Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 18 Fundamental Concepts

Important Points

1. Algebra: Algebra is like a broader version of arithmetic. In arithmetic, we work only with numbers like 3, -8, or 0.63, which have fixed values. In algebra, we use letters alongside numbers.
For example: \( 5x, 3x - 4, 7a + b, 3y - 5x, x + 3y - 9z \), etc.
These letters represent numbers whose values can change. We call them variables, literal numbers, or literals.

2. Signs and Symbols: In algebra, basic signs like addition, subtraction, multiplication, and division work just like they do in standard arithmetic. Here are some other symbols that we often use in math:
= means "is equal to"
\( \neq \) means "is not equal to"
< means "is less than"
> means "is greater than"
\( \not< \) means "is not less than"
\( \not> \) means "is not greater than"
\( \therefore \) means "therefore"
\( \backprime\backprime \) means "because" or "since"
~ means "difference between"

\( \implies \) means "implies that"

3. Writing Statements in Algebraic Form:

Conversely,

Exercise 18(A)

Question 1. Express each of the following statements in algebraic form:
(i) The sum of 8 and x is equal to y.
(ii) x decreased by 5 is equal to y.
(iii) The sum of 2 and x is greater than y.
(iv) The sum of x and y is less than 24.
(v) 15 multiplied by m gives 3n.
(vi) Product of 8 and y is equal to 3x.
(vii) 30 divided by b is equal to p.
(viii) z decreased by 3x is equal to y.
(ix) 12 times of x is equal to 5z.
(x) 12 times of x is greater than 5z.
(xi) 12 times of x is less than 5z.
(xii) 3z subtracted from 45 is equal to y.
(xiii) 8x divided by y is equal to 2z.
(xiv) 7y subtracted from 5x gives 8z.
(xv) 7y decreased by 5x gives 8z.
Answer:
(i) \( 8 + x = y \)
(ii) \( x - 5 = y \)
(iii) \( 2 + x > y \)
(iv) \( x + y < 24 \)
(v) \( 15m = 3n \)
(vi) \( 8y = 3x \)
(vii) \( \frac{30}{b} = p \)
(viii) \( z - 3x = y \)
(ix) \( 12x = 5z \)
(x) \( 12x > 5z \)
(xi) \( 12x < 5z \)
(xii) \( 45 - 3z = y \)
(xiii) \( \frac{8x}{y} = 2z \)
(xiv) \( 5x - 7y = 8z \)
(xv) \( 7y - 5x = 8z \)
In simple words: To turn a word statement into algebra, replace words like 'sum' with addition, 'decreased by' with subtraction, and 'equal to' with the equals sign.

Exam Tip: Watch out for phrases like 'subtracted from' because the order of the numbers reverses. For example, '3z subtracted from 45' means you start with 45 and subtract 3z, which is \( 45 - 3z \).

Question 2. For each of the following algebraic expressions, write a suitable statement in words:
(i) 3x + 8 = 15
(ii) 7 - y > x
(iii) 2y - x < 12
(iv) 5 \(\div\) z = 5
(v) a + 2b > 18
(vi) 2x - 3y = 16
(vii) 3a - 4b > 14
(viii) b + 7a < 21
(ix) (16 + 2a) - x > 25
(x) (3x + 12) - y < 3a
Answer:
(i) The sum of 3x and 8 is equal to 15.
(ii) 7 reduced by y is greater than x.
(iii) 2y decreased by x is less than 12.
(iv) 5 divided by z is equal to 5.
(v) a increased by 2b is greater than 18.
(vi) 2x reduced by 3y is equal to 16.
(vii) 3a decreased by 4b is greater than 14.
(viii) b increased by 7a is less than 21.
(ix) The sum of 16 and 2a, decreased by x, is greater than 25.
(x) The sum of 3x and 12, decreased by y, is less than 3a.
In simple words: You can write algebraic expressions in everyday English by replacing symbols like +, -, >, and < with words like 'added to', 'reduced by', 'is more than', and 'is less than'.

Exam Tip: When writing algebraic expressions in words, make sure to use clear terms like 'increased by' for addition and 'decreased by' for subtraction to avoid any confusion.

Exercise 18(B)

Question 1. Separate the constants and the variables from each of the following:
\( 6, 4y, -3x, \frac{5}{4}, \frac{4}{5}xy, az, 7p, 0, \frac{9x}{y}, \frac{3}{4x}, -\frac{xz}{3y} \)
Answer:
Constants: \( 6, \frac{5}{4}, 0 \)
Variables: \( 4y, -3x, \frac{4}{5}xy, az, 7p, \frac{9x}{y}, \frac{3}{4x}, -\frac{xz}{3y} \)
In simple words: Constants are fixed numbers that never change. Variables are letters or expressions containing letters, as their values can vary.

Exam Tip: A variable can be a single letter, a product of letters, or a combination of numbers and letters. Any term containing a letter is classified as a variable term.

Question 2. Group the like terms together :
(i) \( 4x, -3y, -x, \frac{2}{3}x, \frac{4}{5}y \) and \( y \).
(ii) \( \frac{2}{3}xy, -4yx, 2yz, \frac{-2}{3}yz, \frac{zy}{3} \) and \( yx \).
(iii) \( -ab^2, b^2a^2, 7b^2a, -3a^2b^2 \) and \( 2ab^2 \).
(iv) \( 5ax, -5by, \frac{by}{7}, 7xa \) and \( \frac{2ax}{3} \).
Answer:
(i) Group of x terms: \( 4x, -x, \frac{2}{3}x \); Group of y terms: \( -3y, \frac{4}{5}y, y \)
(ii) Group of xy/yx terms: \( \frac{2}{3}xy, -4yx, yx \); Group of yz/zy terms: \( 2yz, -\frac{2}{3}yz, \frac{zy}{3} \)
(iii) Group of \( ab^2 \) terms: \( -ab^2, 7b^2a, 2ab^2 \); Group of \( a^2b^2 \) terms: \( b^2a^2, -3a^2b^2 \)
(iv) Group of ax/xa terms: \( 5ax, 7xa, \frac{2ax}{3} \); Group of by terms: \( -5by, \frac{by}{7} \)
In simple words: Like terms are terms that have the exact same variables and powers, even if the order of the letters is different or their numerical coefficients are not the same.

Exam Tip: Remember that the order of multiplication does not change like terms. For example, \( xy \) and \( yx \) are like terms, as are \( ab^2 \) and \( b^2a \).

Question 3. State whether true or false :
(i) 16 is a constant and y is a variable but 16y is variable.
(ii) 5x has two terms 5 and x.
(iii) The expression 5 + x has two terms 5 and x.
(iv) The expression \( 2x^2 + x \) is a trinomial.
(v) \( ax^2 + bx + c \) is a trinomial.
(vi) 8 x ab is a binomial.
(vii) 8 + ab is a binomial.
(viii) \( x^3 - 5xy + 6x + 7 \) is a polynomial.
(ix) \( x^3 - 5xy + 6x + 7 \) is a multinomial.
(x) The coefficient of x in 5x is 5x.
(xi) The coefficient of ab in - ab is - 1.
(xii) The coefficient of y in - 3xy is - 3.
Answer:
(i) True
(ii) False
(iii) True
(iv) False
(v) True
(vi) False
(vii) True
(viii) True
(ix) True
(x) False
(xi) True
(xii) False
In simple words: Terms are separated by plus or minus signs. If terms are multiplied together, they form a single term.

Exam Tip: Remember that coefficients do not include the variable itself. For instance, the coefficient of \( x \) in \( 5x \) is just 5, not \( 5x \).

Question 4. State the number of terms in each of the following expressions :
(i) \( 2a - b \)
(ii) \( 3 \times x + \frac{a}{2} \)
(iii) \( 3x - \frac{x}{p} \)
(iv) \( a \div x \times b + c \)
(v) \( 3x \div 2 + y + 4 \)
(vi) \( xy \div 2 \)
(vii) \( x + y \div a \)
(viii) \( 2x + y + 8 \div y \)
(ix) \( 2 \times a + 3 \div b + 4 \)
Answer:
(i) 2 terms
(ii) 2 terms
(iii) 2 terms
(iv) 2 terms
(v) 3 terms
(vi) 1 term
(vii) 2 terms
(viii) 3 terms
(ix) 3 terms
In simple words: Terms are separated only by addition (+) and subtraction (-) signs. Multiplication (\(\times\)) and division (\(\div\)) do not separate terms.

Exam Tip: Always simplify multiplication and division first before counting the terms. For example, \( xy \div 2 \) simplifies to \( \frac{xy}{2} \vec{\rightarrow} \) which is just one single term.

Question 5. State whether true or false:
(i) xy and - yx are like terms.
(ii) \( x^2y \) and \( -y^2x \) are like terms.
(iii) a and - a are like terms.
(iv) - ba and 2ab are unlike terms.
(v) 5 and 5x are like terms.
(vi) 3xy and 4xyz are unlike terms.
Answer:
(i) True
(ii) False
(iii) True
(iv) False
(v) False
(vi) True
In simple words: Like terms must have the exact same variables raised to the exact same powers. The numbers in front of them do not have to match.

Exam Tip: Watch out for the order of letters. For example, \( -ba \) and \( 2ab \) have the same variables with the same powers, so they are like terms, making the statement 'unlike terms' false.

Question 6. For each expression, given below, state whether it is a monomial, or a binomial or a trinomial.
(i) \( xy \)
(ii) \( xy + x \)
(iii) \( 2x \div y \)
(iv) \( -a \)
(v) \( ax^2 - x + 5 \)
(vi) \( -3bc + d \)
(vii) \( 1 + x + y \)
(viii) \( 1 + x \div y \)
(ix) \( x + xy - y^2 \)
Answer:
(i) Monomial
(ii) Binomial
(iii) Monomial
(iv) Monomial
(v) Trinomial
(vi) Binomial
(vii) Trinomial
(viii) Binomial
(ix) Trinomial
In simple words: A monomial has one term, a binomial has two terms, and a trinomial has three terms.

Exam Tip: Simplify any division or multiplication before classifying the expression. For example, \( 1 + x \div y \) simplifies to \( 1 + \frac{x}{y} \), which is a binomial.

Question 7. Write down the coefficient of x in the following monomial:
(i) \( x \)
(ii) \( -x \)
(iii) \( -3x \)
(iv) \( -5ax \)
(v) \( \frac{3}{2}xy \)
(vi) \( \frac{ax}{y} \)
Answer:
(i) 1
(ii) -1
(iii) -3
(iv) -5a
(v) \( \frac{3}{2}y \)
(vi) \( \frac{a}{y} \)
In simple words: To find the coefficient of \( x \), remove \( x \) from the term. Whatever is left over is the coefficient.

Exam Tip: Do not forget the sign of the term. For example, the coefficient of \( x \) in \( -x \) is \( -1 \), and in \( -5ax \) it is \( -5a \).

Question 8. Write the coefficient of :
(i) x in \( -3xy^2 \)
(ii) x in \( -ax \)
(iii) y in \( -y \)
(iv) y in \( \frac{2}{a}y \)
(v) xy in \( -2xyz \)
(vi) ax in \( -axy^2 \)
(vii) \( x^2y \) in \( -3ax^2y \)
(viii) \( xy^2 \) in \( 5axy^2 \)
Answer:
(i) \( -3y^2 \)
(ii) \( -a \)
(iii) -1
(iv) \( \frac{2}{a} \)
(v) \( -2z \)
(vi) \( -y^2 \)
(vii) \( -3a \)
(viii) \( 5a \)
In simple words: To find the coefficient of any variable or group of variables, divide the entire term by those variables.

Exam Tip: Always double-check if there are other letters or constants left behind. For example, for the coefficient of \( xy^2 \) in \( 5axy^2 \), the remaining part is \( 5a \).

Question 9. State the numeral coefficient of the following monomials:
(i) \( 5xy \)
(ii) \( abc \)
(iii) \( 5pqr \)
(iv) \( \frac{-2x}{y} \)
(v) \( \frac{2}{3}xy^2 \)
(vi) \( \frac{-15xy}{2z} \)
(vii) \( -7x \div y \)
(viii) \( -3x \div (2y) \)
Answer:
(i) 5
(ii) 1
(iii) 5
(iv) -2
(v) \( \frac{2}{3} \)
(vi) \( -\frac{15}{2} \)
(vii) -7
(viii) \( -\frac{3}{2} \)
In simple words: The numeral coefficient is just the constant number part of the term, including its sign and any fraction.

Exam Tip: If no number is written in front of the variables, the numeral coefficient is \( 1 \) (or \( -1 \) if there is a minus sign).

Question 10. Write the degree of each of the following polynomials:
(i) \( x + x^2 \)
(ii) \( 5x^2 - 7x + 2 \)
(iii) \( x^3 - x^8 + x^{10} \)
(iv) \( 1 - 100x^{20} \)
(v) \( 4 + 4x - 4x^3 \)
(vi) \( 8x^2y - 3y^2 + x^2y^5 \)
(vii) \( 8z^3 - 8y^2z^3 + 7yz^5 \)
(viii) \( 4y^2 - 3x^3 + y^2x^7 \)
Answer:
(i) 2
(ii) 2
(iii) 10
(iv) 20
(v) 3
(vi) 7
(vii) 6
(viii) 9
In simple words: The degree of a polynomial is the highest sum of the powers of variables in any single term.

Exam Tip: For terms with more than one variable, add the powers of all variables in that term to find its degree. The highest total is the degree of the entire polynomial.

Revision Exercise

Question 1. Express each of the following statements in algebraic form:
(i) The sum of 3x and 4y is 8.
(ii) 5x decreased by 7 gives y.
(iii) 31 added to 4x gives 6x.
(iv) 3x subtracted from 89 gives 44.
Answer:
(i) \( 3x + 4y = 8 \)
(ii) \( 5x - 7 = y \)
(iii) \( 4x + 31 = 6x \)
(iv) \( 89 - 3x = 44 \)
In simple words: Match the English words to mathematical operations like addition (+) and subtraction (-), and use an equals sign (=) for words like 'is' or 'gives'.

Exam Tip: Be careful with the phrase 'subtracted from' - the term after 'from' must be written first in your expression.

Question 2. Group the like terms :
(i) \( 7y, 3x, -8y, -x \) and \( \frac{x}{5} \)
(ii) \( 3x^2, -5x^3, -x^2, 5x^2 \) and \( 8x^3 \)
(iii) \( x^2y^3, -5x^3y^2, 8x^3y^2, -4x^2y^3 \) and \( -x^2y^3 \)
Answer:
(i) Group of x terms: \( 3x, -x, \frac{x}{5} \); Group of y terms: \( 7y, -8y \)
(ii) Group of \( x^2 \) terms: \( 3x^2, -x^2, 5x^2 \); Group of \( x^3 \) terms: \( -5x^3, 8x^3 \)
(iii) Group of \( x^2y^3 \) terms: \( x^2y^3, -4x^2y^3, -x^2y^3 \); Group of \( x^3y^2 \) terms: \( -5x^3y^2, 8x^3y^2 \)
In simple words: Group together terms that have the exact same variables and exponents, even if their starting numbers are different.

Exam Tip: Keep track of negative signs on terms when grouping them, and make sure the exponents on each variable match exactly.

Question 3. Write the number of terms in each of the following polynomials:
(i) \( 5 + 4x \div 2 \)
(ii) \( 5 + 4x + 2y \)
(iii) \( 8x^2 - 4x + 7 \)
(iv) \( \frac{x}{5} + \frac{x^2}{7} - \frac{x^3}{8} - \frac{1}{4} \)
(v) \( 6x^2 \div x - 18 \div 9 + x^2 \)
Answer:
(i) 2 terms
(ii) 3 terms
(iii) 3 terms
(iv) 4 terms
(v) 3 terms
In simple words: Always simplify multiplication and division in an expression first, and then count how many parts are separated by plus or minus signs.

Exam Tip: In terms like \( 6x^2 \div x \), think of division as a single term. Only count individual terms once all division and multiplication operations are worked out.

Question 4. For each expression, given below, state whether it is a monomial, or a binomial or a trinomial:
(i) \( x + y \)
(ii) \( 5x - 4y \)
(iii) \( 7x^2 + 5x + 8 \)
(iv) \( 6a + 3 \div b \)
(v) \( 9 \div a \times b \)
(vi) \( 8a \div b \)
Answer:
(i) Binomial
(ii) Binomial
(iii) Trinomial
(iv) Binomial
(v) Monomial
(vi) Monomial
In simple words: Monomials have one term, binomials have two terms, and trinomials have three terms.

Exam Tip: Remember that \( 9 \div a \times b \) can be written as \( \frac{9b}{a} \). Since it has no plus or minus signs separating the terms, it is a monomial.

Question 5. Write the coefficient of \( x^2y \) in:
(i) \( -7x^2yz \)
(ii) \( 8abx^2y \)
(iii) \( -x^2y \)
Answer:
(i) \( -7z \)
(ii) \( 8ab \)
(iii) -1
In simple words: Remove the parts \( x^2y \) from each expression to find the remaining coefficient.

Exam Tip: When a term like \( -x^2y \) has only a minus sign in front, its coefficient is \( -1 \).

Question 6. Write the coefficient of:
(i) \( x^2 \) in \( -8x^2y \)
(ii) y in \( -4y \)
(iii) x in \( -xy^2 \)
Answer:
(i) \( -8y \)
(ii) -4
(iii) \( -y^2 \)
In simple words: Divide the term by the given variable to find its coefficient.

Exam Tip: Make sure to keep the negative sign with your coefficient when the original term is negative.

Question 7. Write the numeral coefficient in:
(i) \( 7x^2y \)
(ii) \( \frac{2x}{3} \)
(iii) \( -\frac{5}{4}xy^2z \)
Answer:
(i) 7
(ii) \( \frac{2}{3} \)
(iii) \( -\frac{5}{4} \)
In simple words: The numeral coefficient is the numerical value or fraction at the start of the term.

Exam Tip: Fractions like \( \frac{2x}{3} \) can be written as \( \frac{2}{3}x \), which clearly shows that the numeral coefficient is \( \frac{2}{3} \).

Question 8. Write the degree of each of the following polynomials:
(i) \( x^5 - 6x^8 + x \)
(ii) \( 4x^3 - x^4 \)
(iii) \( 4 - x^2 \)
(iv) \( x - 1 \)
(v) \( x^2 + x - x^3 \)
(vi) \( x^3 - 8xy^2 + x^3y^3 \)
(vii) \( x^7 - 6y^4 \)
(viii) \( 3y^3 - 2y^2z^4 \)
(ix) \( 100x^8 - 8x^{100} \)
Answer:
(i) 8
(ii) 4
(iii) 2
(iv) 1
(v) 3
(vi) 6
(vii) 7
(viii) 6
(ix) 100
In simple words: The degree is the largest sum of powers on the variables in any single term of the polynomial.

Exam Tip: If a term has more than one variable, add their powers together. For example, the term \( x^3y^3 \) has a total power of \( 3 + 3 = 6 \).

Question 9. Write each statement, given below in algebraic form:
(i) 28 more than twice of x is equal to 45.
(ii) 3y reduced by 5z is greater than 8x.
(iii) 6x divided by 13y is less than 17.
(iv) 9 multiplied by 5x is equal to 2y.
Answer:
(i) \( 2x + 28 = 45 \)
(ii) \( 3y - 5z > 8x \)
(iii) \( \frac{6x}{13y} < 17 \)
(iv) \( 9 \times 5x = 2y \)
In simple words: Replace mathematical statements with their corresponding algebra symbols.

Exam Tip: 'Twice of x' translates to \( 2x \). Adding 28 to that gives \( 2x + 28 \).

Question 10. State whether true or false:
(i) If 23 is a constant and x is a variable, 23 + x is constant.
(ii) If 23 is a constant and x is a variable, 23x is a variable.
(iii) If y is a variable and 57 is a constant, y - 57 is a variable.
(iv) If 3x and 2y are variable; each of 3x + 2y, 3x - 2y, 3x \(\div\) 2y and 3x x 2y is a variable.
Answer:
(i) False
(ii) True
(iii) True
(iv) True
In simple words: If you combine a variable with a constant using addition, subtraction, multiplication, or division, the final expression remains a variable.

Exam Tip: Any expression that contains at least one variable is always considered a variable because its value depends on the value of that variable.