Exercise 2.2

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Introduction to Algebra

Class 6 — Solutions Key

Exercise 2.2 — Miscellaneous Practice Problems

1. Complete the following pattern.

Given equations and solutions:

•        9 - 1 = 8
•       98 - 21 = 77
•      987 - 321 = 666
•     9876 - 4321 = 5555
•    98765 - 54321 = 44444

What comes next?

The next pattern step subtracts the next sequence digit block:

  987654 - 654321 = 333333

Explanation: Each successive step appends the next decreasing integer to the minuend, the next increasing integer to the subtrahend, and yields a repeating sequence of a digit decremented by 1 compared to the previous step.

2. A piece of wire is '12s' cm long. What will be the length of the side, if it is formed as:

Total length of wire = 12s cm

(i) An equilateral triangle:

• An equilateral triangle has 3 equal sides.
• Length of each side = Total length / 3 = 12s / 3 = 4s cm.

(ii) A square:

• A square has 4 equal sides.
• Length of each side = Total length / 4 = 12s / 4 = 3s cm.

3. Identify the value of the shapes and figures in the table given below and verify their addition horizontally and vertically.

Step 1: Formulate equations to solve for the shapes.

Let Star = x, Smile = y, and Triangle = z.

From Row 1: x + y + y + x = 30 ⇒ 2x + 2y = 30 ⇒ x + y = 15

From Column 1: x + x + x + x = 32 ⇒ 4x = 32 ⇒ x (Star) = 8

Substitute x = 8 into x + y = 15 ⇒ 8 + y = 15 ⇒ y (Smile) = 7

From Row 2: x + x + z + z = 36 ⇒ 2x + 2z = 36 ⇒ 2(8) + 2z = 36 ⇒ 16 + 2z = 36 ⇒ 2z = 20 ⇒ z (Triangle) = 10

Step 2: Calculate missing totals.

• Row 3 total: Star + Triangle + Smile + Smile = 8 + 10 + 7 + 7 = 32
• Row 4 total: Star + Smile + Triangle + Smile = 8 + 7 + 10 + 7 = 32
• Col 2 total: Smile + Star + Triangle + Smile = 7 + 8 + 10 + 7 = 32
• Col 3 total: Smile + Triangle + Smile + Triangle = 7 + 10 + 7 + 10 = 34
• Col 4 total: Star + Triangle + Smile + Smile = 8 + 10 + 7 + 7 = 32
• Grand Total (Diagonal): 32 + 32 + 34 + 32 = 130

Star (8)	Smile (7)	Smile (7)	Star (8)	= 30
Star (8)	Star (8)	Triangle (10)	Triangle (10)	= 36
Star (8)	Triangle (10)	Smile (7)	Smile (7)	= 32
Star (8)	Smile (7)	Triangle (10)	Smile (7)	= 32
= 32	= 32	= 34	= 32	= 130

4. Find the value of all the variables in the Kabaddi Championship table.

Rule: Total Matches Played = Matches Won + Matches Lost

• For Team C: n = 4 + 6 ⇒ n = 10
• For Team D: a = 7 + 2 ⇒ a = 9
• For Team E: 9 = b + 3 ⇒ b = 9 - 3 ⇒ b = 6
• For Team F: 10 = 6 + c ⇒ c = 10 - 6 ⇒ c = 4
• For Team G: 8 = x + 4 ⇒ x = 8 - 4 ⇒ x = 4
• For Team H: y = 3 + 6 ⇒ y = 9
• For Team A (lost matches 'x'): Note: Team G already uses variable 'x'. To solve the entry under Team A, 8 = 5 + k (where k is the variable represented in the table as a modified 'x') ⇒ variable = 3
• For Team B (lost matches 'y'): Note: Team H already uses variable 'y'. To solve the entry under Team B, 7 = 6 + m (where m is the variable represented in the table as a modified 'y') ⇒ variable = 1

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