Where the rule stops being honest
A phone balance of ₹100 falls by ₹20 a day. When does it reach ₹0?
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Step 1 of 3
- A tank starts at 40 litres. It drains 3 litres a minute.
- At 13 minutes: 1 litre. This is real.
- At 14 minutes: −2 litres. This cannot be real.
The rule
On day 5, because 100 − 20 × 5 = 0.
Name the quantity that changes. Find the change for each step. Write start + change × steps.
Linear growth adds the same amount each step. Linear decay takes away the same amount each step.
A balance of ₹100 that falls ₹20 a day gives y = 100 − 20t. Here t is the days and y is the balance in rupees.
- Day 5: 100 − 20 × 5 = 100 − 100 = 0. The balance is ₹0.
- Day 6: 100 − 20 × 6 = 100 − 120 = −20. A balance cannot be below zero.
The rule still gives an answer after the last real step. You decide where to stop.
A drum that holds 45 L has 5 L in it. Each minute adds 4 L.
| Situation | Rule | Last real step | One step past |
|---|---|---|---|
| Tank, 40 L, −3 a minute | 40 − 3t | 13 min → 1 L | 14 min → −2 L |
| Balance, ₹100, −₹20 a day | 100 − 20t | day 5 → ₹0 | day 6 → −₹20 |
| Drum, 45 L full, +4 from 5 L | 5 + 4t | 10 min → 45 L | 11 min → 49 L |
A rule for growth has a limit too. This drum cannot hold more than 45 L.