Analyzing Number Set Relationships

The question asks us to identify a set of three numbers that shares the same mathematical relationship as the given sets: (40, 120, 400) and (18, 20, 78). We are instructed to treat the numbers as whole entities and not break them down into individual digits.

Discovering the Pattern in the Given Sets

Let's examine the relationship between the numbers in the provided sets. Let the three numbers in a set be represented by $a$, $b$, and $c$. We need to find a formula that connects $a$, $b$, and $c$ consistently for both sets.

Set 1: (40, 120, 400)

Here, $a = 40$, $b = 120$, and $c = 400$. Let's explore simple relationships:

• Relationship between $a$ and $b$: $120 = 3 \times 40$.
• Relationship between $a$ and $c$: $400 = 10 \times 40$.
• Relationship involving $a$ and $b$ to get $c$:

Let's try a linear combination of $a$ and $b$, like $c = k_1 a + k_2 b$.

For (40, 120, 400), we have: $400 = k_1 \times 40 + k_2 \times 120$.

Set 2: (18, 20, 78)

Here, $a = 18$, $b = 20$, and $c = 78$. Using the same potential linear relationship:

For (18, 20, 78), we have: $78 = k_1 \times 18 + k_2 \times 20$.

We now have a system of two linear equations with two variables $k_1$ and $k_2$:

\[40k_1 + 120k_2 = 400 \quad \text{(Equation 1)}\] \[18k_1 + 20k_2 = 78 \quad \text{(Equation 2)}\]

We can simplify these equations by dividing by common factors:

\[k_1 + 3k_2 = 10 \quad \text{(Simplified Equation 1)}\] \[9k_1 + 10k_2 = 39 \quad \text{(Simplified Equation 2)}\]

Let's solve this system. From Simplified Equation 1, $k_1 = 10 - 3k_2$. Substitute this into Simplified Equation 2:

\[9(10 - 3k_2) + 10k_2 = 39\] \[90 - 27k_2 + 10k_2 = 39\] \[90 - 17k_2 = 39\] \[17k_2 = 90 - 39\] \[17k_2 = 51\] \[k_2 = \frac{51}{17}\] \[k_2 = 3\]

Now substitute the value of $k_2$ back into $k_1 = 10 - 3k_2$:

\[k_1 = 10 - 3(3)\] \[k_1 = 10 - 9\] \[k_1 = 1\]

So the relationship is $c = 1 \times a + 3 \times b$, which is $c = a + 3b$. Let's verify this relationship with the original sets:

• For (40, 120, 400): $a=40, b=120$. $c = 40 + 3 \times 120 = 40 + 360 = 400$. (Correct)
• For (18, 20, 78): $a=18, b=20$. $c = 18 + 3 \times 20 = 18 + 60 = 78$. (Correct)

The pattern is confirmed: the third number ($c$) is equal to the first number ($a$) plus three times the second number ($b$). The rule is $c = a + 3b$.

Testing the Relationship on the Options

Now we apply the rule $c = a + 3b$ to each of the given options to find the set that follows this pattern.

Option 1: (29, 23, 98)

$a = 29$, $b = 23$.

Calculate $a + 3b$: $29 + 3 \times 23 = 29 + 69 = 98$.

This matches the third number in the set (98). This option follows the pattern.

Option 2: (56, 14, 108)

$a = 56$, $b = 14$.

Calculate $a + 3b$: $56 + 3 \times 14 = 56 + 42 = 98$.

This does not match the third number in the set (108). This option does not follow the pattern.

Option 3: (25, 27, 92)

$a = 25$, $b = 27$.

Calculate $a + 3b$: $25 + 3 \times 27 = 25 + 81 = 106$.

This does not match the third number in the set (92). This option does not follow the pattern.

Option 4: (11, 720, 660)

$a = 11$, $b = 720$.

Calculate $a + 3b$: $11 + 3 \times 720 = 11 + 2160 = 2171$.

This does not match the third number in the set (660). This option does not follow the pattern.

Conclusion

Based on the analysis, only Option 1 follows the relationship $c = a + 3b$ observed in the initial sets (40, 120, 400) and (18, 20, 78).

Revision Table: Number Set Logic

This section summarizes the key aspects of solving number set relationship problems.

• Identify the given sets and their numbers ($a, b, c$).
• Look for mathematical relationships between the numbers ($a$, $b$, and $c$). Common relationships include addition, subtraction, multiplication, division, powers, or linear combinations.
• Test the potential relationship on all given example sets to ensure consistency.
• Once a consistent rule is found, apply it to the options to find the set that follows the same rule.
• Ensure operations are on whole numbers as specified, not individual digits.

Additional Information: Quantitative Reasoning Techniques

Questions involving finding the relationship between numbers in sets fall under quantitative reasoning or logical reasoning. These questions test your ability to identify patterns and apply mathematical rules.

• Common Patterns: Look for arithmetic progressions, geometric progressions, differences, sums, products, ratios, squares, cubes, or combinations of these operations.
• Trial and Error: Often, trying out simple relationships first (like $a+b=c$, $a \times b = c$, $a^2+b^2=c$, $ka+lb=c$, etc.) is the quickest way to find the pattern.
• System of Equations: If the relationship seems linear ($ka+lb=c$), using two given sets allows you to set up a system of equations to solve for the unknown coefficients ($k$ and $l$).
• Practice: Solving various types of number pattern questions helps in quickly recognizing common relationships.