Solution of Chapter 11. Algebra (NCERT - Mathematics Book)

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

(b) A pattern of letter Z as

(c) A pattern of letter U as

(d) A pattern of letter V as

(e) A pattern of letter E as

(f) A pattern of letter S as

(g) A pattern of letter A as

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?

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)

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

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)

A bird flies 1 kilometers 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)

Radhika 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?

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.

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?

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 in a smaller box are taken to be x, what is the number of oranges in the larger box?

Look at the following matchstick pattern of squares (see figure). The squares are not separate. Two neighbouring squares have a common matchstick. Observe the pattern and find the rule that gives the number of matchsticks in terms of the number of squares.

Figure gives a matchstick pattern of triangles. As in question 11 (a) above, find the general rule that gives the number of matchsticks in terms of the number of triangles.

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

The side of a regular hexagon is denoted by l. Express the perimeter of the hexagon using l.

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

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

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.

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, substraction and multiplication.

Which out of the following are expressions with numbers only?

(a) y + 3

(b) (7×20) – 8z

(c) 5(21-7)+7×2

(d) 5 (e) 3x

(f) 5 – 5n

(g) (7×20) – (5×10) – 45 + p

Identify the operations (addition, substraction, division, 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 , , 5z

(c) 2y + 17, 2y – 17

(d) 7m, - 7m + 3, -7m – 3

Give expressions for the following cases:

(a) 7 added to p

(b) 7 stbtracted 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

Give expressions in the following cases:

(a) 11 added to 2m

(b) 11stbtracted from 2m

(c) 5 times yto which 3 is added

(d) 5 times yfrom 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) yis multiplied by 5 and the result is subtracted from 16

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

Form expressions using t and 4.

Use not more then one number operation.

Every expression must have t in it.

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 following:

Take Sarita’s present age to be y years

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

(ii) What was her age 3 years back?

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

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

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

Answer the following:

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?

Answer the following:

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.

Answer the following:

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 yop is 10 less then 4 times what Meena has reached. Express the total number of steps using s.

Answer the following:

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.

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 Rsp. A book costs Rs 3p.

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

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

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

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

Given Munnu’s age to be x years, can you gues what (x – 2) may show?

Can you gues what x + 4 may show? What (3x+7) may show?

Given Sara’s age today to be y years. Think of her age in the future or in the past. What will the following expressions indicate?

Y + 7, y – 3, y + 4, y – 2

Given n students in the class like football, what may 2n show? What may show ?

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

(b) (t-7) > 5

(c) = 2

(d) 7 × 3 – 19 = 8

(e) 5 × 4 – 8 = 2x

(f) x – 2 = 0

(g) 2m< 30

(h) 2n + 1 = 11

(i) 7 = (11×5) – (12×4)

(j) 7 = (11×2) + p

(k) 20 = 5y

(l) < 5

(m) z + 12 > 24

(n) 20 – (10-5) = 3×5

(o) 7 – x = 5

Complete the entires in the third column of the table.

Pick out the solution from the values given in the brackets next to each equation. Show that the other values do not satisfy the equation.

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

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

(c) p – 5 = 5 (0, 10, 5, -5)

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

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

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

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

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

Complete the table and find the solution of the equation = 4 using the table

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

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 !

(ii) For each day of the week

Make an upcount from me

If you make no mistake

You will get twenty three !

(iii) I am a special number

Take away from me a six !

A whole cricket team

You will still be able to fix !

(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 !