - 27 = 169$. The difference is not constant.
Multiplication: $300 / 40 = 7.5$. $196 / 27 \approx 7.26$. The ratio is not constant.

Let's consider a combination of multiplication and addition/subtraction. Suppose the relationship is of the form $y = ax + b$.

For the first pair (40, 300):

$$40a + b = 300 \quad (Equation \ 1)$$

For the second pair (27, 196):

$$27a + b = 196 \quad (Equation \ 2)$$

We can solve these two linear equations to find $a$ and $b$. Subtract Equation 2 from Equation 1:

$$(40a + b) - (27a + b) = 300 - 196$$

$$40a - 27a = 104$$

$$13a = 104$$

$$a = \frac{104}{13} = 8$$

Now substitute the value of $a$ into Equation 2:

$$27(8) + b = 196$$

$$216 + b = 196$$

$$b = 196 - 216$$

$$b = -20$$

So, the relationship is $y = 8x - 20$. Let's verify this relationship with the original pairs:

For 40 ∶ 300: $40 \times 8 - 20 = 320 - 20 = 300$. This matches.
For 27 ∶ 196: $27 \times 8 - 20 = 216 - 20 = 196$. This matches.

The identified relationship is consistent for both given number pairs.

Testing the Options

Now we apply the relationship $y = 8x - 20$ to the first number of each option pair to see if it generates the second number.

Option 1: 55 ∶ 400
For $x = 55$, the calculated $y$ is $55 \times 8 - 20 = 440 - 20 = 420$. This does not match 400.
Option 2: 130 ∶ 940
For $x = 130$, the calculated $y$ is $130 \times 8 - 20 = 1040 - 20 = 1020$. This does not match 940.
Option 3: 200 ∶ 1580
For $x = 200$, the calculated $y$ is $200 \times 8 - 20 = 1600 - 20 = 1580$. This matches 1580.
Option 4: 25 ∶ 140
For $x = 25$, the calculated $y$ is $25 \times 8 - 20 = 200 - 20 = 180$. This does not match 140.

Based on our analysis, only Option 3 follows the same $y = 8x - 20$ relationship as the given pairs.

Pair	First Number (x)	Second Number (y)	Applying Rule ($8x - 20$)	Matches?
Given 1	40	300	$8 \times 40 - 20 = 320 - 20 = 300$	Yes
Given 2	27	196	$8 \times 27 - 20 = 216 - 20 = 196$	Yes
Option 1	55	400	$8 \times 55 - 20 = 440 - 20 = 420$	No
Option 2	130	940	$8 \times 130 - 20 = 1040 - 20 = 1020$	No
Option 3	200	1580	$8 \times 200 - 20 = 1600 - 20 = 1580$	Yes
Option 4	25	140	$8 \times 25 - 20 = 200 - 20 = 180$	No

Conclusion

The pair 200 ∶ 1580 shares the same number relationship as the given pairs 40 ∶ 300 and 27 ∶ 196. The relationship is that the second number is obtained by multiplying the first number by 8 and then subtracting 20.

Revision Table for Number Relationships

Concept	Description	Examples
Number Analogy	Finding a hidden mathematical or logical rule connecting two numbers or sets of numbers.	12 ∶ 144 (Square), 5 ∶ 125 (Cube)
Common Rules	Arithmetic operations (addition, subtraction, multiplication, division), squares, cubes, combinations of operations.	$x \to x+k$, $x \to kx$, $x \to kx+c$, $x \to x^2$, $x \to x^3$.
Solving Steps	1. Analyze the given pairs to find the rule. 2. Test the rule with all given pairs. 3. Apply the rule to the options. 4. Select the option that follows the rule.	As demonstrated in the solution above.

Additional Information on Pattern Recognition

Pattern recognition is a key skill in solving number relationship and analogy problems. It involves observing the relationship between the given elements and identifying a consistent rule. For number-based problems, these patterns are usually mathematical. While linear relationships ($ax+b$) are common, patterns can also involve squares ($x^2$), cubes ($x^3$), square roots ($\sqrt{x}$), or more complex sequences.

Sometimes, the pattern might relate to the position of numbers in a sequence, or properties like prime numbers, odd/even numbers, etc. However, as per the question's note, we focused on operations on the whole numbers themselves, not their constituent digits. Always read the problem constraints carefully.