Expressions Using Letter-Numbers | IT

Question 35

Numbers are written in a particular sequence in this endless 4-column grid.

(a) Give expressions to generate all the numbers in a given column (1, 2, 3, 4).

(b) In which row and column will the following numbers appear:

(i) 124

(ii) 147

(iii) 201

(d) Observe the positions of multiples of 3.

Do you see any pattern in it? List other patterns that you see.

Solution

Understand the Question

The endless grid has $4$ columns, with numbers filled sequentially across rows: Row 1 has $1, 2, 3, 4$ , Row 2 has $5, 6, 7, 8$ , and so on.
Since each row contains $4$ numbers, moving down one row in the same column increases the value by $4$ .
The general formula for the number $N$ at row $r$ and column $c$ is: $N = c + 4(r - 1)$
To find the row $r$ $r$ and column $c$ $c$ for any given number $N$ $N$ :
- Column $c$ corresponds to the remainder when $N$ is divided by $4$ (if remainder is $0$ , $c = 4$ ).
- Row $r$ is given by $r = \dfrac{N - c}{4} + 1$ .

(a) Give expressions to generate all the numbers in a given column (1, 2, 3, 4).

Step 1 · Derive Expressions for Each Column

Let $r$ be the row number ( $r \ge 1$ ). Since each row contains $4$ numbers, numbers in any column increase by $4$ with each successive row. Diagram 1

Column 1: $1, 5, 9, 13, \dots \implies 1 + 4(r - 1)$
Column 2: $2, 6, 10, 14, \dots \implies 2 + 4(r - 1)$
Column 3: $3, 7, 11, 15, \dots \implies 3 + 4(r - 1)$
Column 4: $4, 8, 12, 16, \dots \implies 4 + 4(r - 1)$

Answer

(a) $\begin{aligned}\text{Column 1: } & 1 + 4(r - 1) \\\text{Column 2: } & 2 + 4(r - 1) \\\text{Column 3: } & 3 + 4(r - 1) \\\text{Column 4: } & 4 + 4(r - 1)\end{aligned}$

**(b) In which row and column will the following numbers appear:

(i) 124

(ii) 147

(iii) 201**

Step 1 · Determine Position for Given Numbers

Using $r = \dfrac{N - c}{4} + 1$ , where $c$ is the column determined by $N \pmod 4$ :

(i) For $N = 124$ : $124 \div 4 = 31 \text{ with remainder } 0 \implies c = 4$ $r = \dfrac{124 - 4}{4} + 1 = \dfrac{120}{4} + 1 = 30 + 1 = 31$ $\implies \text{Row } 31, \text{ Column } 4$

(ii) For $N = 147$ : $147 \div 4 = 36 \text{ with remainder } 3 \implies c = 3$ $r = \dfrac{147 - 3}{4} + 1 = \dfrac{144}{4} + 1 = 36 + 1 = 37$ $\implies \text{Row } 37, \text{ Column } 3$

(iii) For $N = 201$ : $201 \div 4 = 50 \text{ with remainder } 1 \implies c = 1$ $r = \dfrac{201 - 1}{4} + 1 = \dfrac{200}{4} + 1 = 50 + 1 = 51$ $\implies \text{Row } 51, \text{ Column } 1$

Answer

(b) (i) Row $31$ , Column $4$

(ii) Row $37$ , Column $3$

(iii) Row $51$ , Column $1$

(c) What number appears in row $r$ and column $c$ ?

Step 1 · General Expression

Before row $r$ , there are $(r - 1)$ rows containing $4$ numbers each. Adding the column offset $c$ gives: $N = c + 4(r - 1)$

Answer

(c) $N = c + 4(r - 1)$

(d) Observe the positions of multiples of 3. Do you see any pattern in it? List other patterns that you see.

Step 1 · Identify Patterns

Listing the positions $(\text{row } r, \text{column } c)$ for the first several multiples of $3$ :

$3 \rightarrow (1, 3)$
$6 \rightarrow (2, 2)$
$9 \rightarrow (3, 1)$
$12 \rightarrow (3, 4)$
$15 \rightarrow (4, 3)$
$18 \rightarrow (5, 2)$
$21 \rightarrow (6, 1)$
$24 \rightarrow (6, 4)$
$27 \rightarrow (7, 3)$
$30 \rightarrow (8, 2)$

Pattern for multiples of 3:

The column numbers repeat in the cyclic sequence 3, 2, 1, 4 every 4 multiples.
The row number increases by $3$ across each complete repetition of this cycle.

Other patterns:

Numbers increase by $1$ horizontally across any row.
Numbers increase by $4$ vertically down any column.
All numbers in Column 4 are multiples of $4$ .
All numbers in Column 1 are of the form $4k + 1$ .

Answer

(d) Multiples of $3$ follow a repeating column cycle of 3, 2, 1, 4. Other patterns include: column numbers increase by $4$ down the grid, row numbers increase by $1$ from left to right, and Column 4 contains all multiples of $4$ .

Common Mistakes

Zero Remainder Confusion: When $N$ is perfectly divisible by $4$ (remainder $0$ ), the number is in Column 4 (not Column $0$ ).
Off-by-One in Row Calculation: Forgetting to add $1$ in $r = \dfrac{N - c}{4} + 1$ or writing $4r$ instead of $4(r - 1)$ in the general term.

More questions in IT

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Context: Consider the number sequence: $4, 8, 12, 16, 20, 24, 28, \dots$

Q. Find an algebraic expression to get the $n\text{th}$ term of this sequence.

Mind the Mistake, Mend the Mistake

Some simplifications are shown below where the letter-numbers are replaced by numbers and the value of the expression is obtained.

Observe each of them and identify if there is a mistake.
If you think there is a mistake, try to explain what might have gone wrong.
Then, correct it and give the value of the expression.

Context: Example 5: Here is a table showing the number of pencils and erasers sold in a shop. The price per pencil is $c$ , and the price per eraser is $d$ .

The total money earned by the sale of pencils is $5c + 3c + 10c = 18c$ .

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Context: Example 5: Here is a table showing the number of pencils and erasers sold in a shop. The price per pencil is $c$ , and the price per eraser is $d$ . Find the total money earned by the shopkeeper during these three days.

Q. Write the expression for the total money earned by selling erasers. Then, simplify the expression.

Q10

Context: In this problem, we saw the expression $5c + 3c + 10c$ getting simplified to the expression $18c$ .

Q. Check that both expressions take the same value when $c$ is replaced by different numbers.

Q11

Context: The expression $(40x + 75y) - (6x + 10y)$ is simplified to $34x + 65y$ , which is the total amount paid in rupees.

Q. Could we have written the initial expression as $(40x + 75y) + (-6x - 10y)$ ?

Q12

Context: The total amount in rupees paid at the beginning is $40x + 75y$ , and the total amount returned is $6x + 10y$ . So, the total amount paid = $(40x + 75y) - (6x + 10y)$ .

Q. Can we simplify this expression? If yes, how? If not, why not?

Q13

What if there is no penalty? What will be the value of $q$ in that situation?

Q14

Context: In a quiz, $p$ represents the score for a correct answer and $q$ represents the penalty for an incorrect answer. Krishita's total score after three rounds is $23p - 7q$ .

Q. Give some possible scores for Krishita in the three rounds so that they add up to give $23p - 7q$ .

Q15

Context: Charu's total score after three rounds is $21p - 9q$ , and Krishita's total score is $23p - 7q$ , where $p$ represents the score for a correct answer and $q$ represents the penalty for an incorrect answer.

Q. Can we say who scored more? Can you explain why?

Q16

Context: Consider the expression representing the difference between Krishita's and Charu's scores: $(23p - 7q) - (21p - 9q)$ .

Q. Simplify this expression further.

Q17

Fill the blanks below by replacing the letter-numbers by numbers; an example is shown. Then compare the values that $5u$ and $5 + u$ take.

Q18

Context: Let us compare the values that the expressions $10y - 3$ and $10(y - 3)$ take for different values of $y$ .

Q. After filling in the two diagrams, do you think the two expressions are equal?

Q19

Mind the Mistake, Mend the Mistake

Some simplifications of algebraic expressions are done below. The expression on the right-hand side should be in its simplest form.

Observe each of them and see if there is a mistake.
If you think there is a mistake, try to explain what might have gone wrong.
Then, simplify it correctly.

Q20

Take a look at all the corrected simplest forms (i.e. brackets are removed, like terms are added, and terms with only numbers are also added). Is there any relation between the number of terms and the number of letter-numbers these expressions have?

Q21

Find the formulas of the number machines below and write the expression for each set of inputs.

Q22

Now, make a formula on your own. Write a few number machines as examples using that formula. Challenge your classmates to figure it out!

Q23

Context: Design C appears at positions that are multiples of 3 (remainder 0). Design B appears at positions that are 1 less than a multiple of 3 (remainder 2). Design A appears at positions that are 2 less than a multiple of 3 (remainder 1).

(a) Can the remainder obtained by dividing the position number by 3 be used for this? Observe the table below.

(b) Use this to find what design appears at positions 99, 122, and 148.

Q24

Context: Let us take the marked $2 \times 2$ square, and consider the numbers lying on the diagonals; 12 and 20; 13 and 19. Find their sums; $12 + 20$ , $13 + 19$ . What do you observe? They are equal. Let us extend the numbers in the calendar beyond 30, creating endless rows.

Q. Will the diagonal sums be equal in every $2 \times 2$ square in this endless grid? How can we be sure?

Q25

Context: Let us consider a $2 \times 2$ square. Its top left number can be any number. Let us call it $a$ .

Q. Given that we know the top left number, how do we find the other numbers in this $2 \times 2$ square?

Q26

Verify this expression for diagonal sums by considering any $2 \times 2$ square and taking its top left number to be $a$ .

Q27

Context: Consider a set of numbers from the calendar (having endless rows) forming under the following shape:

Q. Find the sum of all the numbers. Compare it with the number in the centre: 15. Repeat this for another set of numbers that forms this shape. What do you observe?

Q28

Q. Will this always happen? How do you show this?

[Hint: Consider a general set of numbers that forms this shape. Take the number at the centre to be $a$ . Express the other numbers in terms of $a$ .]

Q29

Context: Consider a set of numbers from the calendar (having endless rows) forming under the following shape:

We see that the total sum is always 5 times the number in the centre.

Q. Find other shapes for which the sum of the numbers within the figure is always a multiple of one of the numbers.

Q30

How many matchsticks will there be in Step 33, Step 84, and Step 108? Of course, we can draw and count, but is there a quicker way to find the answers using the pattern present here?

Q31

The number of matchsticks needed to make 33 triangles (Step 33) is ________. Similarly, find the number of matchsticks needed for Step 84 and Step 108.

Q32

Context: The expressions describing the rule/formula to find the number of matchsticks at step $y$ are $3 + 2 \times (y - 1)$ and $2y + 1$ .

Q. Does the above expression also give the number of matchsticks at each step correctly? Are these expressions the same?

Q33

Context: Matchsticks are placed in two orientations — (a) horizontal ones at the top and bottom, and (b) the ones placed diagonally in the middle. For example, in step 2 there are 2 matchsticks placed horizontally and 3 matchsticks placed diagonally.

Q. What are these numbers in Step 3 and Step 4?

Q34

Q. How does the number of matchsticks change in each orientation as the steps increase? Write an expression for the number of matchsticks at Step $y$ in each orientation. Do the two expressions add up to $2y + 1$ ?

Q35

Numbers are written in a particular sequence in this endless 4-column grid.

(a) Give expressions to generate all the numbers in a given column (1, 2, 3, 4).

(b) In which row and column will the following numbers appear:

(i) 124

(ii) 147

(iii) 201

(d) Observe the positions of multiples of 3.

Do you see any pattern in it? List other patterns that you see.