For a rectangle whose length is 3 cm greater than its width, the perimeter is 4w + 6, where w is the width in centimetres. If the perimeter must be at least 30 cm and less than 45 cm, the possible whole-number widths are 6, 7, 8 and 9 cm.
Translate each limit before solving. ‘At least 30 cm’ includes 30, while ‘less than 45 cm’ excludes 45. The question asks for whole-number widths, so solve for the full interval first and then list the integers it contains.
- Write the length in terms of the width.
- Translate the lower and upper limits separately.
- Substitute the first and last valid widths to check them.
CHAPTER 1 OF 4 · Back to contents
Build the expression for perimeter
Let the width be w cm. The length is w + 3 cm. A rectangle has two widths and two lengths, so its perimeter is 2w + 2(w + 3) = 4w + 6 cm.
Test the expression with w = 5. The rectangle is 5 cm by 8 cm. Walking around its edges gives 5 + 8 + 5 + 8 = 26 cm, which agrees with 4(5) + 6.
CHAPTER 2 OF 4 · Back to contents
Worked example | Include one limit and exclude the other
Write 30 ≤ 4w + 6 < 45. Subtract 6 from all three parts to get 24 ≤ 4w < 39. Divide all three parts by 4 to obtain 6 ≤ w < 9.75.
The whole-number widths are 6, 7, 8 and 9 cm. At 6 cm the perimeter is 30 cm, so it qualifies. At 9 cm it is 42 cm. The next width, 10 cm, gives 46 cm and exceeds the limit.
The interval 6 ≤ w < 9.75 describes all real widths that meet the perimeter condition. The integer list answers the question about whole centimetres.
CHAPTER 3 OF 4 · Back to contents
Practise with an exact upper boundary
Keep the same relationship between the sides. The perimeter must now be more than 30 cm and at most 42 cm. Solve 30 < 4w + 6 ≤ 42 to get 6 < w ≤ 9. The widths are 7, 8 and 9 cm.
Compare this with the first question. A width of 6 cm gives exactly 30 cm. It satisfied the first lower limit and fails the new lower limit.
CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Does dividing by four reverse the signs?
Dividing by a positive number preserves both inequality signs. Dividing by a negative number reverses them.
Should I round 9.75 to 10?
List the whole numbers inside the interval. Ten lies outside it.
How do I draw 6 ≤ w < 9.75?
Use a filled dot at 6 and an open dot at 9.75, with the segment between them shaded. If the diagram asks for integer solutions, mark the four integer points.
Contents · Official syllabus and next practice
Official syllabus and next practice
Read the official SEAB 2027 syllabus
Try finding the corresponding four possible lengths and checking the two smallest perimeters.