Study the given pattern carefully and select the number that can replace the question mark (?) in it. First row : 18, 9, 27 Second row : 40, 36, 140 Third row : 40, 13, ? (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)
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Analyzing the Number Pattern
The question asks us to identify the pattern connecting the numbers in each row and use it to find the missing number in the third row. We are given three rows of numbers:
- First row: 18, 9, 27
- Second row: 40, 36, 140
- Third row: 40, 13, ?
The rule states that operations should be performed on the whole numbers as they are, not by breaking them down into individual digits.
Discovering the Mathematical Rule
Let's examine the relationship between the first number (A), the second number (B), and the third number (C) in each row to find a consistent mathematical rule.
Testing Simple Operations (Row 1)
For the first row (18, 9, 27):
- Addition: $18 + 9 = 27$. This matches the third number.
- Subtraction: $18 - 9 = 9$. This does not match 27.
- Multiplication: $18 \times 9 = 162$. This does not match 27.
The simple sum (A + B = C) works for the first row.
Checking the Rule on Row 2
Let's see if the A + B = C rule works for the second row (40, 36, 140):
- Addition: $40 + 36 = 76$. This does not match 140.
So, the pattern is not simply A + B = C.
Exploring Other Pattern Possibilities
Since simple addition didn't work for the second row, let's look for a more complex pattern involving multiplication, subtraction, or combinations of operations.
We need a rule C = f(A, B) that works for both (18, 9, 27) and (40, 36, 140).
Let's consider patterns of the form $C = k_1 \times A + k_2 \times B$, where $k_1$ and $k_2$ are constants.
Using the first row (A=18, B=9, C=27):
$$18k_1 + 9k_2 = 27$$Using the second row (A=40, B=36, C=140):
$$40k_1 + 36k_2 = 140$$We can simplify these equations by dividing by common factors:
Equation 1 (divide by 9):
$$2k_1 + k_2 = 3 \quad (i)$$Equation 2 (divide by 4):
$$10k_1 + 9k_2 = 35 \quad (ii)$$Now we have a system of two linear equations with two variables $k_1$ and $k_2$. From equation (i), we can express $k_2$ as:
$$k_2 = 3 - 2k_1$$Substitute this expression for $k_2$ into equation (ii):
$$10k_1 + 9(3 - 2k_1) = 35$$ $$10k_1 + 27 - 18k_1 = 35$$ $$-8k_1 + 27 = 35$$ $$-8k_1 = 35 - 27$$ $$-8k_1 = 8$$ $$k_1 = \frac{8}{-8} = -1$$Now substitute the value of $k_1$ back into the expression for $k_2$:
$$k_2 = 3 - 2(-1)$$ $$k_2 = 3 + 2$$ $$k_2 = 5$$So, the constants are $k_1 = -1$ and $k_2 = 5$. The potential pattern rule is $C = -1 \times A + 5 \times B$, which can be written as $C = 5B - A$.
Verifying the Pattern
Let's verify if the rule $C = 5B - A$ holds true for the given rows:
For the first row (18, 9, 27):
$$C = 5 \times 9 - 18$$ $$C = 45 - 18$$ $$C = 27$$This matches the third number in the first row.
For the second row (40, 36, 140):
$$C = 5 \times 36 - 40$$ $$C = 180 - 40$$ $$C = 140$$This matches the third number in the second row.
The pattern $C = 5B - A$ is consistent for both the first and second rows.
Finding the Missing Number
Now, let's apply this pattern to the third row (40, 13, ?). Here, A = 40 and B = 13. We need to find C.
$$C = 5 \times 13 - 40$$ $$C = 65 - 40$$ $$C = 25$$The missing number is 25.
| Row | A | B | Operation ($5B - A$) | C (Result) | Given C |
|---|---|---|---|---|---|
| 1 | 18 | 9 | $5 \times 9 - 18 = 45 - 18$ | 27 | 27 |
| 2 | 40 | 36 | $5 \times 36 - 40 = 180 - 40$ | 140 | 140 |
| 3 | 40 | 13 | $5 \times 13 - 40 = 65 - 40$ | 25 | ? |
Revision Table: Number Pattern Analysis
| Concept | Description | Application in this Problem |
|---|---|---|
| Number Pattern | A sequence or arrangement of numbers following a specific rule. | Identifying the rule $C = 5B - A$ across the rows. |
| Pattern Recognition | The process of finding the underlying rule or relationship in a set of data. | Analyzing rows to deduce the operation connecting A, B, and C. |
| Mathematical Operation | Actions like addition, subtraction, multiplication, division performed on numbers. | The pattern uses multiplication and subtraction ($5 \times B - A$). |
| System of Equations | A set of equations solved together to find the values of unknown variables. | Used implicitly or explicitly to find constant coefficients in the pattern. |
Additional Information: Solving Pattern Puzzles
Solving number pattern puzzles like this involves logical thinking and testing different mathematical operations. Here are some common approaches and tips:
- Look for Simple Relations: First, check for basic patterns like addition, subtraction, multiplication, or division between consecutive numbers or between the first two numbers and the third.
- Consider Combinations: If simple operations don't work, try combinations like A + B, A - B, A * B, and see how they relate to C. Also, consider A + kB, kA + B, kA + mB, etc., where k and m are constants.
- Check Differences and Ratios: Look at the differences or ratios between numbers in a sequence or within a row. Are they constant, or do they follow a pattern?
- Involve Squares or Cubes: Some patterns involve squares, cubes, or their roots, although the note here restricts operations on constituent digits.
- System of Equations: For patterns involving constant coefficients (like $kA + mB = C$), setting up a system of equations from two or more rows can help find the coefficients.
- Examine Options: Sometimes, looking at the answer options can provide clues about the possible nature of the pattern.
- Stay within Rules: Always adhere to any specific rules given in the question, such as the restriction on breaking down numbers into digits.
Practice with various types of number patterns helps in recognizing common structures and developing problem-solving skills.
Similar Questions
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
First row - 8, 28, 53
Second row - 7, 25, 47
Third row - 13, 43, ?
(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)
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
First row: 15, 3, 252
Second row: 11, 5, 246
Third row: 9, 6, ?
(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)
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
(5, 14, 3)
(3, 24, 7)
(9, 30, ?)
(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)
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
(6, 1, 18)
(5, 4, 60)
(6, 2, ?)
(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)
Study the given matrix carefully and select the number from among the given options that can replace the question mark (?) in it.
7 13 174 9 25 104 11 30 ? Study the given pattern carefully and select the number that can replace the question mark (?) in it.
15 9 144 18 12 ? 22 17 195 Study the given matrix carefully and select the number from among the given options that can replace the question mark (?) in it.
12 8 100 18 6 111 15 4 ? Study the given pattern carefully and select the number that can replace the question mark (?) in it.
14 21 165 13 19 ? 10 17 99 Study the given matrix carefully and select the number from among the given options that can replace the question mark (?) in it.
9
7
9
5
4
7
43
26
?
Study the given matrix carefully and select the number from among the given options that can replace the question mark(?) in it.
13 6 75 15 8 ? 18 4 70
Important Questions from Missing Number in Matrix
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
16 295 19 9 144 17 26 ? 16 Study the given pattern carefully and select the number that can replace the question mark (?) in it.
57
28
29
68
?
33
72
37
35
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
13 26 39 30 42 ? 17 16 15 Find the missing number from the below options.
12 16 18 24 32 ? 36 48 54 Study the given pattern carefully and select the number that can replace the question mark (?) in it.
64 12 27 216 ? 343 512 40 125
