Class 7 Mathematics

Chapter 4 Solutions — Expressions using Letter Numbers

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Solutions

Overview

Step-by-step NCERT solutions for Expressions using Letter Numbers (Chapter 4, NCERT Class 7 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Expressions using Letter Numbers textbook chapter.

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What these solutions cover

All 26 questions in Expressions using Letter Numbers are solved in the PDF. Here's what's inside, exercise by exercise:

The Notion of Letter-Numbers

1.
Write formulas for the perimeter of:
- (a) a triangle with all sides equal,
- (b) a regular pentagon,
- (c) a regular hexagon.
Munirathna has a 20 m long pipe. 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 metres of the other pipe.
3.
What is the total amount Krithika has if she has the following numbers of notes of Rs 100, Rs 20, and Rs 5? Complete the table:
- (i) 3 notes of Rs 100, 5 of Rs 20, 6 of Rs 5;
- (ii) 6 of Rs 100, 4 of Rs 20, 3 of Rs 5 (row given: expression = 6×100 + 4×20 + 3×5 = 695);
- (iii) 8 of Rs 100, 4 of Rs 20, z of Rs 5;
- (iv) x of Rs 100, y of Rs 20, z of Rs 5.
4.
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. Which expression describes the time taken to grind 'y' kg of grain (machine is off initially)?
- (a) 10 + 8 + y
- (b) (10 + 8) × y
- (c) 10 × 8 × y
- (d) 10 + 8 × y
- (e) 10 × y + 8
5.
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.
6.
Describe situations corresponding to the following algebraic expressions:
- (a) 8 × x + 3 × y
- (b) 15 × j − 2 × k.
In a calendar month, if any 2 × 3 grid full of dates is chosen, write expressions for the dates in the blank cells if the bottom middle cell has date 'w'. (The grid shows the bottom row with w−1 and w pre-filled.)

Simplification of Algebraic Expressions — Figure it Out

1.
Add the numbers in each picture below. Write their corresponding expressions and simplify them.
- (i) Grid with 5y, −6, x (top row) and x, 2, 5y (bottom row).
- (ii) Grid 4×4 with 2p,3q,−2,3 / 3q,2p,3,−2 / 2p,3q / 3q,2p.
- (iii) Grid 4×4 with −5g,5k,5k,−5g / 5k,5k,5k,5k / 5k,5k,5k,5k / −5g,5k,5k,−5g.
2.
Simplify each of the following expressions:
- (a) p+p+p+p; p+p+p+q
- (b) p+q+p−q
- (c) p−q+p−q
- (d) p+q−p+q
- (e) p+q−(p+q)
- (f) p−q−p−q
- (g) 2d−d−d−d
- (h) 2d−d−d−c
- (i) 2d−d−(d−c)
- (j) 2d−(d−d)−c
- (k) 2d−d−c−c.

Mind the Mistake, Mend the Mistake — Algebraic Simplification

Some simplifications of algebraic expressions are given with claimed simplest forms. Identify mistakes and give the correct simplest form for each: (1) 3a+2b = 5 (2) 3b−2b−b = 0 (3) 6(p+2) = 6p+8 (4) (4x+3y)−(3x+4y) = x+y (5) 5−(2−6z) = 3−6z (6) 2+(x+3) = 2x−6 (7) 2y+(3y−6) = −y+6 (8) 7p−p+5q−2q = 7p+3q (9) 5(2w+3x+4w) = 10w+15x+20w (10) 3j+6k+9h+12 = 3(j+2k+3h+4) (11) 4(2r+3s+5) = −20−8r−12s.

Pick Patterns and Reveal Relationships — Number Machines

Find the formulas for the two number machines and write the expression for each set of inputs. Machine 1 (blue): inputs (5,2)→5; (8,1)→7; (9,11)→18; (10,10)→18; (a,b)→? Machine 2 (purple): inputs (4,1)→5; (6,0)→1; (3,2)→7; (10,3)→?; (a,b)→?

Pick Patterns and Reveal Relationships — Figure it Out

1.
One plate of Jowar roti costs Rs 30 and one plate of Pulao costs Rs 20. If x plates of Jowar roti and y plates of Pulao were ordered in a day, which expression(s) describe the total amount earned?
- (a) 30x+20y
- (b) (30+20)×(x+y)
- (c) 20x+30y
- (d) (30+20)×x+y
- (e) 30x−20y.
2.
Pushpita sells two types of flowers: champak and marigold. 'p' customers bought only champak, 'q' customers bought only marigold, and 'r' customers bought both. She gave a national flag to every customer. How many flags did she give away?
- (a) p+q+r
- (b) p+q+2r
- (c) 2×(p+q+r)
- (d) p+q+r+2
- (e) p+q+r+1
- (f) 2×(p+q).
3.
A snail climbing a well: during the day it climbs u cm, during the night it slips down d cm. This happens for 10 days and 10 nights.
- (a) Write an expression for how far the snail is from its starting position.
- (b) What can we say about the snail's movement if d > u?
Radha practices cycling daily. The first week she cycles 5 km every day. Every week she increases the daily distance by 'z' km. How many kilometres would Radha have cycled after 3 weeks?
In the flow diagram, the expression w+2 is the starting point. One path (given): w+2 → (+3) → w+5 → (×4) → 4w+20. Another path ends at 3w−6. Fill in the missing blanks on all paths. The ovals contain expressions and the boxes contain operations.
6.
A local train from Yahapur to Vahapur stops at three stations at equal distances. Travel time between any two consecutive stations is t minutes. The train stops for 2 minutes at each of the three stations.
- (a) If t = 4, what is the total travel time?
- (b) Write the algebraic expression for the total travel time.
7.
Simplify the following expressions:
- (a) 3a+9b−6+8a−4b−7a+16
- (b) 3(3a−3b)−8a−4b−16
- (c) 2(2x−3)+8x+12
- (d) 8x−(2x−3)+12
- (e) 8h−(5+7h)+9
- (f) 23+4(6m−3n)−8n−3m−18.
8.
Add the expressions given below:
- (a) 4d−7c+9 and 8c−11+9d
- (b) −6f+19−8s and −23+13f+12s
- (c) 8d−14c+9 and 16c−(11+9d)
- (d) 6f−20+8s and 23−13f−12s
- (e) 13m−12n and 12n−13m
- (f) −26m+24n and 26m−24n.
9.
Subtract the expressions given below:
- (a) 9a−6b+14 from 6a+9b−18
- (b) −15x+13−9y from 7y−10+3x
- (c) 17g+9−7h from 11−10g+3h
- (d) 9a−6b+14 from 6a−(9b+18)
- (e) 10x+2+10y from −3y+8−3x
- (f) 8g+4h−10 from 7h−8g+20.
10.
Describe situations corresponding to the following algebraic expressions:
- (a) 8x + 3y
- (b) 15x − 2x.
A straight rope: if cut once → 2 pieces; folded once and cut → 3 pieces. Observe the pattern and find: how many pieces when folded 10 times and cut? What is the expression for pieces when folded r times and cut?
Look at the matchstick pattern of squares placed in a row (each adjacent pair shares one side). How many matchsticks are required to make 10 such squares? How many are required to make w squares?
A traffic signal cycles through colours in the sequence: Red (pos 1), Yellow (pos 2), Green (pos 3), Yellow (pos 4), Red (pos 5), … Find the colour at positions 90, 190, and 343. Write expressions to describe positions for each colour.
Observe the growing cross/plus-shaped square pattern (Step 1 has 5 squares; each step adds 4 squares, one to each arm). How many squares in Step 4, Step 10, Step 50? Write a general formula. How does the formula change for counting vertices of all squares?
15.
Numbers are written in a 4-column endless grid (rows: 1,2,3,4 / 5,6,7,8 / 9,10,11,12 …).
- (a) Give expressions for all numbers in each column (1, 2, 3, 4).
- (b) Find row and column for:
- (i) 124
- (ii) 147
- (iii) 201.
- (c) What number appears in row r and column c?
- (d) Observe the positions of multiples of 3.

Sample

Worked examples

3 questions from Expressions using Letter Numbers, solved step by step.

The Notion of Letter-Numbers

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

(a) 8 × x + 3 × y:

A shopkeeper sells pens at Rs x each and notebooks at Rs y each.

A customer buys 8 pens and 3 notebooks.

Total cost = 8x + 3y rupees.

(b) 15 × j − 2 × k:

A factory produces 15 chairs each day for j days, making 15j chairs in total.

Over k days, 2 chairs per day break, so 2k chairs are broken.

Number of good chairs remaining = 15j − 2k.

Answer: (a) Cost of 8 pens at Rs x each and 3 notebooks at Rs y each (b) Chairs produced in j days minus chairs broken in k days

Pick Patterns and Reveal Relationships — Number Machines

Find the formulas for the two number machines and write the expression for each set of inputs. Machine 1 (blue): inputs (5,2)→5; (8,1)→7; (9,11)→18; (10,10)→18; (a,b)→? Machine 2 (purple): inputs (4,1)→5; (6,0)→1; (3,2)→7; (10,3)→?; (a,b)→?

Machine 1 (blue):

Check: (5,2): 5+2−2=5 ✓; (8,1): 8+1−2=7 ✓; (9,11): 9+11−2=18 ✓; (10,10): 10+10−2=18 ✓

Formula: sum of the two inputs minus 2, i.e., a + b − 2.

Expressions: 5+2−2=5; 8+1−2=7; 9+11−2=18; 10+10−2=18; a+b−2

Machine 2 (purple):

Check: (4,1): 4×1+1=5 ✓; (6,0): 6×0+1=1 ✓; (3,2): 3×2+1=7 ✓

Formula: product of the two inputs plus 1, i.e., a × b + 1 = ab + 1.

Expressions: 4×1+1=5; 6×0+1=1; 3×2+1=7; 10×3+1=31; ab+1

Answer: Machine 1 formula: a + b − 2; (a,b) expression: a+b−2. Machine 2 formula: ab + 1; (10,3)→31; (a,b) expression: ab+1

Pick Patterns and Reveal Relationships — Figure it Out

A snail climbing a well: during the day it climbs u cm, during the night it slips down d cm. This happens for 10 days and 10 nights. (a) Write an expression for how far the snail is from its starting position. (b) What can we say about the snail's movement if d > u?

(a) In each full day-night cycle:

Net movement = climbs up u − slips down d = u − d cm.

After 10 cycles: total displacement = 10 × (u − d) = 10(u − d) cm from starting position.

(Positive means above start; negative means below.)

(b) If d > u, then u − d < 0, so each cycle the snail moves downward.

After 10 cycles the snail is 10(u−d) cm below its starting position.

The snail moves away from the top of the well and will never reach the top.

Answer: (a) 10(u − d) cm from starting position (b) If d > u, the snail slips backward overall and will never reach the top

All 26 questions in Expressions using Letter Numbers are solved step by step in the Solutions PDF above.

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Chapter 4 Solutions — Expressions using Letter Numbers

Overview

What these solutions cover

The Notion of Letter-Numbers

Simplification of Algebraic Expressions — Figure it Out

Mind the Mistake, Mend the Mistake — Algebraic Simplification

Pick Patterns and Reveal Relationships — Number Machines

Pick Patterns and Reveal Relationships — Figure it Out

Worked examples

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

Find the formulas for the two number machines and write the expression for each set of inputs. Machine 1 (blue): inputs (5,2)→5; (8,1)→7; (9,11)→18; (10,10)→18; (a,b)→? Machine 2 (purple): inputs (4,1)→5; (6,0)→1; (3,2)→7; (10,3)→?; (a,b)→?

A snail climbing a well: during the day it climbs u cm, during the night it slips down d cm. This happens for 10 days and 10 nights. (a) Write an expression for how far the snail is from its starting position. (b) What can we say about the snail's movement if d > u?

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