Word Problem: We Distribute, Yet Things Multiply | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

Q1: Rohan’s mother gave him ₹ 3xy² and his father gave him ₹ 5(xy² + 2). Out of this total money he spent ₹ (10 – 3xy²) on his birthday party. How much money is left with him? 
Solution:
Money give by Rohan’s mother = ₹ 3xy²
Money given by his father = ₹ 5(xy² + 2)
Total money given to him = ₹ 3xy² + ₹ 5 (xy² + 2)
= ₹ [3xy² + 5(xy² + 2)]
= ₹ (3xy² + 5xy² + 10)
= ₹ (8xy² + 10).
Money spent by him = ₹ (10 – 3xy)²
Money left with him = ₹ (8xy² + 10) – ₹ (10 – 3xy²)
= ₹ (8xy² + 10 – 10 + 3x²y)
= ₹ (11xy²)
Hence, the required money = ₹ 11xy²​

Q2: The side of a square plot is (a - b) meters. What is the area of the square in terms of a and b?

Solution:
Using the identity (a - b)² = a² - 2ab + b², we can calculate the area of the square:
Area = (a - b)² = a² - 2ab + b²

Thus, the area of the square is a² - 2ab + b².

Q3: A school buys (3x + 5) books of English and (2x – 4) books of Math. If each book costs ₹20, what is the total cost?

Solution:
Total books = (3x + 5) + (2x – 4) = 5x + 1
Total cost = 20(5x + 1) 
= 100x + 20

Q4: Solve (99)² using algebraic identity.

Solution: 

​We can write, 99 = 100 -1

Therefore, (100 – 1 )²

= 100² + 1² – 2 x 100 x 1  [By identity: (a -b)² = a² + b² – 2ab

= 10000 + 1 – 200

= 9801

Q5: A digital marketing agency is creating ads for different businesses. Each ad costs 2x + 8 dollars. If the agency creates ads for 30 businesses, how much will the agency earn in total?

Solution:
Using the distributive property, the total earnings of the agency is:
30 × (2x + 8) 

= 30 × 2x + 30 × 8 

= 60x + 240
Thus, the agency will earn 60x + 240 dollars in total.

Q7: A rectangular garden has a length of (a + b) meters and a width of (a + b) meters. What is the area of the garden in terms of a and b?

Solution:
Using the identity (a + b)² = a² + 2ab + b², we can find the area of the garden:
Area = (a + b)² = a² + 2ab + b²

Thus, the area of the garden is a² + 2ab + b².

Q8: A farmer buys 5 crates of apples, and each crate contains 2x + 3 apples. How many apples does the farmer have in total?

Solution:
Using the distributive property, the total number of apples is:
5 × (2x + 3) 

= 5 × 2x + 5 × 3 

= 10x + 15
Therefore, the farmer has 10x + 15 apples in total.

Q9: The sum of two numbers is (a + b), and the difference is (a - b). What is the difference between the square of the sum and the square of the difference in terms of a and b?

Solution:
Using the identities:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

The difference between the squares is:
(a + b)² - (a - b)² = (a² + 2ab + b²) - (a² - 2ab + b²)
Simplifying the expression:
= a² + 2ab + b² - a² + 2ab - b² = 4ab

Thus, the difference between the square of the sum and the square of the difference is 4ab.

Q10: A rectangular piece of land has a length of (a + b) meters and a width of (a - b) meters. What is the area of the land in terms of a and b?

Solution:
The area of the rectangle is the product of the length and the width:
Area = (a + b)(a - b)
Using the identity:
(a + b)(a - b) = a² - b²

Thus, the area of the land is a² - b².

Q11: A person buys (x + 2) pencils at ₹5 each and (x – 1) pens at ₹10 each. Find the total cost.

Solution:
Cost of pencils = 5(x + 2) = 5x + 10
Cost of pens = 10(x – 1) = 10x – 10
Total cost = (5x + 10) + (10x – 10) = 15x

Q12: The ticket price for adults is (2x + 5) and for children is (x + 3). If 30 adults and 20 children visit a park, find the total money collected.

Solution:
Money from adults = 30(2x + 5) = 60x + 150
Money from children = 20(x + 3) = 20x + 60
Total = (60x + 150) + (20x + 60) = 80x + 210

Q13: A factory produces (3x + 4) toys every day for 15 days, and each toy costs ₹(2x + 1) to make. Find the total production cost.

Solution:
Toys in 15 days = 15(3x + 4) = 45x + 60
Cost per toy = (2x + 1)
Total cost = (45x + 60)(2x + 1)
= 45x(2x + 1) + 60(2x + 1)
= 90x² + 45x + 120x + 60
= 90x² + 165x + 60