Algebraic Expressions and Identities Worksheet

        Fill in the blanks Type Questions

1. Ansh plants two saplings, which will grow into a tree and records their growth annually. Growth of the first tree is represented by the expression 5x² – 4y² + 6y – 3 and that of the second tree is represented as 7x² – 4xy + 8y² + 5x.
   
   To find the average growth, Ansh must first find their sum. So, the total growth is ____________________.
   
   
2. A school has four different divisions i.e. A, B, C & D for each grade. Here, division A has only girls, B has only boys, C has both boys & girls and D has weak students. In grade 8, the number of  students in each division is represented in the form of an expression and it is as given below:
   Division A: 4xy
   Division B: 5x²y²
   Division C: 6x³y³
   Division D: 5y
   Product of number of students in these divisions is ___________________.
   
   
3. Seasonal flu is typical during a monsoon transition. Because Karan has the seasonal flu, his mother measures his temperature in the morning and again in the evening after he has taken his medication. The temperatures measured were as follows:
   Temperature in the morning: (a + 3b)
   Temperature in the evening: (a – 2b)
   After multiplying these temperature values, we get the temperature equal to ___________.
   
   
4. There are 3 different flavours of ice cream available in a shop. 
   
   The consumption of these flavours on the 1st day are as follows: 1.5x, 4y, 3z. This trend is followed for the initial ‘4y’ days. After a gap of few days, the trend repeated again for the next ‘2x’ days. The number of ice creams sold solely in accordance with this trend is ____________.
   
   Multiple Choice Questions
   
   
5. Four cars started from the same point and ends the journey after reaching the same destination. The cars are not necessarily running at the same speed.
   The speed of the cars are given by the expressions as follows:
   
   Car 1: 7a4, Car 2: 6m² – 13m, Car 3: 3pq (p – q), Car 4: y3 – 4y² + 5
   Which of the following car(s)’s speed represent a binomial?
   (a)Car 1 only
   (b)Car 2 only
   (c)Both Car 2 & Car 3
   (d)Both Car 3 & Car 4
   
   
6. Every day, Beena saves more than Rs. 20 of her pocket money. She puts Rs. 10 in the little piggy bank and the rest in the bigger one. 
   
   When she breaks the piggy banks after a few months and calculates the money she has amassed, the amount is much larger than she had anticipated. Money accumulated in the little piggy bank is calculated as 2p²q² – 3pq + 4 and in the other as 5 + 7pq + 3p²q², respectively. How much more amount would be needed in the little piggy bank to make the amounts equal?
   (a)5p²q² – 10pq – 1
   (b)–p²q² + 7pq + 9
   (c)p²q² + 10pq + 1
   (d)p²q² – 7pq – 9
   
   
7. There are two roots available for John to go to school from his house. Route A is shorter but crowded. On the other hand, route B is comparatively longer but less crowded. John tried both the routes and represented the time taken by him to reach the school in the form of an algebraic expression. Time taken by route A is 3x2 and that by route B is 6x. Product of time taken by these routes is
   (a)3x² + 6x
   (b)½ x
   (c)3x²
   (d)18x³
   
   
8. A bookshelf has 3 sections in it and each section has certain number of books.
   
   Ramesh used three variables p, q & r and wrote the number of books in the three sections as 7p, 10q and 4qr. When he multiplied the number of books present in each section, the expression which he gets is equal to
   (a)280pq²r
   (b)7p + 10q + 4qr
   (c)21pqr
   (d)40p²qr
   
   
9. Lucky is refilling the coffee beans in the coffee machine. 
    
   He puts 30 beans at once and does this for ‘x’ number of times. So the total number of beans is 30x. On the next day, he puts 5y beans in the morning and 15xy beans in the evening. What is the product of number of beans filled within 2 days?
   (a)30x(5y + 15xy)
   (b)30x(5y – 15xy)
   (c)225x²y²
   (d)30x + 5y + 15xy
   
   
10. As it has started raining in the city, Kiara looks for an umbrella. She visits a store nearby and asks for an umbrella. According to the shopkeeper, there are ‘3p’ bunches of umbrellas available in the store, each bunch having 3 umbrellas of the following size: 4p², 5p and 7. What is the total size when all the umbrellas are added?
    (a)4p² + 8p + 7
    (b)3p(4p² + 5p + 7)
    (c)420p³
    (d)3p(4p² – 5p – 7)
    
    
11. Rustom runs a pani puri stall which is setup in the market. He has both fixed and varying types of customers.
    
    The number of pani puri consumed by the fixed customers in ‘x’ hours is given by 3x(4x – 5) + 3. How many pani puri did the fixed customers consume in 3 hours?
    (a)66
    (b)21
    (c)10
    (d)5
    
    
12. Suman’s mother used to buy two different kinds of fruits in one combined bucket on the way back from the office, giving the total as (x – y) pieces of fruits. She does this for 3x days and then for the following 5y days. In total, how many pieces of fruits did she purchase?
    (a)4x + 4y
    (b)2x + 6y
    (c)(x – y)(3x + 5y)
    (d)3x + 5y
    
    
13. There is a fixed schedule followed by Krish on weekdays (Monday – Saturday), which is given below:
    
    After 8:00 PM, he eats dinner and go to bed. However, he watched TV for 4 hours on Sunday, and his total watch time for the week is recorded as (t + s²). What is the total amount of time spent by Krish on watching TV if this routine is followed for (t² – s + 4) weeks?
    (a)t – s + s² + t² + 4
    (b)(t + s²) (t² – s + 4)
    (c)t² – s² – s – t + 4
    (d)0
    
    Subjective Questions
    
    
14. Chetana uses a weather app before leaving for any trip. She has planned a trip for next 4 days and thus checks the weather for the coming 4 days. 
    
    Chetana used expressions to write the temperatures recorded for the next 4 days so that other can’t decode it. The temperatures are as follows: 3x², 4xy + 7, –3xz + 5x – 2xy and ab – bc. Classify these temperatures as monomial, binomial and trinomial.
    
    
15. A shop has three types of balls: football, cricket ball and TT ball. All of them are of different sizes and their prices can be represented as follows:
    Football: 20x³, Cricket ball: 15xy, TT ball: 2x.
     
    What is the product of the prices of the football and cricket ball?
    
    
16. Latika purchased few chocolates of two distinct flavours, each flavour having equal number of chocolates. She then gave 9 chocolates to her younger brother Samar and is now left with (a² – 9) chocolates. How many chocolates would Latika have in the end if she repeats this (buy chocolates and give 9 to Samar) for '4a' days?
    
    
17. There are 3 cloth shops located in the same lane of the market. A survey has been done and the monthly earnings for these shops has been recorded. The following expressions 6p2, 7pq, 19p are used to indicate the earnings so that others can’t find it easily. What was the total income over the past 'p' months?
    
    
18. Firoz has got an assignment of writing an essay of 1000 words. He wrote it so that when ‘a’ rows are finished, he completes a(a² + a + 1) + 5 words. How many words will he complete writing after 20 rows?
    
    
19. Ananya loves to eat pasta, so she asks her mother to prepare some for her. Her mother instructs Ananya to take out some uncooked pasta. She fills a bowl with raw pasta and counts the number of pieces as (2x + 5). If she repeats this (4x – 3) times, how many pieces of pasta will there be in total?
    
    
20. There are three pools in a society: one for adults, one for youngsters of ordinary size, and one for young children. The areas covered by these pools are represented as x², xy and y² respectively. The same plan is copied by ‘x’ societies in Mumbai and ‘y’ societies in Pune. How much area will be required both in Mumbai and Pune together to execute this plan?

        Fill in the blanks Type Questions

1. Ansh plants two saplings, which will grow into a tree and records their growth annually. Growth of the first tree is represented by the expression 5x² – 4y² + 6y – 3 and that of the second tree is represented as 7x² – 4xy + 8y² + 5x.
   
   To find the average growth, Ansh must first find their sum. So, the total growth is ____________________.
   Correct Answer: 12x² + 4y² + 6y – 4xy + 5x – 3
   Solution:
   Total growth is same as the sum of the growths of the two trees.
   Total growth = (5x² – 4y² + 6y – 3) + (7x² – 4xy + 8y² + 5x)
    
   To find the average growth, Ansh must first find their sum. So, the total growth is 12x2 + 4y2 + 6y – 4xy + 5x – 3.
   
   
2. A school has four different divisions i.e. A, B, C & D for each grade. Here, division A has only girls, B has only boys, C has both boys & girls and D has weak students. In grade 8, the number of  students in each division is represented in the form of an expression and it is as given below:
   Division A: 4xy
   Division B: 5x²y²
   Division C: 6x³y³
   Division D: 5y
   Product of number of students in these divisions is ___________________.
   Correct Answer: 600 x6y7
   Solution:
   Product of number of students in each division = (4xy) × (5x²y²) × (6x³y³) × (5y)
   = 600 x6y⁷
   Thus, the product of number of students in these divisions is 600 x6y⁷.
   
   
3. Seasonal flu is typical during a monsoon transition. Because Karan has the seasonal flu, his mother measures his temperature in the morning and again in the evening after he has taken his medication. The temperatures measured were as follows:
   Temperature in the morning: (a + 3b)
   Temperature in the evening: (a – 2b)
   After multiplying these temperature values, we get the temperature equal to ___________.
   Correct Answer: a² + ab – 6b²
   Solution:
   Product of temperature = (a + 3b) × (a – 2b)
   = a(a – 2b) + 3b(a – 2b)
   = a² – 2ab + 3ab – 6b²
   = a² + ab – 6b²
   Thus, after multiplying these temperature values, we get the temperature equal to a² + ab – 6b².
   
   
4. There are 3 different flavours of ice cream available in a shop. 
   
   The consumption of these flavours on the 1st day are as follows: 1.5x, 4y, 3z. This trend is followed for the initial ‘4y’ days. After a gap of few days, the trend repeated again for the next ‘2x’ days. The number of ice creams sold solely in accordance with this trend is ____________.
   Correct Answer: 3x² + 16y² + 14xy + 12yz + 6xz
   Solution:
   Consumption of ice creams in 1 day = 1.5x + 4y + 3z
   Consumption of ice creams in the initial “4y” days = 4y (1.5x + 4y + 3z)
   As the trend repeated for 2x days.
   Total consumption = (4y + 2x) × (1.5x + 4y + 3z)
   = 4y (1.5x + 4y + 3z) + 2x (1.5x + 4y + 3z)
   = 6xy + 16y² + 12yz + 3x² + 8xy + 6xz
   = 3x² + 16y² + 14xy + 12yz + 6xz
   So, the number of ice creams sold solely in accordance with this trend is 3x² + 16y² + 14xy + 12yz + 6xz.
   
   Multiple Choice Questions
   
   
5. Four cars started from the same point and ends the journey after reaching the same destination. The cars are not necessarily running at the same speed.
   The speed of the cars are given by the expressions as follows:
   
   Car 1: 7a4, Car 2: 6m² – 13m, Car 3: 3pq (p – q), Car 4: y3 – 4y² + 5
   Which of the following car(s)’s speed represent a binomial?
   (a)Car 1 only
   (b)Car 2 only
   (c)Both Car 2 & Car 3
   (d)Both Car 3 & Car 4
   Correct Option: (c) Both Car 2 & Car 3
   Solution:
   A binomial has only two terms.
   Now, 3pq (p – q) = 3p²q – 3pq²
   So, 6m² – 13m and 3pq (p – q) has only two terms.
   Thus, speeds of both Car 2 & Car 3 represents binomials.
   
   
6. Every day, Beena saves more than Rs. 20 of her pocket money. She puts Rs. 10 in the little piggy bank and the rest in the bigger one. 
   
   When she breaks the piggy banks after a few months and calculates the money she has amassed, the amount is much larger than she had anticipated. Money accumulated in the little piggy bank is calculated as 2p²q² – 3pq + 4 and in the other as 5 + 7pq + 3p²q², respectively. How much more amount would be needed in the little piggy bank to make the amounts equal?
   (a)5p²q² – 10pq – 1
   (b)–p²q² + 7pq + 9
   (c)p²q² + 10pq + 1
   (d)p²q² – 7pq – 9
   Correct Option: (c) p²q² + 10pq + 1
   Solution:
   Money accumulated in the little piggy bank is calculated as (2p²q² – 3pq + 4)
   Money accumulated in the other piggy bank is (5 + 7pq + 3p²q²)
   According to the situation given, little piggy bank has less money than the other.
   Amount of money needed more in the little piggy bank
   = (5 + 7pq + 3p²q²) – (2p²q² – 3pq + 4)
   = 5 + 7pq + 3p²q² – 2p²q² + 3pq – 4
   = 3p²q² – 2p²q2 + 7pq + 3pq + 5 – 4
   = p²q² + 10pq + 1
   
   
7. There are two roots available for John to go to school from his house. Route A is shorter but crowded. On the other hand, route B is comparatively longer but less crowded. John tried both the routes and represented the time taken by him to reach the school in the form of an algebraic expression. Time taken by route A is 3x2 and that by route B is 6x. Product of time taken by these routes is
   (a)3x² + 6x
   (b)½ x
   (c)3x²
   (d)18x³
   Correct Option: (d) 18x³
   Solution:
   Product of time taken by the two routes = 3x² × 6x = 18x³
   
   
8. A bookshelf has 3 sections in it and each section has certain number of books.
   
   Ramesh used three variables p, q & r and wrote the number of books in the three sections as 7p, 10q and 4qr. When he multiplied the number of books present in each section, the expression which he gets is equal to
   (a)280pq²r
   (b)7p + 10q + 4qr
   (c)21pqr
   (d)40p²qr
   Correct Option: (a) 280 pq²r
   Solution:
   Product of number of books present in each section = 7p × 10q × 4qr
   = 280 pq²r
   
   
9. Lucky is refilling the coffee beans in the coffee machine. 
    
   He puts 30 beans at once and does this for ‘x’ number of times. So the total number of beans is 30x. On the next day, he puts 5y beans in the morning and 15xy beans in the evening. What is the product of number of beans filled within 2 days?
   (a)30x(5y + 15xy)
   (b)30x(5y – 15xy)
   (c)225x²y²
   (d)30x + 5y + 15xy
   Correct Option: (a) 30x(5y + 15xy)
   Solution:
   Number of beans filled on 1st day = 30x
   Number of beans filled on the next day = 5y + 15xy
   Product of number of beans filled within 2 days = 30x(5y + 15xy)
   
   
10. As it has started raining in the city, Kiara looks for an umbrella. She visits a store nearby and asks for an umbrella. According to the shopkeeper, there are ‘3p’ bunches of umbrellas available in the store, each bunch having 3 umbrellas of the following size: 4p², 5p and 7. What is the total size when all the umbrellas are added?
    (a)4p² + 8p + 7
    (b)3p(4p² + 5p + 7)
    (c)420p³
    (d)3p(4p² – 5p – 7)
    Correct Option: (b) 3p(4p² + 5p + 7)
    Solution:
    Size of umbrella in 1 bunch = 4p² + 5p + 7
    As there are 3p bunches of umbrellas.
    ∴ Total size when we add all the umbrellas (in 3p bunches) = 3p(4p² + 5p + 7)
    
    
11. Rustom runs a pani puri stall which is setup in the market. He has both fixed and varying types of customers.
    
    The number of pani puri consumed by the fixed customers in ‘x’ hours is given by 3x(4x – 5) + 3. How many pani puri did the fixed customers consume in 3 hours?
    (a) 66
    (b) 21
    (c) 10
    (d) 5
    Correct Option: (a) 66
    Solution:
    Number of pani puri consumed by fixed customers in ‘x’ hours = 3x(4x – 5) + 3
    When x = 3, we get the number of pani puri consumed by fixed customers in 3 hours.
    At x = 3,
    3x(4x – 5) + 3 = 3×3(4×3 – 5) + 3
    = 9(12 – 5) + 3
    = 9 × 7 + 3
    = 63 + 3
    = 66
    
    
12. Suman’s mother used to buy two different kinds of fruits in one combined bucket on the way back from the office, giving the total as (x – y) pieces of fruits. She does this for 3x days and then for the following 5y days. In total, how many pieces of fruits did she purchase?
    (a)4x + 4y
    (b)2x + 6y
    (c)(x – y)(3x + 5y)
    (d)3x + 5y
    Correct Option: (c) (x – y) (3x + 5y)
    Solution:
    Number of pieces of fruits bought in one day = (x – y)
    She bought same number of fruits for total (3x + 5y) days.
    So, the total number of pieces of fruits purchased = (x – y) (3x  +5y)
    
    
13. There is a fixed schedule followed by Krish on weekdays (Monday – Saturday), which is given below:
    
    After 8:00 PM, he eats dinner and go to bed. However, he watched TV for 4 hours on Sunday, and his total watch time for the week is recorded as (t + s²). What is the total amount of time spent by Krish on watching TV if this routine is followed for (t² – s + 4) weeks?
    (a) t – s + s² + t² + 4
    (b) (t + s²) (t² – s + 4)
    (c) t² – s² – s – t + 4
    (d) 0
    Correct Option: (b) (t + s²) (t² – s + 4)
    Solution:
    Amount of time spent by Krish on watching TV in 1 week = (t + s²)
    So, the total amount of time spent by Krish on watching TV = (t + s²) (t² – s + 4)
    
    Subjective Questions
    
    
14. Chetana uses a weather app before leaving for any trip. She has planned a trip for next 4 days and thus checks the weather for the coming 4 days. 
    
    Chetana used expressions to write the temperatures recorded for the next 4 days so that other can’t decode it. The temperatures are as follows: 3x², 4xy + 7, –3xz + 5x – 2xy and ab – bc. Classify these temperatures as monomial, binomial and trinomial.
    Solution:
    Here is the classification of the temperatures.
    Monomial: 3x²
    Binomial: 4xy + 7, ab – bc
    Trinomial: –3xz + 5x – 2xy
    
    
15. A shop has three types of balls: football, cricket ball and TT ball. All of them are of different sizes and their prices can be represented as follows:
    Football: 20x³, Cricket ball: 15xy, TT ball: 2x.
     
    What is the product of the prices of the football and cricket ball?
    Solution:
    Product of the prices of the football and cricket ball = 20x³ × 15xy = 300x⁴y
    
    
16. Latika purchased few chocolates of two distinct flavours, each flavour having equal number of chocolates. She then gave 9 chocolates to her younger brother Samar and is now left with (a² – 9) chocolates. How many chocolates would Latika have in the end if she repeats this (buy chocolates and give 9 to Samar) for '4a' days?
    Solution:
    On the 1^st day, Latika is left with (a² – 9) chocolates.
    This is repeated for 4a days.
    The number of cho.colates that Latika will have after 4a days
    = 4a (a² – 9)
    = 4a³ – 36a
    
    
17. There are 3 cloth shops located in the same lane of the market. A survey has been done and the monthly earnings for these shops has been recorded. The following expressions 6p2, 7pq, 19p are used to indicate the earnings so that others can’t find it easily. What was the total income over the past 'p' months?
    Solution:
    Here is the classification of the temperatures.
    Monthly earnings for all three shops = 6p² + 7pq + 19p
    Total income over the past 'p' months = p (6p² + 7pq + 19p)
    = 6p³ + 7p²q + 19p²
    
    
18. Firoz has got an assignment of writing an essay of 1000 words. He wrote it so that when ‘a’ rows are finished, he completes a(a² + a + 1) + 5 words. How many words will he complete writing after 20 rows?
    Solution:
    Number of words completed after ‘a’ rows = a(a² + a + 1) + 5
    So, the number of words completed after 20 rows is when a = 20.
    At a = 20
    a(a² + a + 1)
    = 20(400 + 20 + 1)
    = 20 × 421
    = 8420
    
    
19. Ananya loves to eat pasta, so she asks her mother to prepare some for her. Her mother instructs Ananya to take out some uncooked pasta. She fills a bowl with raw pasta and counts the number of pieces as (2x + 5). If she repeats this (4x – 3) times, how many pieces of pasta will there be in total?
    Solution:
    Number of pieces of pasta in a bowl = 2x + 5
    The process is repeated (4x – 3) times.
    ∴ Total number of pieces of pasta = (2x + 5)(4x – 3)
        = 2x (4x – 3) + 5(4x – 3)
        = 8x² – 6x + 20x – 15
       = 8x² + 14x – 15
    
    
20. There are three pools in a society: one for adults, one for youngsters of ordinary size, and one for young children. The areas covered by these pools are represented as x², xy and y² respectively. The same plan is copied by ‘x’ societies in Mumbai and ‘y’ societies in Pune. How much area will be required both in Mumbai and Pune together to execute this plan?
    Solution:
    Total area of pool occupied by one society = x² + xy + y²
    Plan copied by total (x + y) societies.
    The area required in Mumbai & Pune together to execute this plan = (x + y) (x² + xy + y²)
    = x(x² + xy + y²) + y(x² + xy + y²)
    = x³ + x²y + xy² + x²y + xy² + y³
    = x³ + 2x²y + 2xy² + y³