Class 6 Maths - Algebra NCERT Solutions

Chapter 11: Algebra (NCERT Solutions)

Exercise 11.1 (Variables and Constants)

Q1. Find the rule which gives the number of matchsticks required to make the following matchstick patterns. Use a variable to write the rule.
(a) A pattern of letter T (b) A pattern of letter Z (c) A pattern of letter U (d) A pattern of letter V (e) A pattern of letter E

(a) Letter T requires 2 matchsticks. For n letters T: 2n matchsticks.

(b) Letter Z requires 3 matchsticks. For n letters Z: 3n matchsticks.

(d) Letter V requires 2 matchsticks. For n letters V: 2n matchsticks.

(e) Letter E requires 5 matchsticks. For n letters E: 5n matchsticks.

Q2. We already know the rule for the pattern of letters L, C and F. Some of the letters from Q1 (given above) give us the same rule as that given by L. Which are these? Why?

Rule for letter L = 2n (2 matchsticks per letter).
Letters T and V also need 2 matchsticks each. So they have the same rule: 2n.
Reason: Each requires exactly 2 matchsticks to form.

Q3. Cadets are marching in a parade. There are 5 cadets in a row. What is the rule which gives the number of cadets, given the number of rows? (Use n for the number of rows.)

Number of cadets = 5 × n = 5n.

For n rows with 5 cadets each, total cadets = 5n.

Q5. To find the sum of three whole numbers, Rita uses the rule: sum = first number + second number + third number (s = a + b + c). If a = 12, b = 8 and c = 3, find s.

s = a + b + c = 12 + 8 + 3 = 23.

Q6. The length of a rectangular hall is 4 metres less than 3 times the breadth of the hall. What is the length, if the breadth is b metres?

Length = 3 × breadth − 4 = 3b − 4 metres.

Q7. A bus travels at v km per hour. It is going from Daspur to Beespur. After the bus has travelled 5 hours, what is the distance still to be covered?
(Note: the distance from Daspur to Beespur is 20 km more than the distance covered in 5 hours.)

Distance covered in 5 hours = 5v km.
Distance still to cover = 5v + 20 km.
(Total distance = 5v + 20)

Exercise 11.2 (Expressions & Equations)

Q1. The side of an equilateral triangle is shown by l. Express the perimeter of the equilateral triangle using l.

Perimeter of equilateral triangle = sum of all three sides = l + l + l = 3l.

Q2. The side of a regular hexagon (6 equal sides) is denoted by l. Express the perimeter of the hexagon using l.

Perimeter = 6 × l = 6l.

Q3. A cube has a side of length l. Write the formula for its total surface area and volume.

Total Surface Area = 6l² (6 square faces each of area l²). Written as 6 × l × l.

Volume = l × l × l = l³.

Q4. Leela is Radha's younger sister. Leela is 4 years younger than Radha. Can you write Leela's age in terms of Radha's age? Take Radha's age to be x years.

Radha's age = x years.
Leela's age = x − 4 = (x − 4) years.

Q5. Mother has made laddus. She gives some laddus to guests and family members; still 5 laddus remain. If she had made n laddus, how many laddus did she give away?

Laddus given away = Total − Remaining = n − 5 = (n − 5) laddus.

Q6. Oranges are to be transferred from larger boxes into smaller boxes. When a large box is emptied, the oranges from it fill two smaller boxes and still 10 oranges remain outside. If the number of oranges in a small box are taken to be x, what is the number of oranges in the larger box?

Oranges in larger box = 2 small boxes + 10 remaining = 2 × x + 10 = 2x + 10.

Exercise 11.3 (Equations & Solution)

Q1. Make two expressions using n, 11, 15 and 3 with arithmetic operations.

Examples: 11n + 15 and 3n − 15 + 11.

Q2. Identify the operations on variables in these expressions:
(a) y + 5 (b) y − 5 (c) 5y (d) y/5 (e) 4y + 5 (f) 4y − 5 (g) y² (h) 2y²

(a) y + 5 → Addition of 5 to y.

(b) y − 5 → Subtraction of 5 from y.

(d) y/5 → Division of y by 5.

(e) 4y + 5 → y multiplied by 4, then 5 added.

(f) 4y − 5 → y multiplied by 4, then 5 subtracted.

(g) y² → y multiplied by itself (y squared).

(h) 2y² → y squared, then multiplied by 2.

Exercise 11.4 (Solving Equations)

Q1. Answer the following:
(a) Take Sarita's present age to be y years. What will be her age 5 years from now?
(b) What was her age 3 years back?
(c) Sarita's grandfather is 6 times her age. What is his age?
(d) Grandmother is 2 years younger than grandfather. What is grandmother's age?

(a) Age after 5 years = y + 5

(b) Age 3 years back = y − 3

(d) Grandmother's age = 6y − 2 = (6y − 2) years

Q2. Solve the following equations by trial and error method:
(a) 5 + x = 9 (b) p − 3 = 7 (c) 4p − 2 = 18 (d) 2b/3 = 6

(a) 5 + x = 9 → x = 4 (try x=4: 5+4=9 ✓)

(b) p − 3 = 7 → p = 10 (try p=10: 10−3=7 ✓)

(d) 2b/3 = 6 → 2b = 18 → b = 9 (try b=9: 2×9/3=6 ✓)

Q5. Raju's father's age is 3 years more than twice Raju's age. Raju's father is 27 years old. Set up an equation for this and solve it to find Raju's age.

Let Raju's age = x years.
Father's age = 2x + 3.
Equation: 2x + 3 = 27.
2x = 27 − 3 = 24.
x = 24 / 2 = 12 years.
Raju is 12 years old.

Q6. There are two types of boxes containing mangoes. Each box of the larger type contains 4 more mangoes than the number of mangoes contained in 8 boxes of the smaller type. Each smaller box has m mangoes. Set up an equation to find the number of mangoes in the larger box.

Mangoes in 8 small boxes = 8m.
Mangoes in large box = 8m + 4 = (8m + 4) mangoes.

Exercise 11.5 (More Equations)

Q1. State which of the following are equations (with a variable). Give reason for your answer:
(a) 17 = x + 7 (b) (t − 7) > 5 (c) 4/2 = 2 (d) (7 × 3) − 19 = 8 (e) 5 × 4 − 8 = 2x (f) x − 2 = 0

(a) 17 = x + 7 → Yes, equation (has variable x, LHS = RHS with =).

(b) (t − 7) > 5 → No (it's an inequality, not an equation).

(d) (7×3) − 19 = 8 → No variable (purely numerical).

(e) 5×4 − 8 = 2x → Yes, equation (has variable x).

(f) x − 2 = 0 → Yes, equation (has variable x).

Q2. Complete the table and find by inspection the solution to the equation m + 10 = 16.

m	m + 10
1	11
2	12
5	15
6	16 ✓ (solution)
10	20

Solution: m = 6.

Class 6 Maths - Algebra Practice Questions

Chapter 11: Algebra (Practice Questions)

RD Sharma / HOT Practice

Q1. Write expressions for the following statements:
(a) 7 added to p (b) 7 subtracted from p (c) p multiplied by 7 (d) p divided by 7 (e) 7 subtracted from −p (f) p multiplied by −5 (g) p divided by 5 added to 5

(a) p + 7 (b) p − 7 (c) 7p (d) p/7 (e) −p − 7 (f) −5p (g) p/5 + 5

Q2. Solve (by inspection/trial):
(a) x + 5 = 13 (b) y − 4 = 9 (c) 3z = 36 (d) n/4 = 3 (e) 2m + 3 = 15 (f) 5t − 1 = 19

(a) x + 5 = 13 → x = 8

(b) y − 4 = 9 → y = 13

(d) n/4 = 3 → n = 12

(e) 2m + 3 = 15 → 2m = 12 → m = 6

(f) 5t − 1 = 19 → 5t = 20 → t = 4

Q3. Aman's age is twice his sister's age. If his sister is x years old, write Aman's age. If the sum of their ages is 30, find their individual ages.

Aman's age = 2x. Sum = x + 2x = 3x = 30 → x = 10.
Sister = 10 years. Aman = 20 years.

Q4. Evaluate the following expressions for n = 3:
(a) 2n + 5 (b) n² + 1 (c) 4n − 7 (d) 3n² − 2n + 1

(a) 2(3) + 5 = 6 + 5 = 11

(b) 3² + 1 = 9 + 1 = 10

(d) 3(9) − 2(3) + 1 = 27 − 6 + 1 = 22

Q5. A wire of length 4l cm is made into a square. Write the expression for the side of the square.

Total wire = 4l. Square has 4 equal sides. Side = 4l/4 = l cm.

Q6. The cost of 1 pen is Rs. 5. Write an expression for the cost of p pens. If cost of 1 notebook is Rs. 15, write the total cost of p pens and q notebooks.

Cost of p pens = 5p. Cost of q notebooks = 15q.
Total cost = 5p + 15q rupees.

Q7. Which of the following are expressions and which are equations?
(a) 3x + 5 (b) 3x + 5 = 11 (c) 2y − 7 (d) 4 = 2m

(a) 3x + 5 → Expression (no = sign).

(b) 3x + 5 = 11 → Equation (has = sign).

(d) 4 = 2m → Equation.

Q8. A number when doubled and then increased by 9 gives 33. What is the number?

Let number = n. 2n + 9 = 33 → 2n = 24 → n = 12.

Q9. There are 6 rows of chairs. Each row has p chairs. 4 chairs are extra. How many chairs in total? If total is 40, find p.

Total chairs = 6p + 4. If 6p + 4 = 40 → 6p = 36 → p = 6 chairs per row.

Q10. The pattern of letter X uses 5 matchsticks. Write the rule for n such figures. How many matchsticks are needed for 15 X figures?

Rule = 5n. For n = 15: 5 × 15 = 75 matchsticks.

Q11. If a = 5 and b = 3, find: (a) 2a + 3b (b) a² − b² (c) (a + b)²

(a) 2(5) + 3(3) = 10 + 9 = 19

(b) 25 − 9 = 16

Q12. Write the following algebraic statement in words:
(a) 3m + 7 (b) 2x − 5 (c) n/3 + 4

(a) 7 added to 3 times m (or: Product of 3 and m, then 7 added).

(b) 5 subtracted from 2 times x.

Q13. Solve the equation: n/2 + n/3 = 10

LCD = 6. 3n/6 + 2n/6 = 10 → 5n/6 = 10 → 5n = 60 → n = 12.

Q14. Identify constants and variables in: (a) 5 + x (b) 3abc (c) 7 (d) −2y + 6

(a) Constant: 5; Variable: x.

(b) Constants: 3; Variables: a, b, c.

(d) Constants: −2, 6; Variable: y.

Q15. The sum of three consecutive integers is 48. Find them.

Let integers = n, n+1, n+2.
n + (n+1) + (n+2) = 48 → 3n + 3 = 48 → 3n = 45 → n = 15.
The integers are 15, 16, 17.

Q16. Arjun is 3 years older than Bunty. If their total age is 25 years, find their ages.

Let Bunty's age = x. Arjun's age = x + 3.
x + (x + 3) = 25 → 2x = 22 → x = 11.
Bunty = 11 years. Arjun = 14 years.

Q17. A train covers a distance of d km. If it travels at speed s km/h, write the expression for time taken. If d = 360 and s = 60, find the time.

Time = d/s. For d = 360, s = 60: Time = 360/60 = 6 hours.

Q18. What is the difference between an expression and an equation? Give one example of each.

Expression: A mathematical phrase with numbers, variables and operations but NO equal sign. Example: 3x + 7.

Equation: A statement that two expressions are equal (has an equal sign). Example: 3x + 7 = 16.

Q19. If 2 apples and 3 bananas cost Rs. 28, and each apple costs Rs. 8, find the cost of one banana.

2(8) + 3b = 28 → 16 + 3b = 28 → 3b = 12 → b = Rs. 4 per banana.

Q20. A number is multiplied by 5 and then 12 is subtracted. The result is 38. Find the number.

Let the number = n. 5n − 12 = 38 → 5n = 50 → n = 10.

Class 6 Maths - Algebra Summary

Chapter 11: Algebra (Concepts & Summary)

1. Key Definitions

Term	Definition	Example
Variable	A symbol (usually a letter) that can take different values.	x, y, n
Constant	A fixed value that does not change.	5, −3, 100
Expression	A combination of variables, constants and operations. No equal sign.	3x + 5
Equation	Two expressions connected by an equal sign (=).	3x + 5 = 11
Solution	The value of the variable that makes the equation true.	x = 2 for 3x + 5 = 11
Coefficient	The number multiplied with the variable.	In 3x, coefficient is 3

2. Writing Algebraic Expressions

Statement	Expression
Sum of x and 7	x + 7
7 less than y	y − 7
5 times z	5z
p divided by 4	p/4
3 more than twice n	2n + 3
Half of m reduced by 5	m/2 − 5

3. Matchstick Pattern Rules

Letter	Matchsticks per letter	Rule for n letters
L, T, V	2	2n
C, F, U, Z	3	3n
H, N, S, I	4	4n
E, F (alt), M	5	5n
Square pattern	4 for first, 3 for each next (shared side)	3n + 1

4. Solving Simple Equations

Balance Method: What you do to one side, do to the other to keep the equation balanced.

Equation Type	How to Solve	Example
x + a = b	Subtract a from both sides	x + 5 = 12 → x = 7
x − a = b	Add a to both sides	x − 3 = 8 → x = 11
ax = b	Divide both sides by a	4x = 20 → x = 5
x/a = b	Multiply both sides by a	x/3 = 6 → x = 18

5. Key Points

Variables can take different values; constants are fixed numbers.
An equation is satisfied when LHS = RHS for a specific value of the variable.
The solution is the value of the variable that makes the equation true.
An expression is NOT an equation (no = sign).
An inequality (using <, >) is NOT an equation.
Perimeter of a square = 4s, rectangle = 2(l + b) are expressed using variables.
Simple equations can be solved by trial and error or by the balance method.

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