Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 21 Framing Algebraic Expressions

Question 1. Write in the form of an algebraic expression :
(i) Perimeter (P) of a rectangle is two times the sum of its length (l) and its breadth (b).
(ii) Perimeter (P) of a square is four times its side.
(iii) Area of a square is square of its side.
(iv) Surface area of a cube is six times the square of its edge.
Answer:
(i) If we denote \( P \) as the perimeter, \( l \) as the length, and \( b \) as the breadth, the formula is:
\( P = 2(l + b) \)
(ii) Let \( P \) represent the perimeter and \( a \) denote the side length. The equation is:
\( P = 4a \)
(iii) Let \( A \) be the area and \( a \) be the side of the square. The relationship is:
\( A = a^2 \)
(iv) Let \( S \) be the total surface area and \( a \) be the edge length. The expression is:
\( S = 6a^2 \)
In simple words: We can write word rules using short math symbols and letters instead of long sentences.
Exam Tip: Always state clearly what each letter in your formula represents so that anyone reading it understands your work.

Question 2. Express each of the following as an algebraic expression :
(i) The sum of x and y minus m.
(ii) The product of x and y divided by m.
(iii) The subtraction of 5m from 3n and then adding 9p to it.
(iv) The product of 12, x, y and z minus the product of 5, m and n.
(v) Sum of p and 2r - s minus sum of a and 3n + 4x.
Answer:
(i) \( x + y - m \)
(ii) \( \frac{xy}{m} \)
(iii) \( 3n - 5m + 9p \)
(iv) \( 12xyz - 5mn \)
(v) \( p + 2r - s - (a + 3n + 4x) \)
In simple words: We can turn words like "sum" into plus, "product" into times, and "minus" into subtraction signs to make math sentences.

Exam Tip: Be very careful with brackets when subtracting a whole group of terms, like putting parentheses around \( a + 3n + 4x \).

Question 3. Construct a formula for the following : Total wages (Rs. W) of a man whose basic wage is (Rs. B) for t hours week plus (Rs. R) per hour, if he Works a total of T hours.
Answer: The amount earned for the standard \( t \) hours is Rs. \( B \).
The remaining overtime hours worked are calculated as \( T - t \).
The money earned during these overtime hours is Rs. \( R(T - t) \).
Adding these two amounts together gives the formula for the complete wage:

\( \implies \) Rs. \( W = B + R(T - t) \)
In simple words: The total pay equals the standard weekly wage plus the extra rate multiplied by the overtime hours.

Exam Tip: Ensure you subtract the basic hours \( t \) from the total hours \( T \) to find the correct overtime duration.

Question 4. If x = 4, evaluate :
(i) 3x + 8
(ii) x^2 - 2x
(iii) x^2 / 2
Answer:
(i) Replace \( x \) with \( 4 \):
\( 3(4) + 8 = 12 + 8 = 20 \)
(ii) Replace \( x \) with \( 4 \):
\( (4)^2 - 2(4) = 16 - 8 = 8 \)
(iii) Replace \( x \) with \( 4 \):
\( \frac{4^2}{2} = \frac{16}{2} = 8 \)
In simple words: Swap the letter x with the number four and calculate the answer using standard order of operations.

Exam Tip: Solve the powers first before you perform operations like division, following the BODMAS rule.

Question 5. If m = 6, evaluate :
(i) 5m - 6
(ii) 2m^2 + 3m
(iii) (2m)^2
Answer:
(i) Substituting \( m = 6 \) gives:
\( 5(6) - 6 = 30 - 6 = 24 \)
(ii) Substituting \( m = 6 \) gives:
\( 2(6)^2 + 3(6) = 2(36) + 18 = 72 + 18 = 90 \)
(iii) Substituting \( m = 6 \) gives:
\( (2 \times 6)^2 = 12^2 = 144 \)
In simple words: Put the number 6 in place of m, then solve the multiplication and powers.

Exam Tip: For expressions like \( (2m)^2 \), always multiply inside the bracket before squaring the result.

Question 6. If x = 4, evaluate :
(i) 12x + 7
(ii) 5x^2 + 4x
(iii) x^2 / 8
Answer:
(i) Substitute \( x = 4 \) into the term:
\( 12(4) + 7 = 48 + 7 = 55 \)
(ii) Substitute \( x = 4 \) into the term:
\( 5(4)^2 + 4(4) = 5(16) + 16 = 80 + 16 = 96 \)
(iii) Substitute \( x = 4 \) into the term:
\( \frac{4^2}{8} = \frac{16}{8} = 2 \)
In simple words: Swap x for 4 and calculate the value of each expression.

Exam Tip: Be careful not to multiply \( 5 \) by \( x \) before squaring in the term \( 5x^2 \). Only \( x \) is squared.

Question 7. If m = 2, evaluate :
(i) 16m - 7
(ii) 15m^2 - 10m
(iii) 1/4 * m^3
Answer:
(i) Substituting \( m = 2 \) gives:
\( 16(2) - 7 = 32 - 7 = 25 \)
(ii) Substituting \( m = 2 \) gives:
\( 15(2)^2 - 10(2) = 15(4) - 20 = 60 - 20 = 40 \)
(iii) Substituting \( m = 2 \) gives:
\( \frac{1}{4}(2)^3 = \frac{1}{4}(8) = 2 \)
In simple words: Put 2 wherever you see the letter m, then calculate the answer.

Exam Tip: Remember that \( m^3 \) means multiplying the number by itself three times: \( 2 \times 2 \times 2 = 8 \).

Question 8. If x = 10, evaluate :
(i) 100x + 225
(ii) 6x^2 - 25x
(iii) 1/50 * x^3
Answer:
(i) Putting \( x = 10 \) gives:
\( 100(10) + 225 = 1000 + 225 = 1225 \)
(ii) Putting \( x = 10 \) gives:
\( 6(10)^2 - 25(10) = 6(100) - 250 = 600 - 250 = 350 \)
(iii) Putting \( x = 10 \) gives:
\( \frac{1}{50}(10)^3 = \frac{1000}{50} = 20 \)
In simple words: Use 10 instead of x, then perform the simple math operations.

Exam Tip: Keep track of the number of zeros when working with powers of 10, such as \( 10^3 = 1000 \).

Question 9. If a = -10, evaluate :
(i) 5a
(ii) a^2
(iii) a^3
Answer:
(i) Using \( a = -10 \) gives:
\( 5(-10) = -50 \)
(ii) Using \( a = -10 \) gives:
\( (-10)^2 = 100 \)
(iii) Using \( a = -10 \) gives:
\( (-10)^3 = -1000 \)
In simple words: Replace a with negative ten. Multiplying two negative numbers gives a positive answer.

Exam Tip: Remember that squaring a negative number yields a positive result, while cubing it yields a negative result.

Question 10. If x = -6, evaluate :
(i) 11x
(ii) 4x^2
(iii) 2x^3
Answer:
(i) Put \( x = -6 \) to get:
\( 11(-6) = -66 \)
(ii) Put \( x = -6 \) to get:
\( 4(-6)^2 = 4(36) = 144 \)
(iii) Put \( x = -6 \) to get:
\( 2(-6)^3 = 2(-216) = -432 \)
In simple words: Replace x with negative six, then work out the math step by step.

Exam Tip: Write down your steps when evaluating powers with negative bases to prevent sign errors.

Question 11. If m = -7, evaluate :
(i) 12m
(ii) 2m^2
(iii) 2m^3
Answer:
(i) Substitute \( m = -7 \):
\( 12(-7) = -84 \)
(ii) Substitute \( m = -7 \):
\( 2(-7)^2 = 2(49) = 98 \)
(iii) Substitute \( m = -7 \):
\( 2(-7)^3 = 2(-343) = -686 \)
In simple words: Swap m for negative seven, calculate the powers first, and then multiply.

Exam Tip: Ensure that the exponent is solved before multiplying by the coefficient, as in \( 2 \times (-7)^2 \).

Question 12. Find the average (A) of four quantities p, q, r and s. If A = 6, p = 3, q = 5 and r = 7 ; find the value of s.
Answer: The average \( A \) of the four numbers is computed as:
\( A = \frac{p + q + r + s}{4} \)
Substituting the values we know into this equation:
\( 6 = \frac{3 + 5 + 7 + s}{4} \)
Multiplying both sides by 4:
\( 24 = 15 + s \)
Solving for \( s \):
\( s = 24 - 15 = 9 \)
In simple words: To find the average, we add the four numbers and divide by four. We use this to solve for the missing number.

Exam Tip: Start by multiplying the average by the count of terms to get the sum of all values first.

Question 13. If a = 5 and b = 6, evaluate :
(i) 3ab
(ii) 6a^2b
(iii) 2b^2
Answer:
(i) Substituting \( a = 5 \) and \( b = 6 \):
\( 3(5)(6) = 15(6) = 90 \)
(ii) Substituting \( a = 5 \) and \( b = 6 \):
\( 6(5)^2(6) = 6(25)(6) = 150(6) = 900 \)
(iii) Substituting \( b = 6 \):
\( 2(6)^2 = 2(36) = 72 \)
In simple words: Put 5 for a and 6 for b, then multiply the numbers together.

Exam Tip: Always evaluate exponents first before performing multiplication across multiple variables.

Question 14. If x = 8 and y = 2, evaluate :
(i) 9xy
(ii) 5x^2y
(iii) (4y)^2
Answer:
(i) Substitute \( x = 8 \) and \( y = 2 \) into the term:
\( 9(8)(2) = 144 \)
(ii) Substitute \( x = 8 \) and \( y = 2 \) into the term:
\( 5(8)^2(2) = 5(64)(2) = 640 \)
(iii) Substitute \( y = 2 \) into the term:
\( (4 \times 2)^2 = 8^2 = 64 \)
In simple words: Use 8 for x and 2 for y, then simplify using basic arithmetic.

Exam Tip: In expressions with parentheses like \( (4y)^2 \), make sure to multiply inside the brackets before squaring.

Question 15. If x = 5 and y = 4, evaluate :
(i) 8xy
(ii) 3x^2y
(iii) 3y^2
Answer:
(i) Putting \( x = 5 \) and \( y = 4 \) gives:
\( 8(5)(4) = 160 \)
(ii) Putting \( x = 5 \) and \( y = 4 \) gives:
\( 3(5)^2(4) = 3(25)(4) = 300 \)
(iii) Putting \( y = 4 \) gives:
\( 3(4)^2 = 3(16) = 48 \)
In simple words: Replace x with 5 and y with 4, then carry out the multiplication.

Exam Tip: Keep your steps clear and substitute one variable at a time to avoid simple calculation mistakes.

Question 16. If y = 5 and z = 2, evaluate :
(i) 100yz
(ii) 9y^2z
(iii) 5y^2
(iv) (5z)^3
Answer:
(i) Substitute \( y = 5 \) and \( z = 2 \):
\( 100(5)(2) = 1000 \)
(ii) Substitute \( y = 5 \) and \( z = 2 \):
\( 9(5)^2(2) = 9(25)(2) = 450 \)
(iii) Substitute \( y = 5 \):
\( 5(5)^2 = 5(25) = 125 \)
(iv) Substitute \( z = 2 \):
\( (5 \times 2)^3 = 10^3 = 1000 \)
In simple words: Replace the letters with 5 and 2, then calculate the final numbers.

Exam Tip: For the cubed expression \( (5z)^3 \), solve the inner multiplication first before raising it to the power of three.

Question 17. If x = 2 and y = 10, evaluate :
(i) 30xy
(ii) 50xy^2
(iii) (10x)^2
(iv) 5y^2
Answer:
(i) Substituting \( x = 2 \) and \( y = 10 \):
\( 30(2)(10) = 600 \)
(ii) Substituting \( x = 2 \) and \( y = 10 \):
\( 50(2)(10)^2 = 100(100) = 10000 \)
(iii) Substituting \( x = 2 \):
\( (10 \times 2)^2 = 20^2 = 400 \)
(iv) Substituting \( y = 10 \):
\( 5(10)^2 = 5(100) = 500 \)
In simple words: Swap x with 2 and y with 10, then perform the simple math operations.

Exam Tip: Pay close attention to which variables have exponents; only \( y \) is squared in \( 50xy^2 \).

Question 18. If m = 3 and n = 7, evaluate :
(i) 12mn
(ii) 5mn^2
(iii) (10m)^2
(iv) 4n^2
Answer:
(i) Using \( m = 3 \) and \( n = 7 \) gives:
\( 12(3)(7) = 252 \)
(ii) Using \( m = 3 \) and \( n = 7 \) gives:
\( 5(3)(7)^2 = 15(49) = 735 \)
(iii) Using \( m = 3 \) gives:
\( (10 \times 3)^2 = 30^2 = 900 \)
(iv) Using \( n = 7 \) gives:
\( 4(7)^2 = 4(49) = 196 \)
In simple words: Put 3 for m and 7 for n, then multiply the terms together.

Exam Tip: Write down your multiplication steps clearly on paper to avoid mental math mistakes during exams.

Question 19. If a = -10, evaluate :
(i) 3a - 2
(ii) a^2 + 8a
(iii) 1/5 * a^2
Answer:
(i) Replace \( a \) with \( -10 \):
\( 3(-10) - 2 = -30 - 2 = -32 \)
(ii) Replace \( a \) with \( -10 \):
\( (-10)^2 + 8(-10) = 100 - 80 = 20 \)
(iii) Replace \( a \) with \( -10 \):
\( \frac{1}{5}(-10)^2 = \frac{1}{5}(100) = 20 \)
In simple words: Replace the letter a with negative ten, and then calculate the result step by step.

Exam Tip: Remember that adding a negative number is the same as subtraction: \( 100 + (-80) = 20 \).

Question 20. If x = -6, evaluate :
(i) 4x - 9
(ii) 3x^2 + 8x
(iii) x^2 / 2
Answer:
(i) Substituting \( x = -6 \):
\( 4(-6) - 9 = -24 - 9 = -33 \)
(ii) Substituting \( x = -6 \):
\( 3(-6)^2 + 8(-6) = 3(36) - 48 = 108 - 48 = 60 \)
(iii) Substituting \( x = -6 \):
\( \frac{(-6)^2}{2} = \frac{36}{2} = 18 \)
In simple words: Put negative six in place of x, and then solve the expressions carefully.

Exam Tip: Be sure to keep negative bases inside parentheses when squaring them, like \( (-6)^2 \), to ensure a positive result.

Question 21. If m = -8, evaluate :
(i) 2m + 21
(ii) m^2 + 9m
(iii) m^2 / 4
Answer:
(i) Substituting \( m = -8 \) gives:
\( 2(-8) + 21 = -16 + 21 = 5 \)
(ii) Substituting \( m = -8 \) gives:
\( (-8)^2 + 9(-8) = 64 - 72 = -8 \)
(iii) Substituting \( m = -8 \) gives:
\( \frac{(-8)^2}{4} = \frac{64}{4} = 16 \)
In simple words: Swap the letter m for negative eight, then calculate the answer following basic math rules.

Exam Tip: When subtracting a larger value from a smaller value, the final answer will be negative: \( 64 - 72 = -8 \).

Question 22. If p = -10, evaluate :
(i) 6p + 50
(ii) 3p^2 - 20p
(iii) p^2 / 50
Answer:
(i) If we substitute \( p = -10 \):
\( 6(-10) + 50 = -60 + 50 = -10 \)
(ii) If we substitute \( p = -10 \):
\( 3(-10)^2 - 20(-10) = 3(100) + 200 = 300 + 200 = 500 \)
(iii) If we substitute \( p = -10 \):
\( \frac{(-10)^2}{50} = \frac{100}{50} = 2 \)
In simple words: Replace the letter p with negative ten, and then calculate the final value.

Exam Tip: Multiplying two negative values results in a positive value, as in \( -20 \times (-10) = +200 \).

Question 23. If y = -8, evaluate :
(i) 6y + 53
(ii) y^2 + 12y
(iii) y^3 / 4
Answer:
(i) Putting \( y = -8 \) gives:
\( 6(-8) + 53 = -48 + 53 = 5 \)
(ii) Putting \( y = -8 \) gives:
\( (-8)^2 + 12(-8) = 64 - 96 = -32 \)
(iii) Putting \( y = -8 \) gives:
\( \frac{(-8)^3}{4} = \frac{-512}{4} = -128 \)
In simple words: Swap y with negative eight, calculate the powers first, and then divide or add.

Exam Tip: Keep in mind that raising a negative number to an odd power, like \( (-8)^3 \), yields a negative result.

Question 24. If x = 2 and y = -4, evaluate :
(i) 11xy
(ii) 5x^2y
(iii) (5y)^2
(iv) 8x^2
Answer:
(i) Substituting \( x = 2 \) and \( y = -4 \):
\( 11(2)(-4) = -88 \)
(ii) Substituting \( x = 2 \) and \( y = -4 \):
\( 5(2)^2(-4) = 5(4)(-4) = -80 \)
(iii) Substituting \( y = -4 \):
\( (5 \times -4)^2 = (-20)^2 = 400 \)
(iv) Substituting \( x = 2 \):
\( 8(2)^2 = 8(4) = 32 \)
In simple words: Swap x with 2 and y with negative four, then evaluate the terms step by step.

Exam Tip: Squaring a negative product such as \( (-20)^2 \) will always result in a positive value.

Question 25. If m = 9 and n = -2, evaluate :
(i) 4mn
(ii) 2m^2n
(iii) (2n)^3
Answer:
(i) Substituting \( m = 9 \) and \( n = -2 \):
\( 4(9)(-2) = -72 \)
(ii) Substituting \( m = 9 \) and \( n = -2 \):
\( 2(9)^2(-2) = 2(81)(-2) = -324 \)
(iii) Substituting \( n = -2 \):
\( (2 \times -2)^3 = (-4)^3 = -64 \)
In simple words: Put 9 for m and negative two for n, then solve each math equation.

Exam Tip: Be mindful of negative bases with odd exponents, which remain negative, as in \( (-4)^3 = -64 \).

Question 26. If m = -8 and n = -2, evaluate :
(i) 12mn
(ii) 3m^2n
(iii) (4n)^2
Answer:
(i) Put \( m = -8 \) and \( n = -2 \) to get:
\( 12(-8)(-2) = 192 \)
(ii) Put \( m = -8 \) and \( n = -2 \) to get:
\( 3(-8)^2(-2) = 3(64)(-2) = -384 \)
(iii) Put \( n = -2 \) to get:
\( (4 \times -2)^2 = (-8)^2 = 64 \)
In simple words: Use negative eight for m and negative two for n. Multiplying two negatives makes a positive.

Exam Tip: Work out the exponent on the variable before multiplying it by the outer coefficient.

Question 27. If x = -5 and y = -8, evaluate :
(i) 4xy
(ii) 2xy^2
(iii) 4x^2
(iv) 3y^2
Answer:
(i) Substitute \( x = -5 \) and \( y = -8 \) into the terms:
\( 4(-5)(-8) = 160 \)
(ii) Substitute \( x = -5 \) and \( y = -8 \) into the terms:
\( 2(-5)(-8)^2 = 2(-5)(64) = -640 \)
(iii) Substitute \( x = -5 \) into the terms:
\( 4(-5)^2 = 4(25) = 100 \)
(iv) Substitute \( y = -8 \) into the terms:
\( 3(-8)^2 = 3(64) = 192 \)
In simple words: Replace the letters with negative five and negative eight, then calculate the results.

Exam Tip: Even though squaring a negative number yields a positive, multiplying by another negative value will make the final product negative, as in \( 2 \times (-5) \times 64 \).

Question 28. Find T, if T = 2a - b, a = 7 and b = 3.
Answer: Substitute \( a = 7 \) and \( b = 3 \) into the given formula:
\( T = 2(7) - 3 \)
\( T = 14 - 3 \)
\( T = 11 \)
In simple words: Use 7 for a and 3 for b to work out the value of T.

Exam Tip: Double check that you substitute the correct values from the question text into the formula.

Question 29. From the formula B = 2a^2 - b^2, calculate the value of B when a = 3 and b = -1.
Answer: Substitute \( a = 3 \) and \( b = -1 \) into the equation:
\( B = 2(3)^2 - (-1)^2 \)
\( B = 2(9) - 1 \)
\( B = 18 - 1 \)
\( B = 17 \)
In simple words: Swap a with 3 and b with negative one, then solve the exponents and subtract.

Exam Tip: Be sure to write \( (-1)^2 = 1 \) with brackets, as dropping them might lead to sign errors in subtraction.

Question 30. The wages Rs. W of a man earning Rs. x per hour for t hours are given by the formula W = xt. Find his wages for working 40 hours at a rate of Rs. 39.45 per hour.
Answer: Using the given formula: \( W = xt \)
Here, the time \( t = 40 \) hours and the rate \( x = \text{Rs. } 39.45 \) per hour.
Substitute these values into the formula:
\( W = 39.45 \times 40 \)
\( W = \text{Rs. } 1578 \)
In simple words: Multiply the hourly rate by the total hours worked to find the complete wage.

Exam Tip: Always make sure to state the final answer with its correct currency unit, such as Rs.

Question 31. The temperature in Fahrenheit scale is represented by F and the temperature in Celsius scale is represented by C. If F = 9/5 * C + 32, find F when C = 40.
Answer: We substitute \( C = 40 \) into the conversion formula:
\( F = \frac{9}{5}C + 32 \)
\( F = \frac{9}{5} \times 40 + 32 \)
\( F = 9 \times 8 + 32 \)
\( F = 72 + 32 \)
\( F = 104^\circ\text{F} \)
In simple words: Replace C with 40. Divide 40 by 5 to get 8, multiply by 9, and then add 32 to get Fahrenheit.

Exam Tip: Simplify the fraction multiplication first (dividing 40 by 5) before multiplying to keep calculations simple and avoid mistakes.