# NCERT Solutions for Class 6 Maths Chapter - 11: Algebra

### Exercise 11.1

1. 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 as

Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

Letter T requires two numbers of matchsticks.

For n =1

Number of matchsticks required to make the pattern = 2

For n = 2,

Number of matchsticks required to make the pattern = 4

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of T's formed 1 2 3 4 5 6 7 8 n Number of matchsticks required 2 4 6 8 10 12 14 16 2n

Thus,

2n gives the number of matchsticks required to make the matchstick patterns of letter T.

(b) A pattern of letter Z as

Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

Letter Z requires three numbers of matchsticks.

For n = 1

Number of matchsticks required to make the pattern = 3

For n = 2,

Number of matchsticks required to make the pattern = 6

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of Z's formed 1 2 3 4 5 6 7 8 n Number of matchsticks required 3 6 9 12 15 18 21 24 3n

Thus,

3n gives the number of matchsticks required to make the matchstick patterns of letter Z.

(c) A pattern of letter U as

Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

Letter U requires three numbers of matchsticks.

For n = 1

Number of matchsticks required to make the pattern = 3

For n = 2,

Number of matchsticks required to make the pattern = 6

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of U's formed 1 2 3 4 5 6 7 8 n Number of matchsticks required 3 6 9 12 15 18 21 24 3n Thus,

3n gives the number of matchsticks required to make the matchstick patterns of letter U.

(d) A pattern of letter V as

Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

Letter V requires two numbers of matchsticks.

For n =1

Number of matchsticks required to make the pattern = 2

For n = 2,

Number of matchsticks required to make the pattern = 4

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of V's formed 1 2 3 4 5 6 7 8 n Number of matchsticks required 2 4 6 8 10 12 14 16 2n

Thus,

2n gives the number of matchsticks required to make the matchstick patterns of letter V.

(e) A pattern of letter E as

Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

Letter E requires five numbers of matchsticks. For n =1

Number of matchsticks required to make the pattern = 5

For n = 2,

Number of matchsticks required to make the pattern = 10

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of E's formed 1 2 3 4 5 6 7 8 n Number of matchsticks required 5 10 15 20 25 30 35 40 5n

Thus,

5n gives the number of matchsticks required to make the matchstick patterns of letter E.

(f) A pattern of letter S as

Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

Letter S requires five numbers of matchsticks. For n =1

Number of matchsticks required to make the pattern = 5

For n = 2,

Number of matchsticks required to make the pattern = 10

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of S's formed 1 2 3 4 5 6 7 8 n Number of matchsticks required 5 10 15 20 25 30 35 40 5n

Thus,

5n gives the number of matchsticks required to make the matchstick patterns of letter S.

(g) A pattern of letter A as

Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

Letter A requires six numbers of matchsticks.

For n =1

Number of matchsticks required to make the pattern = 6

For n = 2,

Number of matchsticks required to make the pattern = 12

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of A's formed 1 2 3 4 5 6 7 8 n Number of matchsticks required 6 12 18 24 30 36 42 48 6n

Thus,

6n gives the number of matchsticks required to make the matchstick patterns of letter A.

2. We already know the rule for the pattern of letters L, C and F. Some of the letters from Q.1 (given above) give us the same rule as that given by L. Which are these? Why does this happen?

Answer: T, V / (a) and (d)

Explanation: L requires 2 number of matchsticks.

Similarly, letter T and V from the options (a) and (d) also requires two numbers of matchsticks.

It is shown in the below table:

 Number of V's formed 1 2 3 4 5 6 7 8 n Number of matchsticks required 2 4 6 8 10 12 14 16 2n

Thus,

2n gives the number of matchsticks required to make the matchstick patterns of letter L, T and V.

3. 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.)

Explanation: A single row has 5 numbers of cadets.

Let the number of rows be n.

For n = 1,

For n = 2,

Similarly, for different values of n, the numbers of cadets are shown in the below table:

 Number of rows 1 2 3 4 5 6 7 8 n Number of cadets 5 10 15 20 25 30 35 40 5n

4. If there are 50 mangoes in a box, how will you write the total number of mangoes in terms of the number of boxes? (Use b for the number of boxes.)

Explanation: A single box has 50 numbers of mangoes.

Let the number of boxes be b.

For b = 1,

Number of mangoes = 50

For b = 2,

Number of mangoes = 100

Similarly, for different values of b, the numbers of mangoes are shown in the below table:

 Number of boxes 1 2 3 4 5 6 7 8 b Number of mangoes 50 100 150 200 250 300 350 400 50b

5. The teacher distributes 5 pencils per student. Can you tell how many pencils are needed, given the number of students? (Use s for the number of students.)

Explanation: Let the number of students be s.

The pencils distributed per student = 5

For s = 1,

Number of pencils distributed = 5

For s = 2,

Number of pencils distributed = 10

Similarly, for different values of s, the numbers of pencils distributed are shown in the below table:

 Number of students 1 2 3 4 5 6 7 8 s Number of pencils distributed 5 10 15 20 25 30 35 40 5s

6. A bird flies 1 kilometer in one minute. Can you express the distance covered by the bird in terms of its flying time in minutes? (Use t for flying time in minutes.)

Explanation: Let the time required by the bird for flying in minutes = t

Distance travelled by bird in 1 minute = 1 kilometer

For t = 1,

Distance = 1 km

For t = 2,

Distance = 2 km

 Time 1 2 3 4 5 6 7 8 t Distance 1 km 2 km 3 km 4 km 5 km 6 km 7 km 8 km t km

7. Radha is drawing a dot Rangoli (a beautiful pattern of lines joining dots) with chalk powder. She has 9 dots in a row. How many dots will her Rangoli have for r rows? How many dots are there if there are 8 rows? If there are 10 rows?

Explanation: A single row has 9 numbers of dots.

Let the number of rows be r.

For r = 1,

Number of dots = 9

For r = 2,

Similarly, for different values of r, the numbers of dots are shown in the below table:

 Number of rows 1 2 3 4 5 6 7 8 r Number of cadets 9 18 27 36 45 54 63 72 9r

If there are 8 rows,

Number of dots = 9r

Number of dots = 9 � 8

Number of dots = 72

Thus, there are 72 dots in 8 rows.

If there are 10 rows,

Number of dots = 9r

Number of dots = 9 � 10

Number of dots = 90

Thus, there are 90 dots in 10 rows.

8. 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.

Explanation: Let the Radha's age be x.

Leela is gour year younger than Radha.

Thus, leela's age = (x - 4) years

9. Mother has made laddus. She gives some laddus to guests and family members; still 5 laddus remain. If the number of laddus mother gave away is l, how many laddus did she make?

Explanation: The laddus given by the mother = l

10. 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?

Explanation: The number of oranges in a small box = X

Number of oranges in larger box = 2 � Number of oranges in smaller box + 10

Number of oranges in larger box = 2X + 10

11. Look at the following matchstick pattern of squares (Fig 11.6). The squares are not separate. Two neighbouring squares have a common matchstick. Observe the patterns and find the rule that gives the number of matchsticks Fig 11. in terms of the number of squares. (Hint: If you remove the vertical stick at the end, you will get a pattern of Cs.) Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

n = Number of squares

The above pattern is drawn using different number of matchsticks.

For n = 1

Number of matchsticks required to make the pattern = 4

For n = 2,

Number of matchsticks required to make the pattern = 7

For n = 3

Number of matchsticks required to make the pattern = 10

Thus, the pattern forms triplet the number of the squares and one extra matchstick.

Pattern = 3n + 1

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of letters in pattern 1 2 3 4 5 6 7 8 n Number of matchsticks required 4 7 10 13 16 19 22 25 3n + 1

Thus,

3n + 1 give the number of matchsticks required to make the matchstick pattern.

(b) Fig 11.7 gives a matchstick pattern of triangles. As in Exercise 11 (a) above, find the general rule that gives the number of matchsticks in terms of the number of triangles. Explanation: Let the variable be n.

Where,

n = 1, 2, 3, 4, 5, 6, 7, 8, 9 ... n

n = Number of triangles

The above pattern is drawn using different number of matchsticks.

For n = 1

Number of matchsticks required to make the pattern = 3

For n = 2,

Number of matchsticks required to make the pattern = 5

For n = 3

Number of matchsticks required to make the pattern = 7

Thus, the pattern forms double the number of the triangles and one extra matchstick.

Pattern = 2n + 1

Similarly, the following numbers of matchsticks are required for different number of n. It is shown in the below table:

 Number of letters in pattern 1 2 3 4 5 6 7 8 n Number of matchsticks required 3 5 7 9 11 13 15 17 2n + 1

Thus,

2n + 1 give the number of matchsticks required to make the matchstick pattern.

### Exercise 11.2

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

Explanation: Perimeter is equal to the sum of all sides of the given figure.

Perimeter of a triangle = Sum of all its three sides

An equilateral triangle has all its three sides equal = l

Perimeter of an equilateral triangle = l + l + l

Perimeter of an equilateral triangle = 3l

2. The side of a regular hexagon (Fig 11.10) is denoted by l. Express the perimeter of the hexagon using l. (Hint: A regular hexagon has all its six sides equal in length.) Explanation: Perimeter is equal to the sum of all sides of the given figure.

Perimeter of a hexagon = Sum of all its six sides

The hexagon has all its six sides equal = l

Perimeter of the hexagon = l + l + l + l + l + l

Perimeter of the hexagon = 6l

3. A cube is a three-dimensional figure as shown in Fig 11.11. It has six faces and all of them are identical squares. The length of an edge of the cube is given by l. Find the formula for the total length of the edges of a cube. Fig 11.11 Explanation: Perimeter is equal to the sum of all sides of the given figure.

A cube has twelve edges or twelve sides

Perimeter of a cube = Sum of all its twelve sides

The cube has all its twelve sides equal = l

Perimeter of the cube = l + l + l + l + l + l + l + l + l + l + l + l

Perimeter of the cube = 12l

4. The diameter of a circle is a line which joins two points on the circle and also passes through the centre of the circle. (In the adjoining figure (Fig 11.12) AB is a diameter of the circle; C is its centre.) Express the diameter of the circle (d) in terms of its radius (r). Explanation: A diameter passes through the centre of the circle.

CA = CP

It is because both are the radius of the circle.

The centre point equally divides the diameter into two equal parts.

Thus, diameter is equal to twice the radius of the circle.

d = 2 r

5. To find sum of three numbers 14, 27 and 13, we can have two ways:

(a) We may first add 14 and 27 to get 41 and then add 13 to it to get the total sum 54 or

(b) We may add 27 and 13 to get 40 and then add 14 to get the sum 54. Thus, (14 + 27) + 13 = 14 + (27 + 13)

This can be done for any three numbers. This property is known as the associativity of addition of numbers. Express this property which we have already studied in the chapter on Whole Numbers, in a general way, by using variables a, b and c.

Answer: (a + b) + c = a + (b + c)

### Exercise 11.3

1. Make up as many expressions with numbers (no variables) as you can from three numbers 5, 7 and 8. Every number should be used not more than once. Use only addition, subtraction and multiplication. (Hint: Three possible expressions are 5 + (8 - 7), 5 - (8 - 7), (5 � 8) + 7; make the other expressions.)

The possible expressions using the three numbers, 5, 7, and 8 are as follows:

• 5 + (8 - 7)
• 5 + (8 + 7)
• 5 � (8 - 7)
• 5 - (8 - 7)
• 5 - (8 + 7)
• 5 � (8 + 7)
• 8 - (5 + 7)
• 8 - (5 - 7)
• 8 � (5 + 7)
• 8 + (5 + 7)
• 8 + (5 - 7)
• 8 � (5 - 7)
• 7 - (8 - 5)
• 7 - (8 + 5)
• 7 + (8 + 5)
• 7 + (8 - 5)
• 7 � (8 - 5)
• 7 � (8 + 5)

2. Which out of the following are expressions with numbers only?

Let's discuss each expression individually.

(a) y + 3

Answer: No. It is not an expression with only numbers.

It has a variable 'y'.

(b) (7 � 20) - 8z

Answer: No. It is not an expression with only numbers.

It has a variable 'z'.

(c) 5 (21 - 7) + 7 � 2

Answer: Yes. It is an expression with only numbers.

It has no variable.

(d) 5

Answer: Yes. It is an expression with only number.

It has no variable.

(e) 3x

Answer: No. It is not an expression with only numbers.

It has a variable 'x'.

(f) 5 - 5n

Answer: No. It is not an expression with only numbers.

It has a variable 'n'.

(g) (7 � 20) - (5 � 10) - 45 + p 3.

Answer: No. It is not an expression with only numbers.

It has a variable 'p'.

Thus, option (c) and (d) are the only expressions with only numbers.

3. Identify the operations (addition, subtraction, division, multiplication) in forming the following expressions and tell how the expressions have been formed.

(a) z +1, z - 1, y + 17, y - 17

Z is increased by 1.

z - 1: Subtraction

Z is subtracted by 1

Y is increased by 17.

y - 17: Subtraction

Y is subtracted by 17.

(b) 17y, y 17, 5z

17y: Multiplication

Y is multiplied by 17.

y/17: Division

Y is divided by 17.

5z: Multiplication

Y is multiplied by 5.

(c) 2y + 17, 2 y - 17

2y + 17: Multiplication and addition

Y is multiplied by 2 and then increased by 17.

2 y - 17: Multiplication and subtraction

Y is multiplied by 2 and then decreased by 17.

(d) 7 m, - 7 m + 3, - 7 m - 3

7 m: Multiplication

M is multiplied by 7.

- 7 m + 3: Multiplication and addition

Y is multiplied by -7 and then increased by 3.

- 7 m - 3: Multiplication and subtraction

Y is multiplied by -7 and then decreased by 3.

4. Give expressions for the following cases.

(b) 7 subtracted from p

(c) p multiplied by 7

(d) p divided by 7

(e) 7 subtracted from - m

(f) - p multiplied by 5

(g) - p divided by 5

(h) p multiplied by - 5

5. Give expressions in the following cases.

(b) 11 subtracted from 2m

(c) 5 times y to which 3 is added

(d) 5 times y from which 3 is subtracted

(e) y is multiplied by - 8

(f) y is multiplied by - 8 and then 5 is added to the result

(g) y is multiplied by 5 and the result is subtracted from 16

(h) y is multiplied by - 5 and the result is added to 16.

6.

(a) Form expressions using t and 4. Use not more than one number operation. Every expression must have t in it.

Answer: The expressions are as follows:

• t - 4
• t + 4
• 4 - t
• 4 + t
• t/4
• 4t
• 4/t

(b) Form expressions using y, 2 and 7. Every expression must have y in it. Use only two number operations. These should be different.

Answer: The expressions are as follows:

• 2y - 7
• 2y + 7
• y/2 - 7
• y - 7/2
• 7y - 2
• 7y + 2
• y/7 - 2

### Exercise 11.4

(a) Take Sarita's present age to be y years

(i) What will be her age 5 years from now?

Explanation: Let the Sarita's age = y years

Age of Sarita 5 years from now = current age + 5 years

= y + 5

(ii) What was her age 3 years back?

Explanation: Let the Sarita's age = y years

Age of Sarita 3 years back = current age - 3 years

= y - 3

(iii) Sarita's grandfather is 6 times her age. What is the age of her grandfather?

Explanation: Let the Sarita's age = y years

Age of her grandfather = 6 time her age

(Times means multiplication)

Age of her grandfather = 6y

(iv) Grandmother is 2 years younger than grandfather. What is grandmother's age?

Explanation: Let the Sarita's age = y years

Age of her grandfather = 6 time her age

(Times means multiplication)

Age of her grandfather = 6y

Age of her grandmother = 2 years younger than grandfather

Age of her grandmother = Age of grandfather - 2

= 6y - 2

(v) Sarita's father's age is 5 years more than 3 times Sarita's age. What is her father's age?

Explanation: Let the Sarita's age = y years

Age of her father = 3 times Sarita's age + 5 years

Age of her father = 3y + 5

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

Explanation: The breadth of the rectangular hall = b meters

Length of the rectangular hall = 4 meters less than 3 times the breadth

Length of the rectangular hall = 3 times he breadth - 4

= (3b - 4) meters

Thus, the length of the rectangular hall is 3b - 4 and the breadth is b meters.

(c) A rectangular box has height h cm. Its length is 5 times the height and breadth is 10 cm less than the length.

Express the length and the breadth of the box in terms of the height.

Breadth = (5h - 10) cm

Height = h cm

Explanation: The height of the box = h cm

Length of the box = 5 times the height

Length of the box = 5h cm

Breadth of the box = 10 cm less than the length

= Length of the box - 10

= 5h - 10

Thus, the dimensions of the box are:

Length = 5h cm

Breadth = (5h - 10) cm

Height = h cm

(d) Meena, Beena and Leena are climbing the steps to the hill top. Meena is at step s, Beena is 8 steps ahead and Leena 7 steps behind. Where are Beena and Meena? The total number of steps to the hill top is 10 less than 4 times what Meena has reached. Express the total number of steps using s.

Answer: Beena = s + 8

Leena = s- 7

Total number of steps to the hill top = 4s - 10

Explanation: Steps of the Meena = s

Steps of Beena = 8 steps ahead of Meena

Steps of Beena = steps of Meena + 8

= s + 8

Steps of Leena = 8 steps behind of Meena

Steps of Leena = steps of Meena - 7

= s - 7

Total number of steps to the hill top = 10 less than 4 times of Meena

= 4 time Meena steps - 10

= 4s - 10

(e) A bus travels at v km per hour. It is going from Daspur to Beespur. After the bus has travelled 5 hours, Beespur is still 20 km away. What is the distance from Daspur to Beespur? Express it using v.

Explanation: The distance travelled by the bus in 1 hour = v km

The distance travelled by the bus in 5 hours = 5v km

Total distance between Daspur to Beespur = distance travelled in 5 hours + 20 km

Total distance between Daspur to Beespur = (5v + 20) km

2. Change the following statements using expressions into statements in ordinary language. (For example, Given Salim scores r runs in a cricket match, Nalin scores (r + 15) runs. In ordinary language - Nalin scores 15 runs more than Salim.)

(a) A notebook costs rupee p. A book costs rupee 3 p.

Answer: A book cost three times the cost of a notebook.

Explanation: Cost of a book = 3p

Cost of a notebook = p

Thus, it signifies that cost of book is 3 times p, i.e., 3 times the cost of a notebook.

(b) Tony puts q marbles on the table. He has 8 q marbles in his box.

Answer: Tony's box contains marbles eight times the marbles on the table.

Explanation: Marbles on the table = q

Marbles in the Tony's box = 8q

Thus, it signifies that ton's box has the marbles equal to 8 times the marbles on the table.

(c) Our class has n students. The school has 20 n students.

Answer: Total number of students in the school is twenty times the number of students in our class.

Explanation: Total number of students in the class = n

Total number of students in the school = 20n

Thus, it signifies that school has 20 times the students in the class.

(d) Jaggu is z years old. His uncle is 4 z years old and his aunt is (4z - 3) years old.

Answer: Jaggu's uncle is four times older than Jaggu and his aunt is 3 times younger than the uncle's age.

Explanation: Age of the Jaggu = z

Age of his uncle = 4z

Thus, the age of uncle is 4 times the age of Jaggu.

Age of her aunt = 4z - 3

Age of her aunt = Uncle's age - 3

Thus, the age of Jaggu's aunt is 3 times less than the age of his uncle.

(e) In an arrangement of dots there are r rows. Each row contains 5 dots.

Answer: The total number of dots in an arrangement is five times the number of rows.

Explanation: Total number of rows = r

Total number of dots in a row = 5

Total number of dots in an arrangement = Total number of rows x 5

Thus, thetotal number of dots in an arrangement is five times the number of rows.

3.

(a) x years, can you guess what (x - 2) may show? (Hint: Think of Munnu's younger brother.) Can you guess what (x + 4) may show? What (3 x + 7) may show?

Answer: Munnu's age = x years

X - 2 = 2 years less than the age of Munnu

X - 2 =Munnu's younger brother/sister

X + 4 = 4 years older than the age of Munnu

X + 4 = Elder sister/brother

3X + 7 = 7 added to 3 times the age of Munnu

Thus, 3X + 7 can be the age of Munnu's mother/father.

(b) Given Sara's age today to be y years. Think of her age in the future or in the past. What will the following expression indicate? y + 7, y - 3, y + 4 1/2 , y - 2 1/2 .

Answer: Sara's age = y years

Y + 7 = Seven years after the current age

Y + 7 = Age of Sara in the future

Y - 3 = Three years before the current age

Y - 3 = Age of Sara in the past

y + 4 ½ = Four and a half years after the current age

y + 4 ½ = Age of Sara in the future

y - 2 ½ =Two and a half years before the current age

y - 2 ½ = Age of Sara in the past

(c) Given n students in the class like football, what may 2n show? What may 2 n show? (Hint: Think of games other than football).

Answer: Total number of students who like football = n

2n may be any other game like basketball, cricket, etc.

Total number of students who like basketball = 2n

Thus, total numbers of students who like other game (for example, basketball) are two times the number of students who like football.

### Exercise 11.5

1. State which of the following are equations (with a variable). Give reason for your answer. Identify the variable from the equations with a variable.

(a) 17 = x + 7

Yes, it is an equation with a variable x.

Variable: x

(b) (t - 7) > 5

No, it is an equation with any variable. An equation always has an equal sign (=) sign between them.

Variable: none

(c) 4/2 = 2

No, it is not an equation with any variable.

Variable: None

(d) (7 � 3) - 19 = 8

No, it is not an equation with any variable.

Variable: None

(e) 5 � 4 - 8 = 2 x

Yes, it is an equation with a variable x.

Variable: x

(f ) x - 2 = 0

Yes, it is an equation with a variable x.

Variable: x

(g) 2m < 30

No, it is an equation with any variable. An equation always has an equal sign (=) sign between them.

Variable: none

(h) 2n + 1 = 11

Yes, it is an equation with a variable n.

Variable: n

(i) 7 = (11 � 5) - (12 � 4)

No, it is not an equation with any variable.

Variable: None

( j) 7 = (11 � 2) + p

Yes, it is an equation with a variable p.

Variable: p

(k) 20 = 5y

Yes, it is an equation with a variable y.

Variable: y

( l) 3q/2 < 5

No, it is an equation with any variable. An equation always has an equal sign (=) sign between them.

Variable: none

(m) z + 12 > 24

No, it is an equation with any variable. An equation always has an equal sign (=) sign between them.

Variable: none

(n) 20 - (10 - 5) = 3 � 5

No, it is not an equation with any variable.

Variable: None

(o) 7 - x = 5

Yes, it is an equation with a variable x.

Variable: x

2. Complete the entries in the third column of the table.

(a) Equation: 10y = 80

Value of variable: y = 10

Equation satisfied: No

Reason:

For y = 10,

LHS = 10(10)

LHS = 100

100 is not equal to 80

Thus, the above equation is not satisfied

Correct:

10y = 80

Y = 80/10

Y = 8

Thus, the value of 'y' for which the equation is satisfied is 8.

(b) Equation: 10y = 80

Value of variable: y = 8

Equation satisfied: Yes

Reason: 10y = 80

Y = 80/10

Y = 8

Thus, the value of 'y' for which the equation is satisfied is 8.

(c) Equation: 10y = 80

Value of variable: y = 5

Equation satisfied: No

Reason: For y = 5,

LHS = 10(5)

LHS = 50

50 is not equal to 80

Thus, the above equation is not satisfied

Correct:

10y = 80

Y = 80/10

Y = 8

Thus, the value of 'y' for which the equation is satisfied is 8.

(d) Equation: 4l = 20

Value of variable: l = 20

Equation satisfied: No

Reason: For l = 20,

LHS = 4(20)

LHS = 80

80 is not equal to 20

Thus, the above equation is not satisfied

Correct:

4l = 20

L = 20/4

L = 5

Thus, the value of 'l' for which the equation is satisfied is 5.

(e) Equation: 4l = 20

Value of variable: l = 80

Equation satisfied: No

Reason: For l = 80,

LHS = 4(80)

LHS = 320

320 is not equal to 20

Thus, the above equation is not satisfied

Correct:

4l = 20

L = 20/4

L = 5

Thus, the value of 'l' for which the equation is satisfied is 5.

(f) Equation: 4l = 20

Value of variable: l = 5

Equation satisfied: Yes

Reason: 4l = 20

L = 20/4

L = 5

Thus, the value of 'l' for which the equation is satisfied is 5.

(g) Equation: b + 5 = 9

Value of variable: b = 5

Equation satisfied: No

Reason: For b = 5,

LHS = 5 + 5

LHS = 10

10 is not equal to 9

Thus, the above equation is not satisfied

Correct:

b + 5 = 9

b = 9 - 5

b = 4

Thus, the value of 'b' for which the equation is satisfied is 4.

(h) Equation: b + 5 = 9

Value of variable: b = 9

Equation satisfied: No

Reason: For b = 9,

LHS = 9 + 5

LHS = 14

14 is not equal to 9

Thus, the above equation is not satisfied

Correct:

b + 5 = 9

b = 9 - 5

b = 4

Thus, the value of 'b' for which the equation is satisfied is 4.

(i) Equation: b + 5 = 9

Value of variable: b = 4

Equation satisfied: Yes

Reason: b + 5 = 9

b = 9 - 5

b = 4

Thus, the value of 'b' for which the equation is satisfied is 4.

(j) Equation: h - 8 = 5

Value of variable: h = 13

Equation satisfied: Yes

Reason: h - 8 = 5

h = 5 + 8

h = 13

Thus, the value of 'h' for which the equation is satisfied is 13.

(k) Equation: h - 8 = 5

Value of variable: h = 8

Equation satisfied: No

Reason: For h = 8,

LHS = 8 - 8

LHS = 0

0 is not equal to 5

Thus, the above equation is not satisfied

Correct:

h - 8 = 5

h = 5 + 8

h = 13

Thus, the value of 'h' for which the equation is satisfied is 13.

(l) Equation: h - 8 = 5

Value of variable: h = 0

Equation satisfied: No

Reason: For h = 0,

LHS = 0 - 8

LHS = -8

-8 is not equal to 5

Thus, the above equation is not satisfied

Correct:

h - 8 = 5

h = 5 + 8

h = 13

Thus, the value of 'h' for which the equation is satisfied is 13.

(m) Equation: p + 3 = 1

Value of variable: p = 3

Equation satisfied: No

Reason: For p = 3,

LHS = 3 + 3

LHS = 6

6 is not equal to 1

Thus, the above equation is not satisfied

Correct:

p + 3 = 1

p = 1 - 3

p = -2

Thus, the value of 'p' for which the equation is satisfied is -2.

(n) Equation: p + 3 = 1

Value of variable: p = 1

Equation satisfied: No

Reason: For p = 1,

LHS = 1 + 3

LHS = 4

4 is not equal to 1

Thus, the above equation is not satisfied

Correct:

p + 3 = 1

p = 1 - 3

p = -2

Thus, the value of 'p' for which the equation is satisfied is -2.

(o) Equation: p + 3 = 1

Value of variable: p = 0

Equation satisfied: No

Reason: For p = 0,

LHS = 0 + 3

LHS = 3

3 is not equal to 1

Thus, the above equation is not satisfied

Correct:

p + 3 = 1

p = 1 - 3

p = -2

Thus, the value of 'p' for which the equation is satisfied is -2.

(p) Equation: p + 3 = 1

Value of variable: p = -1

Equation satisfied: No

Reason: For p = -1,

LHS = -1 + 3

LHS = 2

2 is not equal to 1

Thus, the above equation is not satisfied

Correct:

p + 3 = 1

p = 1 - 3

p = -2

Thus, the value of 'p' for which the equation is satisfied is -2.

(q) Equation: p + 3 = 1

Value of variable: p = -2

Equation satisfied: Yes

Reason: p + 3 = 1

p = 1 - 3

p = -2

Thus, the value of 'p' for which the equation is satisfied is -2.

(r) Pick out the solution from the values given in the bracket next to each equation. Show that the other values do not satisfy the equation.

(a) 5m = 60 (10, 5, 12, 15)

Explanation:

For m = 10,

LHS = 5 (10)

LHS = 50

50 is not equal to 60

Hence, the value for m = 10 is not satisfied.

For m = 5,

LHS = 5 (5)

LHS = 25

25 is not equal to 60

Hence, the value for m = 5 is not satisfied.

For m = 12,

LHS = 5 (12)

LHS = 60

60 is equal to 60

Hence, the value for m = 12 is satisfied.

For m = 15,

LHS = 5 (15)

LHS = 75

75 is not equal to 60

Hence, the value for m = 15 is not satisfied.

(b) n + 12 = 20 (12, 8, 20, 0)

Explanation:

For n = 12,

LHS = 12 + 12

LHS = 24

24 is not equal to 20

Hence, the value for n = 12 is not satisfied.

For n = 8,

LHS = 8 + 12

LHS = 20

20 is equal to 20

Hence, the value for n = 8 is satisfied.

For n = 20,

LHS = 20 + 12

LHS = 32

32 is not equal to 20

Hence, the value for n = 20 is not satisfied.

For n = 0,

LHS = 0 + 12

LHS = 12

12 is not equal to 20

Hence, the value for n = 0 is not satisfied.

(c) p - 5 = 5 (0, 10, 5 - 5)

Explanation:

For p = 0,

LHS = 0 - 5

LHS = -5

-5 is not equal to 5

Hence, the value for p = 0 is not satisfied.

For p = 10,

LHS = 10 - 5

LHS = 5

5 is equal to 5

Hence, the value for p = 10 is satisfied.

For p = 5,

LHS = 5 - 5

LHS = 0

0 is not equal to 5

Hence, the value for p = 5 is not satisfied.

For p = -5,

LHS = -5 - 5

LHS = -10

-10 is not equal to 5

Hence, the value for p = -5 is not satisfied.

(d) q/2 = 7 (7, 2, 10, 14)

Explanation:

For q = 7,

LHS = 7/2

LHS = 3.5

3.5 is not equal to 7

Hence, the value for q = 7 is not satisfied.

For q = 2,

LHS = 2/2

LHS = 1

1 is not equal to 7

Hence, the value for q = 2 is not satisfied.

For q = 10,

LHS = 10/2

LHS = 5

5 is not equal to 7

Hence, the value for q = 10 is not satisfied.

For q = 14,

LHS = 14/2

LHS = 7

7 is equal to 7

Hence, the value for q = 14 is satisfied.

(e) r - 4 = 0 (4, - 4, 8, 0)

Explanation:

For r = 4,

LHS = 4 - 4

LHS = 0

0 is equal to 0

Hence, the value for r = 4 is satisfied.

For r =- 4,

LHS = -4 - 4

LHS = -8

-8 is not equal to 0

Hence, the value for r = -4 is not satisfied.

For r = 8,

LHS = 8 - 4

LHS = 4

4 is not equal to 0

Hence, the value for r = 8 is not satisfied.

For r = 0,

LHS = 0 - 4

LHS = -4

-4 is not equal to 0

Hence, the value for r = 0 is not satisfied.

(f) x + 4 = 2 (- 2, 0, 2, 4)

Explanation:

For x = - 2,

LHS = -2 + 4

LHS = 2

2 is equal to 2

Hence, the value for x = -2 is satisfied.

For x = 0,

LHS = 0 + 4

LHS = 4

4 is not equal to 2

Hence, the value for x = 0 is not satisfied.

For x = 2,

LHS = 2 + 4

LHS = 6

6 is not equal to 2

Hence, the value for x = 2 is not satisfied.

For x = 4,

LHS = 4 + 4

LHS = 8

8 is not equal to 2

Hence, the value for x = 4 is not satisfied.

4.

(a) Complete the table and by inspection of the table find the solution to the equation m + 10 = 16.

m + 10 = 16

Let's find the solution for different values of m.

LHS:

 m 1 2 3 4 5 6 7 8 9 m + 10 11 12 13 14 15 16 17 18 19

At m = 6, the value of equation is equal to 16.

(b) Complete the table and by inspection of the table find the solution to the equation 5t = 35.

5t = 35

Let's find the solution for different values of t.

LHS:

 t 1 2 3 4 5 6 7 8 9 5t 5 10 15 20 25 30 35 40 45

At t = 7, the value of equation is equal to 35.

(c) Complete the table and find the solution of the equation z/3 = 4 using the table

z/3 = 4

Let's find the solution for different values of z.

LHS:

 z 8 9 10 11 12 13 14 15 16 z/3 2 2/3 3 3 1/3 3 2/3 4 4 1/3 4 2/3 5 5 1/3

At z = 12, the value of equation is equal to 4.

(d) Complete the table and find the solution to the equation m - 7 = 3.

m - 7 = 3

Let's find the solution for different values of m.

LHS:

 m 5 6 7 8 9 10 11 12 13 m - 7 -2 -1 0 1 2 3 4 5 6

At m = 10, the value of equation is equal to 3.

4. Solve the following riddles; you may yourself construct such riddles.

Who am I?

(i) Go round a square

Counting every corner

Thrice and no more! Add the count to me

To get exactly thirty four!

Explanation: There are four corners in a square.

Let the riddles be x.

Thrice every corner = 3 (4) = 12

Add the count to get thirty four.

Thus, the equation can be written as:

12 + x = 34

X =34 - 12

X = 22

(ii) For each day of the week

Make an up count from me

If you make no mistake

You will get twenty three!

Explanation: There are seven days in a week.

Let the riddles be X.

For every day of a week, the count can be written as X + 7.

Thus, the equation can be written as:

X + 7 = 23

X = 23 - 7

X = 16

(iii) I am a special number

Take away from me a six!

A whole cricket team

You will still be able to fix!

Explanation: Let the special number be X.

Take from a special number signifies subtraction.

Six taken from a number = X - 6

Total number of players in a cricket team = 11

Thus, the equation can be written as:

X - 6 = 11

X = 11 + 6

X = 17

(iv) Tell me who I am I shall give a pretty clue!

You will get me back

If you take me out of twenty two!

Explanation: Let the riddles be X.

You will get me back means twice.

It represents a double sided count.

Thus, the equation can be written as:

2X = 22

X = 22/2

X = 11

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