RBSE Solutions Class 5 Maths Chapter 8 Patterns More Ques

Patterns In Text Exercise

 

Question 1. Proceed the given pattern
Answer: The given pattern is a repeating block of 4x4 elements. To proceed, we simply repeat this block.
The repeating 4x4 block is:
Row 1: D, half-circle (right), full-circle, half-circle (left)
Row 2: right arrow, right arrow, left arrow, left arrow
Row 3: S, S, S, S
Row 4: right arrow, up arrow, left arrow, down arrow
The completed pattern, by repeating this 4x4 block twice to fill the 4x8 grid, is:

In simple words: Look at the first block of shapes and arrows. The complete pattern is made by simply drawing that same block again right next to it.

🎯 Exam Tip: When dealing with repeating visual patterns, identify the smallest repeating unit first. Then, replicate that unit to extend the pattern.

 

Question 2. Recognize the pattern and proceed further.
Answer: We need to find the pattern in each row and then extend it. The numbers are arranged in a grid, and each row follows a different rule.
The initial numbers are:

(i) First row: \( 2, 5, 8, 11 \) - Each number increases by 3. \( (5-2=3, 8-5=3, 11-8=3) \). So, the next number is \( 11+3 = 14 \).
(ii) Second row: \( 145, 125, 105 \) - Each number decreases by 20. \( (125-145=-20, 105-125=-20) \). So, the next number is \( 105-20 = 85 \).
(iii) Third row: \( 2, 4, 8, 16 \) - Each number is multiplied by 2. \( (4=2\times2, 8=4\times2, 16=8\times2) \). So, the next number is \( 16\times2 = 32 \).
(iv) Fourth row: \( 7, 14, 21, 28 \) - Each number increases by 7 (or is a multiple of 7). \( (14-7=7, 21-14=7, 28-21=7) \). So, the next number is \( 28+7 = 35 \).
The completed pattern, with the next number in each row, is:

In simple words: Look at each line of numbers separately. Find out if numbers are being added, subtracted, or multiplied by the same amount each time. Then, follow that rule to find the next number in the sequence.

🎯 Exam Tip: When extending multiple patterns, analyze each sequence independently to find its specific rule (addition, subtraction, multiplication, etc.) before applying it.

 

You also make your own pattern-
Answer: We need to create a pattern based on a rule. The example shows a pattern with a base and numbers building upwards. Let's consider the initial pattern given and then the completed pattern.
Initial partial pattern:

Based on the concept that a number is formed by a rule from the numbers below it, we can form a pattern like a pyramid. For instance, if we add 3 to each number horizontally, and also vertically with varying steps, or based on multiplication tables. A simpler example would be to just fill in the given partial pattern.
The numbers provided `3, 9, 12`, `6`, `15, 25, 35` and the solution suggests a pattern where `6` is below `3`, `9` is below `3` and `6`, etc. Let's assume the question meant a multiplication table based pattern, like the one presented in the solution.
Based on multiplication tables, a possible pattern is formed where numbers increase by a steady difference horizontally and vertically, like a multiplication grid. The completed pattern could be:

In simple words: This pattern shows a grid of numbers. Each row and column follows its own rule, usually by adding a fixed number. For example, the first row adds 3 each time, and the first column also adds different numbers to create a larger pattern.

🎯 Exam Tip: When asked to create a pattern, choose a simple arithmetic rule (addition, subtraction, multiplication) and apply it consistently across rows or columns to generate the numbers.

 

Question 1. Complete the pattern
Answer: This pattern shows the result of multiplying numbers made of only the digit '1'. The result is a palindromic number that first increases and then decreases.
\( 1 \times 1 = 1 \)
\( 11 \times 11 = 121 \)
\( 111 \times 111 = 12321 \)
Following this pattern, the missing lines are:
\( 1111 \times 1111 = 1234321 \)
\( 11111 \times 11111 = 123454321 \)
\( 111111 \times 111111 = 12345654321 \)
\( 1111111 \times 1111111 = 1234567654321 \)
This type of pattern often appears in number puzzles, where the number of '1's in the multiplier determines the highest digit in the middle of the product.
In simple words: When you multiply numbers made of only '1's, the answer goes up to a peak number and then comes back down. The highest number it reaches is the same as how many '1's are in each number you multiplied.

🎯 Exam Tip: For numerical patterns like these, observe how the number of digits in the input relates to the number of digits and the sequence in the output. Look for symmetry or arithmetic progressions.

 

Question 2.
1. Think of a number =
2. Add 5 to it = + 5 =
3. Multiply by 2 = x 2 =
4. Subtract 10 from it = - 10 =
5. Now divide it by 2 = ÷ 2 =
6. Is this the same number you thought of ?
Answer: Let's follow the steps by picking an example number, say 4.
1. Think of a number: Let the number be \( 4 \).
2. Add 5 to it: \( 4 + 5 = 9 \).
3. Multiply by 2: \( 9 \times 2 = 18 \).
4. Subtract 10 from it: \( 18 - 10 = 8 \).
5. Now divide it by 2: \( 8 \div 2 = 4 \).
6. Is this the same number you thought of? Yes, it is the same number, \( 4 \).
This trick always returns the original number because the operations cancel each other out: adding 5 then multiplying by 2 (which gives \( 2x + 10 \)) and then subtracting 10 (giving \( 2x \)) and finally dividing by 2 (giving \( x \)).
In simple words: Pick any number you like. Follow all the steps carefully. You will find that you always end up with the same number you started with.

🎯 Exam Tip: For number puzzles, test with a small, easy number first to understand the steps. Algebraically, this type of puzzle is designed for the operations to cancel out, returning the original value.

 

Question 3. Understand the following patterns and fill in the blanks.
Multiplication of two number written below is written above.
Answer: We need to complete the pyramid patterns by multiplying the two numbers below to get the number above them. This rule builds the pyramid upwards.
The first partial pattern:

Using the rule, `2 x 3 = 6`, `1 x 2 = 2`. Then `2 x 6 = 12`.
The completed pattern is:

The second partial pattern has a base of `2, 3, 1, 5`.

Let's fill the pyramid using multiplication:
Bottom row: \( 2, 3, 1, 5 \)
Second row (from bottom):
\( 2 \times 3 = 6 \)
\( 3 \times 1 = 3 \)
\( 1 \times 5 = 5 \)
So, the second row is \( 6, 3, 5 \).
Third row (from bottom):
\( 6 \times 3 = 18 \)
\( 3 \times 5 = 15 \)
So, the third row is \( 18, 15 \).
Top row:
\( 18 \times 15 = 270 \)
The completed pattern is:

In simple words: For these pyramid puzzles, you find the number above by multiplying the two numbers directly below it. Start from the bottom layer and work your way up to fill all the empty spaces.

🎯 Exam Tip: Always start filling pyramid patterns from the base (the lowest given row). Double-check your calculations for each step to avoid errors that can affect the entire pyramid.

 

Question 4. Similarly, fill in the blanks by doing addition instead of multiplication.
Answer: We need to complete the pyramid pattern by adding the two numbers below to get the number above them. This is the opposite of the previous question, using addition instead of multiplication.
The partial pattern:

Let's fill the pyramid using addition:
Bottom row: \( 5, 8, 7, 6 \)
Second row (from bottom):
\( 5 + 8 = 13 \)
\( 8 + 7 = 15 \)
\( 7 + 6 = 13 \)
So, the second row is \( 13, 15, 13 \).
Third row (from bottom):
\( 13 + 15 = 28 \)
\( 15 + 13 = 28 \)
So, the third row is \( 28, 28 \).
Top row:
\( 28 + 28 = 56 \)
The completed pattern is:

In simple words: To fill this pyramid, you add the two numbers directly below a box to get the number inside that box. Start from the bottom layer and keep adding upwards until all the boxes are filled.

🎯 Exam Tip: Carefully follow the operation specified (addition or multiplication). A common mistake is to confuse the operations between different pyramid patterns.

 

Question 1. Which pattern are you able to see in numbers of rectangular block. Extend it 2, 9, 16, 23, 30, .........
Answer: The rectangular block mentioned is likely a calendar snippet. The sequence of numbers \( 2, 9, 16, 23, 30 \) shows a clear pattern where each number is obtained by adding 7 to the previous number. This is a common pattern for days of the week in a calendar, where the same day repeats every 7 days.
To extend the sequence, we continue adding 7 to the last number:
\( 30 + 7 = 37 \)
\( 37 + 7 = 44 \)
\( 44 + 7 = 51 \)
\( 51 + 7 = 58 \)
So, the extended sequence is: \( 2, 9, 16, 23, 30, 37, 44, 51, 58, ... \).
This pattern is an arithmetic progression with a common difference of 7. Calendar numbers often follow this rule for days in the same column.
In simple words: The numbers in the pattern always go up by 7. To continue the pattern, just keep adding 7 to the last number you have.

🎯 Exam Tip: For sequences, always find the difference or ratio between consecutive terms. If it's constant, you've found the rule. Calendar patterns often involve adding or subtracting 7.

 

Question 2. Write sum of numbers, which lie on horizontal, vertical and diagonal line of square block.
Answer: The square block refers to a 3x3 grid of numbers, likely a calendar segment, where the central number is 14. We can reconstruct this block as:

Now, we calculate the sums:
(i) Sum of horizontal numbers: \( 13 + 14 + 15 = 42 \).
(ii) Sum of vertical numbers: \( 7 + 14 + 21 = 42 \).
(iii) Sum of diagonal numbers (top-left to bottom-right): \( 6 + 14 + 22 = 42 \).
(iv) Sum of diagonal numbers (top-right to bottom-left): \( 8 + 14 + 20 = 42 \).
This square block exhibits a property similar to a magic square where the sum across rows, columns, and main diagonals is the same. This happens because the numbers are arranged with consistent differences.
In simple words: For the group of 9 numbers, if you add up the numbers in any straight line (across, down, or corner-to-corner), the answer is always the same. Here, that sum is 42.

🎯 Exam Tip: When given a grid of numbers for sums, always clearly identify the specific numbers for each requested line (horizontal, vertical, diagonal) before performing the addition.

 

Question 3. Total of all 9 numbers in the square block = ... is it equal to 9 x 14?
Answer: Using the same square block from Question 2:

To find the total of all 9 numbers, we add them up:
\( 6 + 7 + 8 + 13 + 14 + 15 + 20 + 21 + 22 \)
\( = (6+22) + (7+21) + (8+20) + (13+15) + 14 \)
\( = 28 + 28 + 28 + 28 + 14 \) (This is one way to group them for quick addition)
\( = 84 + 14 = 98 + 14 = 126 \).
Alternatively, sum all numbers directly: \( 6+7+8+13+14+15+20+21+22 = 126 \).
Now, let's check if this is equal to \( 9 \times 14 \):
\( 9 \times 14 = 126 \).
Yes, the total of all 9 numbers in the square block is equal to \( 9 \times 14 \). This property is characteristic of these specific types of number grids, where the sum of all numbers is the product of the count of numbers and the middle number.
In simple words: If you add all 9 numbers in the box, the total is 126. If you multiply the number of boxes (9) by the middle number (14), you also get 126. So, they are the same.

🎯 Exam Tip: For magic squares or arithmetic grids, the sum of all numbers is often the product of the number of cells and the central value (if odd number of cells). Always verify the sum with a direct calculation.

 

Question 4. Recognize the pattern of horizontal, vertical and both diagonal numbers of square block and extend them.
Answer: We will use the 3x3 square block from the previous questions:

(i) Horizontal numbers: The pattern for numbers like \( 13, 14, 15 \) is obtained by adding 1 to each previous number. To extend this pattern, we continue adding 1:
\( 13, 14, 15, 16, 17, 18, 19, 20, ... \)
(ii) Vertical numbers: The pattern for numbers like \( 7, 14, 21 \) is obtained by adding 7 to each previous number (or they are multiples of 7). To extend this:
\( 7, 14, 21, 28, 35, 42, 49, 56, ... \)
(iii) First diagonal numbers (top-left to bottom-right): The pattern for \( 6, 14, 22 \) is obtained by adding 8 to each previous number. To extend this:
\( 6, 14, 22, 30, 38, 46, 54, 62, ... \)
(iv) Second diagonal numbers (top-right to bottom-left): The pattern for \( 8, 14, 20 \) is obtained by adding 6 to each previous number. To extend this:
\( 8, 14, 20, 26, 32, 38, 44, 50, 56, ... \)
Each line of numbers follows a simple arithmetic progression, making it easy to predict the next terms. This is why calendar grids are often used for pattern recognition exercises.
In simple words: Look at the numbers in a straight line (across, down, or diagonal). Find out how much is added each time. Then, keep adding that same amount to extend the list of numbers.

🎯 Exam Tip: Pay close attention to the direction (horizontal, vertical, diagonal) for each pattern. The common difference changes based on the line being considered.

 

Question 5. Make square blocks of Four x Four. Recognize the pattern between horizontal, vertical and diagonal numbers of square block and extend it.
Answer: Let's create a 4x4 block of numbers, typical of a calendar segment. We'll include the days of the week for clarity, although the pattern is in the numbers. A sample 4x4 block could be:

(i) Two patterns of horizontal numbers (found by adding 1 to each number) and their extensions:
(a) For a row like \( 9, 10, 11, 12 \), the pattern is adding 1. Extended: \( 9, 10, 11, 12, 13, 14, 15, 16, ... \)
(b) For a row like \( 16, 17, 18, 19 \), the pattern is adding 1. Extended: \( 16, 17, 18, 19, 20, 21, 22, 23, ... \)
(ii) Two patterns of vertical numbers (from the 4x4 block) and their extensions:
(a) For a column like (e.g., Tuesday): \( 2, 9, 16, 23 \). The numbers are obtained by adding 7 to each number. Extended: \( 2, 9, 16, 23, 30, 37, 44, 51, 58, ... \)
(b) For another column (e.g., Thursday): \( 4, 11, 18, 25 \). The numbers are obtained by adding 7 to each number. Extended: \( 4, 11, 18, 25, 32, 39, 46, 53, ... \)
(iii) Two patterns of diagonal numbers (from the 4x4 block) and their extensions:
(a) First diagonal (e.g., starting top-left: 1, 8, 15, 22). These numbers are obtained by adding 7 to each number. Extended: \( 1, 8, 15, 22, 29, 36, 43, 50, ... \)
(b) Second diagonal (e.g., starting top-right: 5, 11, 17, 23). These numbers are obtained by adding 6 to each number. Extended: \( 5, 11, 17, 23, 29, 35, 41, 47, 53, ... \)
Calendar grids offer many predictable arithmetic patterns both horizontally, vertically, and diagonally, which are fun to discover.
In simple words: Draw a 4x4 calendar grid. Then, for lines of numbers going across, down, or diagonally, find out how much is added each time. Keep adding that amount to make the number patterns longer.

🎯 Exam Tip: When exploring patterns in larger grids like 4x4, pick distinct horizontal, vertical, and diagonal lines to demonstrate a variety of pattern types and their extensions.