NCERT Solutions Ganita Prakash (Part 1) Chapter 4 –85Section 4.1 The Notion of Letter-Numbers — Figure it Out

Book page 84 Updated on2026-09-05

Q1.

Write formulas for the perimeter of: (a) triangle with all sides equal. (b) a regular pentagon (as we have learnt last year, we use the word ‘regular’ to say that all sidelengths and angle measures are equal) (c) a regular hexagon

Answer

In each case the perimeter is the sidelength added as many times as there are sides. Let the sidelength be a units.

Shape	Number of equal sides	Formula for perimeter
Equilateral triangle	3	3a
Regular pentagon	5	5a
Regular hexagon	6	6a

(a) Perimeter = a + a + a = 3a units
(b) Perimeter = a + a + a + a + a = 5a units
(c) Perimeter = 6a units

Why it happens: In a regular polygon every side is the same length. Adding the same number several times is multiplication, so a polygon with k equal sides of length a has perimeter ka.

Q2.

Munirathna has a 20 m long pipe. However, he wants a longer watering pipe for his garden. He joins another pipe of some length to this one. Give the expression for the combined length of the pipe. Use the letter-number ‘k’ to denote the length in meters of the other pipe.

Answer

Joining two pipes end to end adds their lengths.

Combined length = 20 + k
= (20 + k) metres

Check it yourself: If the second pipe is 15 m, k = 15 and the total is 20 + 15 = 35 m. If it is 8 m, the total is 28 m — one expression covers every case.

Q3.

What is the total amount Krithika has, if she has the following numbers of notes of ₹100, ₹20 and ₹5? Complete the following table: (rows — 3, 5, 6 | 6 × 100 + 4 × 20 + 3 × 5 = 695 | 8, 4, z | x, y, z)

Answer

Each row follows the same rule: (number of ₹100 notes × 100) + (number of ₹20 notes × 20) + (number of ₹5 notes × 5).

No. of ₹100 notes	No. of ₹20 notes	No. of ₹5 notes	Expression and total amount
3	5	6	3 × 100 + 5 × 20 + 6 × 5 = 430
6	4	3	6 × 100 + 4 × 20 + 3 × 5 = 695
8	4	z	8 × 100 + 4 × 20 + z × 5 = 880 + 5z
x	y	z	x × 100 + y × 20 + z × 5 = 100x + 20y + 5z

Row 1: 300 + 100 + 30 = ₹430
Row 2: 600 + 80 + 15 = ₹695
Row 3: 800 + 80 + 5z = ₹(880 + 5z)
Row 4: ₹(100x + 20y + 5z)

Why it happens: The last row is the general formula. Put x = 8, y = 4 into it and you get 800 + 80 + 5z — exactly row 3. The rows above are only special cases of the same expression.

Q4.

Venkatalakshmi owns a flour mill. It takes 10 seconds for the roller mill to start running. Once it is running, each kg of grain takes 8 seconds to grind into powder. Which of the expressions below describes the time taken to complete grind ‘y’ kg of grain, assuming the machine is off initially? (a) 10 + 8 + y (b) (10 + 8) × y (c) 10 × 8 × y (d) 10 + 8 × y (e) 10 × y + 8

Answer

The answer is (d) 10 + 8 × y.

Starting time = 10 seconds (only once)
Grinding time = 8 seconds for each kg = 8 × y
Total = 10 + 8y seconds

Why it happens: The 10 seconds of starting up happens once, no matter how much grain there is, so it is added, not multiplied. The 8 seconds repeats for every kilogram, so it is multiplied by y.

Check it yourself: For y = 5 kg, time = 10 + 40 = 50 s. Option (b) would give 90 s and option (c) 400 s — both far too long.

Q5.

Write algebraic expressions using letters of your choice. (a) 5 more than a number (b) 4 less than a number (c) 2 less than 13 times a number (d) 13 less than 2 times a number

Answer

Let the number be n.

In words	Expression
(a) 5 more than a number	n + 5
(b) 4 less than a number	n – 4
(c) 2 less than 13 times a number	13n – 2
(d) 13 less than 2 times a number	2n – 13

Why it happens: (c) and (d) use the same two numbers but in opposite roles. For n = 10, 13n – 2 = 128 while 2n – 13 = 7. Read the sentence carefully — ‘times’ tells you what to multiply, ‘less than’ tells you what to subtract at the end.

Q6.

Describe situations corresponding to the following algebraic expressions: (a) 8 × x + 3 × y (b) 15 × j – 2 × k

Answer

(a) 8 × x + 3 × y

A stationery shop sells a pen for ₹x and a notebook for ₹y. Abha buys 8 pens and 3 notebooks. Her bill is 8x + 3y rupees.

(b) 15 × j – 2 × k

A carpenter makes 15 stools every day and works for j days, so he makes 15j stools. On k of those days, 2 stools got damaged. The number of good stools left is 15j – 2k.

Try This: Many different stories fit one expression. Write a cricket story for 8x + 3y — say x runs for every boundary and y runs for every six.

Q7.

In a calendar month, if any 2 × 3 grid full of dates is chosen as shown in the picture, write expressions for the dates in the blank cells if the bottom middle cell has date ‘w’.

Answer

In a calendar, moving one step right adds 1 and moving one step up subtracts 7. Starting from w in the bottom-middle cell:

w – 8	w – 7	w – 6
w – 1	w	w + 1

Left of w = w – 1, right of w = w + 1
Directly above w = w – 7
Above-left = w – 7 – 1 = w – 8
Above-right = w – 7 + 1 = w – 6

Why it happens: A calendar row holds 7 days, so the cell just above any date is exactly 7 less.

Check it yourself: In the picture the marked block is 12, 13, 14 over 19, 20, 21. Here w = 20, so w – 1 = 19, w + 1 = 21, w – 7 = 13, w – 8 = 12 and w – 6 = 14. Every one matches.

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