Syafi has decided to cut off 15% of the length of his hair today and will continue tocut off 15% every 2 months thereafter. Knowing that his hair is presently 15centimetres long and that it will grow about 8 centimetres in 2 months, how long willhis hair be after his haircut one year from now?
The Correct Answer and Explanation is:
To solve this, we need to model Syafi’s hair length every two months, accounting for 8 cm growth and then a 15% haircut.
Step-by-Step Breakdown:
- Initial hair length: 15 cm
- Every 2 months:
- Hair grows 8 cm
- Then, 15% is cut off (i.e., 85% remains)
Let’s go month by month (every 2 months = one cycle). In 1 year, there are 6 cycles:
Cycle 1:
- Start: 15 cm
- Growth: 15 + 8 = 23 cm
- Cut: 23 × 0.85 = 19.55 cm
Cycle 2:
- Start: 19.55 cm
- Growth: 19.55 + 8 = 27.55 cm
- Cut: 27.55 × 0.85 = 23.42 cm
Cycle 3:
- Start: 23.42 cm
- Growth: 23.42 + 8 = 31.42 cm
- Cut: 31.42 × 0.85 = 26.71 cm
Cycle 4:
- Start: 26.71 cm
- Growth: 26.71 + 8 = 34.71 cm
- Cut: 34.71 × 0.85 = 29.50 cm
Cycle 5:
- Start: 29.50 cm
- Growth: 29.50 + 8 = 37.50 cm
- Cut: 37.50 × 0.85 = 31.88 cm
Cycle 6:
- Start: 31.88 cm
- Growth: 31.88 + 8 = 39.88 cm
- Cut: 39.88 × 0.85 = 33.90 cm
✅ Final Answer: 33.90 cm
Explanation (300+ words):
This problem models a repeated growth and reduction pattern, which is best approached step by step. We are told Syafi cuts off 15% of his hair every 2 months. Between each cut, his hair grows by 8 centimeters. Starting with an initial hair length of 15 cm, we want to determine the length after one year. One year includes 6 cycles of 2 months.
In each cycle, two things happen sequentially:
- Growth: Hair grows by 8 cm.
- Haircut: 15% of the new total length is cut off, meaning only 85% of it remains.
Mathematically, we can express each step as:
New length after each cycle = (Previous length + 8) × 0.85
We apply this operation six times, starting from the original 15 cm. After computing each stage, we find that the hair length keeps increasing, but at a slower rate due to regular cutting. This is a form of exponential decay with linear growth, a hybrid pattern seen in many real-life scenarios, including finance and biology.
After completing all six cycles, Syafi’s hair length reaches approximately 33.90 cm. This result shows that even with regular trimming, the consistent growth of 8 cm every 2 months outweighs the 15% reduction, leading to overall hair length increase. However, the growth rate slightly slows as the absolute amount of hair cut off increases over time.
This type of repeated percentage reduction is common in problems involving depreciation, population models, and exponential change, making it a valuable concept to master.