A rectangular garden of length l metres and width w metres has to be fenced and decorated. A wire fence is to be laid along the length costing ₹100 per metre and a wooden fence is to be built along the width costing ₹80 per metre. Special seeds have to be sown throughout the garden which will cost ₹50 per square metre. What will be the total cost incurred?
Step-by-step solution
To find: The total cost as an algebraic expression
Idea: A rectangle has two lengths and two widths, so each fence covers 2l or 2w metres. The seeds cover the area, l × w. Cost = quantity × rate for each part; then add.
- Wire fence along both lengths: 2l metres. Cost = 2l × 100 = ₹200l.1 mark
- Wooden fence along both widths: 2w metres. Cost = 2w × 80 = ₹160w.½ mark
- Seeds depend on the area. Area = l × w = lw square metres. Cost = 50 × l × w = ₹50lw.1 mark
- Total cost = ₹(200l + 160w + 50lw).½ mark
Check: Take a garden 10 m by 6 m. Wire: 20 m × ₹100 = ₹2000. Wood: 12 m × ₹80 = ₹960. Seeds: 60 m2 × ₹50 = ₹3000. Total ₹5960. The expression: 200(10) + 160(6) + 50(10)(6) = 2000 + 960 + 3000 = ₹5960 ✓.
Answer to write in the exam
Cost of wire fencing = 2l × 100 = ₹200l
Cost of wooden fencing = 2w × 80 = ₹160w
Area of garden = lw m2; cost of seeds = 50 × l × w = ₹50lw
∴ Total cost = ₹(200l + 160w + 50lw)
Common mistakes that cost marks
- Using l instead of 2l for the fence. A rectangle has two lengths, so the wire fence is 2l metres long, not l.
- Using the perimeter for the seeds. Seeds are sown over the whole garden, so we need the area lw, not 2(l + w).
- Adding 200l and 160w to get 360lw. They are unlike terms and must stay separate.
How this can come in the exam
A rectangular park is a m long and b m wide. Grass costs ₹40 per square metre. The cost of grassing the park is
- ₹40(a + b)
- ₹80(a + b)
- ₹40ab
- ₹(40 + ab)
Show answer
(C) ₹40ab
Grass covers the area ab m2, and each square metre costs ₹40, so the cost is ₹40ab.
A school plans a rectangular vegetable patch p m long and q m wide. A bamboo fence along the two lengths costs ₹60 per metre, a net along the two widths costs ₹45 per metre, and manure costs ₹20 per square metre.
(i) Write the cost of the bamboo fence. (ii) Write the cost of the net. (iii) Write the total cost as an expression. (iv) Find the total cost when p = 8 and q = 5.
Show answer
(i) 2p × 60 = ₹120p (1 mark). (ii) 2q × 45 = ₹90q (1 mark). (iii) Manure 20pq, so total ₹(120p + 90q + 20pq) (1 mark). (iv) 960 + 450 + 800 = ₹2210 (1 mark).Try one yourself
A rectangular hall is x m long and y m wide. Skirting along all four walls costs ₹150 per metre and flooring costs ₹400 per square metre. Write the total cost.
Show answer
Skirting: 2(x + y) × 150 = 300x + 300y. Flooring: 400xy. Total ₹(300x + 300y + 400xy).
More questions like this
- Thus, 200l + 160w + 50lw is the algebraic expression for the total cost.
1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. How is it different from the algebraic expression 4x + 5y + 3 for Raju’s pens and pencils? - A wire of length 20 cm is bent in different ways to form rectangles. For example, we can have a rectangle with length 7 cm and width 3 cm. We can also have one of length 5.5 cm and width 4.5 cm. (Think of a few more ways of forming such rectangles.) Can you write an expression for the area of such rectangles?
- The expression for the area of these rectangles is x(10 − x) or 10x − x2.
1. Can you identify the terms, variables and coefficients of this algebraic expression?
2. Can you point out any similarity or difference between the algebraic expressions obtained for Raju’s pens and pencils (4x + 5y + 3) and for these rectangles? - Find the degrees of the following polynomials:
- Write polynomials of degrees 1, 2 and 3.