If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle?
Answer: Width = 3 cm, length = 9 cm.
Step-by-step solution
Given: Length = 2 × width + 3; Perimeter = 24 cm
To find: The length and the width
To find: The length and the width
Idea: “Three more than twice the width” means 2w + 3. Perimeter of a rectangle = 2(length + width).
- Let the width be w cm. Then the length is (2w + 3) cm.½ mark
- Perimeter: 2[(2w + 3) + w] = 24, so 2(3w + 3) = 24.1 mark
- 3w + 3 = 12, so 3w = 9 and w = 3.1 mark
- Width = 3 cm; length = 2 × 3 + 3 = 9 cm.½ mark
The rectangle is 9 cm long and 3 cm wide.
Check: 2(9 + 3) = 24 ✓ and 9 = 2 × 3 + 3 ✓.
Answer to write in the exam
Let width = w cm; length = (2w + 3) cm
2[(2w + 3) + w] = 24
3w + 3 = 12 ⇒ 3w = 9 ⇒ w = 3
Length = 2(3) + 3 = 9
∴ Length = 9 cm, width = 3 cm
Common mistakes that cost marks
- Writing the length as 3(2w) or 2(w + 3). “Three more than twice the width” is 2w + 3.
- Using perimeter = length + width = 24. The perimeter has two lengths and two widths.
- Giving only w = 3 and forgetting to find the length.
How this can come in the exam
MCQ (1 mark)
The length of a rectangle is 4 cm more than its width and its perimeter is 40 cm. Its width is
- 8 cm
- 12 cm
- 18 cm
- 9 cm
Show answer
(A) 8 cm
2(w + 4 + w) = 40, so 2w + 4 = 20 and w = 8 cm.
Short answer (3 marks)
The length of a rectangular field is 5 m less than three times its width. The perimeter is 150 m. Find the dimensions.
Show answer
Length = 3w − 5 (½ mark). 2(3w − 5 + w) = 150, so 4w − 5 = 75 (1 mark). w = 20 m (1 mark), length = 55 m (½ mark).Try one yourself
The length of a rectangle is one more than three times its width, and its perimeter is 42 cm. Find its dimensions.
Show answer
2(3w + 1 + w) = 42, so 4w + 1 = 21 and w = 5. Width 5 cm, length 16 cm.
More questions like this
- Observe the following growing pattern of square tiles.
Predict the number of squares in the next three stages of the pattern and write the sequence of numbers up to Stage 7 of the pattern. - This leads us to conclude that the number of squares at Stage n is given by 2n − 1.
Using the expression 2n − 1, can you find out how many tiles will be there in the 15th stage and the 26th stage of the pattern? Also, which stage will contain 21 tiles and 47 tiles? - Bela has ₹100 for pocket money. She spends ₹5 every day. After how many days will she be left with ₹40?
- Bela has ₹100 for pocket money. She spends ₹5 every day. Observe that the amount left on the nth day will be ₹(100 − 5n).
What amount will be left on the 15th day? How many days will it take for the entire amount to be spent? - An auto-rikshaw fare starts at ₹25 and remains the same for the initial 2 km. Then it increases by ₹15 per km. What will be the fare for a travel of 10 km?