Chapter 11 – Algebra

Question 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 T

(b) A pattern of letter Z as Z

(c) A pattern of letter U as U

(d) A pattern of letter V as V

(e) A pattern of letter E as E

(f) A pattern of letter S as S

(g) A pattern of letter A as A

(a)

From the figure, it can be observed that it will require two matchsticks to make a T. Therefore, the pattern is 2n.

(b)

From the figure, it can be observed that it will require three matchsticks to make a Z. Therefore, the pattern is 3n.

(c)

From the figure, it can be observed that it will require three matchsticks to make a U. Therefore, the pattern is 3n.

(d)

From the figure, it can be observed that it will require two matchsticks to make a V. Therefore, the pattern is 2n.

(e)

From the figure, it can be observed that it will require five matchsticks to make an E. Therefore, the pattern is 5n.

(f)

From the figure, it can be observed that it will require five matchsticks to make a S. Therefore, the pattern is 5n.

(g)

From the figure, it can be observed that it will require six matchsticks to make an A. Therefore, the pattern is 6n.

Question 2:

We already know the rule for the pattern of letters L, C and F. Some of the letters from some of the letters out of (a) T, (b) Z, (c) U, (d) V, (e) E, (f) S, (g) R give us the same rule as that given by L. Which are these? Why does this happen?

It is known that L requires only two matchsticks. Therefore, the pattern for L is 2n. Among all the letters given above in question 1, only T and V are the two letters which require two matchsticks.

Hence, (a) and (d)

Question 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.)

Let number of rows be n.

Number of cadets in one row = 5

Total number of cadets = Number of cadets in a row × Number of rows

= 5n

Question 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.)

Let the number of boxes be b.

Number of mangoes in a box = 50

Total number of mangoes = Number of mangoes in a box × Number of boxes

= 50b

Question 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.)

Let the number of students be s.

Pencils given to each student = 5

Total number of pencils

= Number of pencils given to each student × Number of students

= 5s

Question 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.)

Let the flying time be t minutes.

Distance covered in one minute = 1 km

Distance covered in t minutes = Distance covered in one minute × Flying time

= 1 × t = t km

Question 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?

Number of dots in 1 row = 9

Number of rows = r

Total number of dots in r rows = Number of rows × Number of dots in a row

= 9r

Number of dots in 8 rows = 8 × 9 = 72

Number of dots in 10 rows = 10 × 9 = 90

Question 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.

Let Radha’s age be x years.

Leela’s age = Radha’s age − 4

= (x − 4) years

Question 9:

Number of laddus given away = l

Number of laddus remaining = 5

remaining

= l + 5

Question 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?

Number of oranges in one small box = x

Number of oranges in two small boxes = 2x

Number of oranges left = 10

Number of oranges in the large box = Number of oranges in two small boxes

+ Number of oranges left

= 2x + 10

Question 11:

(a) Look at the following matchstick pattern of squares. 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 in terms of the number of squares. (Hint: if you remove the vertical stick at the end, you will get a pattern of Cs.)

(b) The given figure gives a matchstick pattern of triangles. Find the general rule that gives the number of matchsticks in terms of the number of triangles.

(a) It can be observed that in the given matchstick pattern, the number of

matchsticks are 4, 7, 10, and 13, which is 1 more than thrice of the number of squares in the pattern.

Hence, the pattern is 3n + 1, where n is the number of squares.

(b) It can be observed that in the given matchstick pattern, the number of

matchsticks are 3, 5, 7, and 9, which is 1 more than twice of the number of triangles in the pattern.

Hence, the pattern is 2n + 1, where n is the number of triangles.

Chapter 11.2

Question 1:

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

Side of equilateral triangle = l

Perimeter = l + l + l = 3l

Question 2:

The Side of a regular hexagon (see the given figure) is denoted by l. Express the perimeter of the hexagon using l.

(Hint: A regular hexagon has all its six sides equal in length.)

Side of regular hexagon = l

Perimeter = 6l

Question 3:

A cube is a three-dimensional figure as shown in the given figure. 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.

Length of edge = l

Number of edges = 12

Total length of the edges = Number of edges × Length of one edge

= 12l

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Question 4:

The diameter of a circle is a line which joins two points on the circle and also passed through the center of the circle. (In the adjoining figure AB is a diameter of the circle; C is its center.) Express the diameter of the circle (d) in terms of its radius(r).

Diameter = AB = AC + CB = r + r = 2r

d = 2r

Question 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.

For any three whole numbers a, b, and c,

(a + b) + c = a + (b + c)

Page No 233:

Chapter 11.3

Question 1:

Question 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, subtractions and multiplication.
Solution:
Given numbers are 5, 7 and 8.
Expressions are:
(i) 8 + (5 + 7)
(ii) 5 + (8 – 7)
(iii) 8 + (5 x 7)
(iv) 7 – (8 – 5)
(v) 7 x (8 + 5)
(vi) 5 x (8 + 7)
(vii) 8 x (5 + 7)
(viii) 7 + (8 – 5)
(ix) (5 x 7) – 8
(x) 7 + (8 x 5)

Question 2.
Which out of the following are expressions with numbers only?
(a) y + 3
(b) (7 x 20) – 8z
(c) 5(21 – 7) + 7 x 2
(d) 5
(e) 3x
(f) 5 – 5n
(g) (7 x 20) – (5 x 10) – 45 +p
Solution:
(a) y + 3. This expression has variable ‘y’.
(b) (7 x 20) – 8z. This expression has a variable ‘z’.
(c) 5(21 -7) + 7 x 2. This expression has no variable. So it is with numbers only.
(d) 5. This expression is with numbers only.
(e) 3x. This expression has a variable ‘x’.
(f) 5 – 5n. This expression has a variable ‘n’.
(g) (7 x 20) – (5 x 10) – 45 + p. This expression has a variable ‘p’.

Question 3.
Identify the operations (addition, subtraction, division and multiplication) in forming the following
expressions and tell how the expressions have been formed.
(a) z + 1, z – 1,y + 17, y – 17
(b) 17y,y17 , 5z
(c) 2y + 17, 2y – 17
(d) 7m, -7m + 3, -7m – 3
Solution:

Expressions Operations used Formation of expression

(a)

(i)z + 1    Addition                     z is increased by 1

(ii)z – 1   Subtraction                 z is decreased by 1

(iii)y +17 Addition                     y is increased by 17

(iv)y -17  Subtraction                 y is decreased by 17

(b)

(i)17y         Multiplication          y is multiplied by 17

(ii)y/17       Division                   y is Divided by 17

(iii)5z         Multiplication          z is Multiplied by 5

(c)

(i)2y + 17   Multiplication and    y is multiplied by 2 and

(ii)2y -17     Multiplication and   Twice of y is

subtraction           decreased by 17

(d)

(i)7 m Multiplication m is multiplied by 7

(ii)--7m + 3Multiplication and addition m is multiplied by -7 and then increased by 3

(iii)-7m – 3Multiplication and subtraction

M is multiplied by -7 and then decreased by 3

Question 4.
Give expressions for the follow
(b) 7 subtracted from p
(c) p multiplied by 7
(d) p divided by 7
(e) 7 subtracted from -m
(j) -p multiplied by 5
(g) -p divided by 5
(h) p multiplied by -5
Solution:
(a) p + 7
(b) p – 7
(c) 7p
(d) p7
(e) -m – 7
(f) -5p
(g) −p5
(h) 5p

Question 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.
Solution:
(a) 2m + 11
(b) 2m – 11
(e) 5y + 3
(d) 5y – 3
(e) -8y
(f) -8y+5
(g) 16 – 5y
(h) -5y + 16

Question 6.
(a) Form expressions using t and 4. Use not more than one number operation. Every expression must have t in it.
(b) Form expressions using y, 2 and 7. Every expression must have y in it. Use only two number operations. These should, be different.
Solution:
(a) The possible expressions are:
(i) t + 4
(ii) t – 4
(iii) 4t
(iv) t4
(v) 4 + t
(vi) 4 + t, etc.

(b) The possible expressions are:
(i) 2y + 7
(ii) 7y – 2
(iii) 7 – 2y
(iv) 7y + 2
(v) 7y2
(vi) 2y7
(vii) y7 + 2
(viii) y2 – 7,etc.