Math 1 · Unit 8 · Lesson 5 · part 2 of 5
Depreciation: the factor is 1 minus the rate
Everything so far grew.
Depreciation: the factor is 1 minus the rate
Everything so far grew. The same form works when value goes down. A new car is worth less every year. Losing value over time is called depreciation.
35,000(0.87)t
Read it the way you read a savings account in Lesson 2.
- 1The number in front is the starting value.The car cost $35,000 new.
- 2The factor is 1 minus the rate.0.87 = 1 − 0.13. Keeping all of the value would be multiplying by 1; this car keeps only 0.87 of it, so it loses 13% of its value each year.
- 3The exponent counts the years.After 3 years: 35,000 × 0.873 = 35,000 × 0.658503 = about $23,048.
| year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| value | $35,000.00 | $30,450.00 | $26,491.50 | $23,047.60 | $20,051.42 | $17,444.73 |
This is decay from Unit 7, Lesson 3, with money names:
A = P(1 − r)t
The decay factor is 1 − r, the number you multiply by each year. The decay rate is r, the percent lost each time.
Be careful with the rate. A factor of 0.85 does not mean losing 85%. It means keeping 85% and losing 15%, because 0.85 = 1 − 0.15. Always start from 1, which is keeping all of it, and ask how far the factor is from 1.
Growth: factor above 11.05 = 1 + 0.05: grows 5%.
1.12 = 1 + 0.12: grows 12%.
Grows 3.5%: factor 1.035.
1.12 = 1 + 0.12: grows 12%.
Grows 3.5%: factor 1.035.
Decay: factor below 10.87 = 1 − 0.13: loses 13%.
0.85 = 1 − 0.15: loses 15%.
Loses 40%: factor 0.60.
0.85 = 1 − 0.15: loses 15%.
Loses 40%: factor 0.60.
Factor exactly 1Keeps all of it, adds nothing: no change. The do-nothing number from Unit 1.
Slide the rate below zero to make a decay factor, and watch the car's value fall.