Olympiad Notes: Algebra | Maths Olympiad Class 6 PDF Download

Introduction

• Algebra is a branch of mathematics, which involves using letters and symbols to represent numbers in equations.
• It's particularly useful for solving problems involving unknown quantities.
• Algebra provides a structured approach to tackling mathematical puzzles with variables.

Matchstick Patterns

• No. of matchsticks used to make 1^st square = 4.
• No. of matchsticks used to make 2^nd square = 7.
• No. of matchsticks used to make 3^rd square = 10.
• So, the pattern that we observe here is 3n + 1.

With this pattern, we can easily find the number of matchsticks required in any number of squares.

Example: Find the number of matchsticks that will be used in pattern 32.

Sol: No. of matchsticks used in 1^st  Pattern = 4
No. of matchsticks used in 2^nd  Pattern = 8
No. of matchsticks used in 3^rd  Pattern = 12
So, the pattern we observe here is = 4n, (n = 1, 2, 3, 4, ...)
Hence, no. of matchsticks that will be used in 32^nd  Pattern = 4 x 32 = 128

Variable

• A variable is a quantity whose value can vary depending on the situation.
• It represents an unknown value that can change.
• For instance, in the expression 2x+5, the letter x is the variable.

Constant

• A quantity whose value does not vary is called a constant.
• For instance, in the expression 2x+5, 5 is a constant.

Expressions

• Arithmetic expressions involve numbers and basic math operations like adding, subtracting, multiplying, and dividing.
• They're like the building blocks of math, helping us solve problems and make sense of numbers in real life.

Example: 2 + (9 – 3), (4 × 6) – 8
(4 × 6) – 8 = 24 – 8
= 16

Expressions with variable

• Expressions in algebra often include variables, such as "2m" or "5 + t."
• These variables represent unknown quantities that can take on different values.
• Until we assign specific values to the variables, we can't fully evaluate or analyze the expression.

Example: Find 3x – 12 if x = 6.
Sol: (3 × 6) – 12
= 18 – 12
= 5
Thus, 3x – 12 = 5.

Formation of Expressions

• Algebraic expressions are combinations of variables (like x or y), constants (numbers), and operators (like + or -).
• They're used to represent mathematical ideas or formulas.
  Some examples have been given below:-
  

Practical use of Expressions

Example: Write down the equation justify the given statement.
'The age of Riya is five more than three times the age of her son, where age of Riya is Q and her son is P.'
Sol: Given that,
Age of Riya's son = P
Age of Riya = Q
According to question,  Riya's age (Q) is 5 more than three times the age of her son(P).

• 3 times Riya's son's age = 3 x P = 3P
• 5 more than 3 times Riya's son's age = 5 + 3P

Hence, Riya's age , Q =5 + 3P.

Like and Unlike Terms

• Like terms are the terms have the same variables in them.
• For example, 8xy and 3xy are like terms.
  These terms share the same variables, which are 'x' and 'y'.

Unlike Terms

• When terms in an expression have different variables, they are called unlike terms.
• For instance, 7xy and -3x are unlike terms.
• This means they don't have the same combination of variables.

Monomial, Binomial, Trinomial and Polynomial Terms

Addition and Subtraction of Algebraic Expressions

Addition and Subtraction of Like Terms

• Sum of two or more like terms is a like term.
• Its numerical coefficient will be equal to the sum of the numerical coefficients of all the like terms.

          Example:  8y+7y=?           

                                                   8y
                                                +7y
                              _______________
                              (8+7)y = 15y
                              _______________

•  Difference between two like terms is a like term.
• Its numerical coefficient will be equal to the difference between the numerical coefficients of the two like terms.
  Example: 11z−8z=? 
                                      11z
                                     −8z
                         _____________
                         (1-8)z = 3z
                         _____________                                                                                                                    

Addition and Subtraction of Unlike Terms

• For adding or subtracting two or more algebraic expressions, like terms of both the expressions are grouped together and unlike terms are retained as it is.

Addition of −5x2+12xy and 7x2+xy+7x is shown below:

                                    −5x²+12xy
                                      7x²+xy+7x
                                     ______________
                                  2x²+13xy+7x
                                     ______________

Subtraction of −5x2+12xy and 7x2+xy+7x is shown below:

                                   −5x²+12xy
                                   −7x²+xy+7x
                                   _____________
                              12x²+11xy−7x
                               __________________

Algebra as Patterns

Number Patterns

• If a natural number is denoted by n, then its successor is (n + 1).
  Example: Successor of n=10 is n+1=11.
• If a natural number is denoted by n, then 2n is an even number and (2n+1) is an odd number.
  Example: If n=10, then 2n=20 is an even number and 2n+1=21 is an odd number.

Patterns in Geometry

• Some geometrical figures follow patterns which can be represented by algebraic expressions.
  Example: Number of diagonals we can draw from one vertex of a polygon of n sides is (n – 3) which is an algebraic expression.

Practice Questions

Q1: Write the algebraic expression for the statement 'One-fifth of a number x is subtracted from the sum of b and thrice of c'.
Sol: Sum of b and thrice of c = b + 3c.
One - fifth of x =

Required expression: b + 3c - .

Q2: Find the value of m in the given equation.

Sol:

Q3: The present age of Reena is 1/4 of her father’s age. If the present age of her father is 52 years, then what will be the age of Reena after 7 years? Finally, express the present ages of Reena and her father in ratio form.
Sol: Given that,
 Present Age of Reena's Father = 52 years
∴ Present Age of Reena = (1/4) * Father's age
                                                      = (1/4) * 52
                                                      = 13 years
Reena's age after 7 years = Reena's present age + 7
                                                        = 13 + 7
                                                        = 20 years

Q4: The ratio of the present age of Sonam and Priya be 10 : 9. If sum of their ages be 57 years, then find their present ages respectively 
Sol: Let's denote the present age of Sonam as 10𝑥.
And the present age of Priya as 9x, where x is a common multiplier.

A common multiplier is a number that can be multiplied by each term in a ratio or equation to produce equivalent ratios or equations.
In this context, the common multiplier refers to a number that can be multiplied by both parts of the ratio (in this case, the ages of Sonam and Priya) to maintain the same ratio relationship between them.

According to question,
10x+9x=57
19x=57 (Combining like terms)

Now, solve for x:
x = 57/19
x = 3

Now that we have found the value of 𝑥, we can find their present ages:
Sonam's age: 10x=10×3=30 years
Priya's age: 9x=9×3=27 years
So, Sonam's present age is 30 years, and Priya's present age is 27 years.

Q5: The breadth of a rectangular bed sheet is 5 m more than half the length of the bed sheet. What is the perimeter of the bed sheet, if the
length is x m?
Sol: