RS Aggarwal Solutions for Class 6 Chapter 8 Algebraic Expressions

Exercise 8.1

Question 1. Write the following using numbers, literals and signs of basic operations. State what each letter represents:
(i) The diameter of a circle is twice its radius.
(ii) The area of a rectangle is the product of its length and breadth.
(iii) The selling price equals the sum of the cost price and profit.
(iv) The total amount equals the sum of the principal and the interest.
(v) The perimeter of a rectangle is two times the sum of its length and breadth.
(vi) The perimeter of a square is four times its side.
Answer:
(i) Let r and d be the radius and diameter of the circle, respectively.

Therefore, d = 2r

(ii) Let l and b be the length and breadth of the rectangle, respectively.

Therefore, area of rectangle = lb

(iii) Let s, c and p be the selling price, cost price and profit, respectively.

Therefore, s = c + p

(iv) Let T, p and i be the total amount, principal and interest, respectively.

Therefore, T = p + i

(v) Let l and b be the length and breadth of the rectangle, respectively.

Therefore, perimeter of rectangle = 2(l + b)

(vi) Let a be the side of the square.

Therefore, perimeter of the square = 4a

Exam Tip: Clearly state what each variable represents before writing the algebraic expression - examiners expect variable definitions for every problem.

Question 2. Write the following using numbers, literals and signs of basic operations:
(i) The sum of 6 and x.
(ii) 3 more than a number y.
(iii) One-third of a number x.
(iv) One-half of the sum of number x and y.
(v) Number y less than a number 7.
(vi) 7 taken away from x.
(vii) 2 less than the quotient of x by y
(viii) 4 times x taken away from one-thirds of y.
(ix) Quotient of x by 3 is multiples by y.
Answer:
(i) The sum of 6 and x is 6 + x.

(ii) 3 more than a number y means y + 3.

(iii) One-third of a number x is x/3.

(iv) One-half of the sum of numbers x and y is (x + y)/2.

(v) Number y less than a number 7 means 7 - y.

(vi) 7 taken away from x means x - 7.

(vii) 2 less than the quotient of x by y is x/y - 2.

(viii) 4 times x taken away from one-third of y is y/3 - 4x.

(ix) Quotient of x by 3 is multiplied by y means: xy/3

Exam Tip: Pay close attention to the order of operations - "taken away from" means subtraction with the subtracted quantity at the end, not the beginning.

Question 3. Think of a number. Multiply it by 5. Add 5 to the result. Subtract y from this result. What is the result?
Answer: Let the number be x.

When we multiply the number by 5, we get: 5x

When we add 6 to 5x, we get: 5x + 6

Finally, when we subtract y from 5x + 6,

We get: 5x + 6 - y

Exam Tip: Always define the variable first and then perform each step in the order given - this helps avoid algebraic mistakes.

Question 4. The number of rooms on the ground floor of a building is 12 less than the twice of the numbers of rooms on the first floor. If the first floor has x rooms, how many rooms does the ground floor has?
Answer: Let the number of rooms on the ground floor be y.

It is given that the number of rooms on the first floor is x; therefore, we have:

y = 2 × x - 12

= 2x - 12

Thus, the number of rooms on the ground floor is 2x - 12.

Exam Tip: Identify the reference value (first floor rooms = x) and build the expression for the ground floor based on the given relationship "12 less than twice".

Question 5. Binny spends Rs. A daily and saves Rs. B per week. What is her income for 2 weeks?
Answer: It is given that Binny spends Rs. a in one day.

Money spent by him in one week = 7 × a = 7a

It is further given that he saves Rs. b in one week; therefore we have:

Total income in one week = Total expenditure in one week + Total saving in one week

= 7a + b

Therefore, Binny's total income in 2 weeks = 2 X (7a + b)

= Rs. (14a + 2b)

Exam Tip: Income equals expenditure plus savings - make sure to multiply the weekly total by 2 to get the two-week total.

Question 6. Rahul score 80 marks in English and x marks in Hindi. What is his total scores in the two subjects?
Answer: Marks obtained in English = 80

Marks obtained in Hindi = x

Total marks obtained = 80 + x

Exam Tip: When combining known and variable quantities, simply add them in the required order.

Question 7. Rohit covers x centimeters in one step. How much distance does he covers in y steps?
Answer: It is given that Rohit covers x cm in one step.

Therefore, distance covered by him in y steps = x × y = xy cm

Exam Tip: When a quantity is repeated multiple times, multiply the per-unit value by the number of repetitions.

Question 8. One apple weighs 75 grams and one orange weighs 40 grams. Determine the weight of x apples an y oranges.
Answer: Weight of an apple = 75 grams

Weight of an orange = 40 grams

Weight of x apples = 75 X x = 75x grams

Weight of y oranges = 40 X y = 40y grams

Total weight of x apples and y oranges = (75x + 40y) grams

Exam Tip: Multiply each item's unit weight by its quantity, then add them together for the total weight.

Question 9. One pencil costs Rs. 2 and one fountain pen costs Rs. 15. What is the cost of x pencils and y fountain pens?
Answer: Cost of one pencil = Rs. 2

Cost of x pencils = Rs. 2x

Cost of one fountain pen = Rs. 15

Cost of y fountain pens = Rs. 15y

Total cost of x pencils and y fountain pens = Rs. (2x + 15y)

Exam Tip: Always check the unit price and multiply by quantity for each item separately before combining them.

Exercise 8.2

Question 1. Write each of the following products into exponential form:
(i) a × a × a × a × ... 15 times
(ii) 8 × b × b × b × a × a × a × a
(iii) 5 × a × a × a × b × b × c × c × c
(iv) 7 × a × a × a... 8 times × b × b × b × ... 5 times
(v) 4 × a × a × a × ... 5 times × b × b × ... 12 times × c × c × c × ... 15 times
Answer:
(i) a¹⁵

(ii) 8a⁴b³

(iii) 5a³b²c³

(iv) 7a⁸b⁵

(v) 4a⁵b¹²c¹⁵

Exam Tip: Count how many times each factor is multiplied and express it as the exponent - the numerical coefficient stays outside the exponents.

Question 2. Write each of the following in the product form:
(i) a²b⁵
(ii) 8x³
(iii) 7a³b⁴
(iv) 15a⁹b⁸c⁶
(v) 30x⁴y⁴z⁵
(vi) 43p¹⁰q⁵r¹⁵
(vii) 17p¹²q²⁰
Answer:
(i) a × a × b × b × b × b × b

(ii) 8 × x × x × x

(iii) 7 × a × a × a × b × b × b

(iv) 15 × a × a × a × a × a × a × a × a × a × b × b × b × b × b × b × b × b × ... 9 times × b × b × b × ... 8 times

(v) 30 × x × x × x × x × x × y × y × y × y × y × z × z × z × z × z ... 5 times

(vi) 43 × p × p ..... 10 times × q × q ... 5 times × r × r ..... 15 times

(vii) 17 × p × p ... 12 times × q × q ... 20 times

Exam Tip: Expand each exponent by writing the base as many times as the exponent indicates - the numerical coefficient comes first, then the factors in order.

Question 3. Write down each of the following in the exponential form:
(i) 4a³ × 6ab² × c²
(ii) 5xy × 3x²y × 7y²
(iii) a³ × 3ab² × 2a²b²
Answer:
(i) 24a⁴b²c²

(ii) 105x³y⁴

(iii) 6a⁶b⁴

Exam Tip: Multiply the numerical coefficients first, then add the exponents of like bases using the rule a^m × aⁿ = a^m+n.

Question 4. The number of bacteria in a culture is x now. It becomes square of itself after one week. What will be its number after two weeks?
Answer: Present number of bacteria in a culture = x

Number of bacteria in the culture after one week = x²

Number of bacteria in the culture after two weeks = (x²)² = x⁴

Exam Tip: Apply the power rule (a^m)ⁿ = a^mn when dealing with exponential growth over multiple time periods.

Question 5. The area of a rectangle is given by the product of its length and breadth. The length of a rectangle is two-thirds of its breadth. Find its area if its breadth is x cm.
Answer: Breadth of the given rectangle = x cm

Length of the rectangle = (2/3)x cm

Area of the rectangle = (2/3)x × x = (2/3)x² cm²

Exam Tip: When the length is expressed as a fraction of the breadth, multiply both dimensions together carefully to get the area in terms of the given variable.

Question 6. If there are x rows of chairs and each row contains x² chairs. Determine the total numbers of chairs.
Answer: Total number of chairs = Number of rows × Number of chairs in each row

= x × x² = x³

Exam Tip: When finding a total by multiplying quantities with exponents, add the exponents of the same base to simplify.

Exercise 8.3

Question 1. 5 more than twice a number x is written as:
(a) 5 + x + 2
(b) 2x + 5
(c) 2x - 5
(d) 5x + 2
Answer: (b) 2x + 5
In simple words: Twice the number x means 2x. Adding 5 more gives us 2x + 5.

Exam Tip: Parse word problems carefully - "more than" indicates addition, and "twice" means multiply by 2.

Question 2. The quotient of x by 2 is written as:
(a) x/2 + 5
(b) 2/x + 5
(c) (x + 2)/5
(d) x/(2 + 5)
Answer: (a) x/2 + 5
In simple words: Quotient means division, so x divided by 2 is x/2. Adding 5 more gives x/2 + 5.

Exam Tip: The term "quotient" always refers to the result of division - place the dividend first and divisor second.

Question 3. The quotient of x by 3 is multiplied by y is written as:
(a) x/3y
(b) 3x/y
(c) 3y/x
(d) xy/3
Answer: (d) xy/3
In simple words: First divide x by 3 to get x/3. Then multiply this by y to get (x/3) × y = xy/3.

Exam Tip: Follow the order of operations in word problems - perform division first, then multiplication, and arrange the final result in simplified form.

Question 4. 9 taken away from the sum of x and y is
(a) x + y - 9
(b) 9 - (x + y)
(c) (x + y)/9
(d) 9/(x + y)
Answer: (a) x + y - 9
In simple words: The sum of x and y is x + y. When 9 is taken away (subtracted) from it, we get x + y - 9.

Exam Tip: "Taken away from" means subtraction, with the quantity being subtracted placed at the end of the expression.

Question 5. The quotient of x by y added to the product of x and y is written as:
(a) x/y + xy
(b) y/x + xy
(c) (xy + x)/y
(d) (xy + y)/x
Answer: (a) x/y + xy
In simple words: The quotient of x by y is x/y. The product of x and y is xy. When added together, we get x/y + xy.

Exam Tip: Identify each operation separately (quotient and product) before combining them with the final operation (addition).

Question 6. a²b³ × 2ab² is equal to
(a) 2a³b⁴
(b) 2a³b⁵
(c) 2ab
(d) a³b⁵
Answer: (b) 2a³b⁵
In simple words: When you multiply terms with the same variable base, you keep the base and add the exponents together. Here, a² times a gives a³, and b³ times b² gives b⁵, making the product 2a³b⁵.

Exam Tip: Always apply the rule: when multiplying powers with the same base, add the exponents. Double-check your arithmetic with the exponents to avoid errors.

Question 7. 4a²b³ × 3ab² × 5a³b is equal to
(a) 60a³b⁵
(b) 60a⁶b⁵
(c) 60a⁶b⁶
(d) a⁵b⁶
Answer: (c) 60a⁶b⁶
In simple words: First, multiply the numerical coefficients: 4 × 3 × 5 = 60. Then combine the powers of a by adding the exponents: 2 + 1 + 3 = 6. For b, add the exponents: 3 + 2 + 1 = 6. The result is 60a⁶b⁶.

Exam Tip: Organize your work by separating coefficients from variables. Add exponents carefully for each variable separately to avoid mixing them up.

Question 8. If 2x²y and 3xy² denote the length and breadth of a rectangle, then its area is
(a) 6xy
(b) 6x²y²
(c) 6x³y³
(d) x³y³
Answer: (c) 6x³y³
In simple words: The area of a rectangle is found by multiplying length and breadth together. Multiplying 2x²y by 3xy² gives 6x³y³.

Exam Tip: Remember that area = length × breadth. Always multiply the entire expressions, including both coefficients and variables with their exponents.

Question 9. In a room there are x² rows of chairs and each rows contains 2x² chairs. The total number of chairs in the room is
(a) 2x²
(b) 2x⁴
(c) x⁴
(d) \( \frac{x^4}{4} \)
Answer: (b) 2x⁴
In simple words: To find the total number of chairs, multiply the number of rows by the number of chairs in each row. This gives x² × 2x², which equals 2x⁴.

Exam Tip: Read word problems carefully to identify what you're multiplying. Here, it's rows × chairs per row. Apply exponent rules correctly when combining like bases.

Question 10. a³ × 2a²b × 3ab⁵ is equal to
(a) a⁶b⁶
(b) 23a⁶b⁶
(c) 6a⁶b⁶
(d) None of these
Answer: (c) 6a⁶b⁶
In simple words: Multiply the coefficients: 1 × 2 × 3 = 6. For a, add the exponents: 3 + 2 + 1 = 6, giving a⁶. For b, add the exponents: 0 + 0 + 5 = 5, but b appears only in the last term, so you get b⁵ total. Wait — rechecking: a³ has no b, 2a²b has b¹, and 3ab⁵ has b⁵, so b¹ × b⁵ = b⁶. The answer is 6a⁶b⁶.

Exam Tip: Keep track of which variables appear in each term and add exponents only for matching variables. A systematic layout helps prevent arithmetic mistakes.