SME
Grade 9 Math · Problem type
Decay by a percent
Use this for “A $18,000 car loses 15% of its value each year. In which year is it first worth under $9,000?” Read down the table.
Table
Year (t)
Value after t years (A)
($)
This is an example. Type your own values to start a new problem.
A
=
P
(1
−
%)
g
Decay factor
worked out (not typed)
Formulas
g
=
1
−
r
100
0.85
=
1
−
15
100
A
=
P
×
g
t
7,986.70
=
18,000
×
0.85
5
Assumptions
•
The amount loses the same percent r each period, so it keeps 1 − r of itself.
•
Write r as a decimal: 15% is 0.15, so the decay factor is 0.85.
•
Read down the table for the first period past a target.
Step-by-step
WE KNOW
t = 5, P = $18,000, r = 15%
FIND
decay factor (g), value after t years (A)
Step 1 · Find decay factor (g)
Use g = 1 − r ÷ 100
Write
the
rate
as
a
decimal
and
take
it
from
1:
each
period
keeps
g
of
the
amount.
g
=
1
−
15
100
15% = 0.15
g
=
0.85
Step 2 · Find value after t years (A)
Use A = P × gᵗ
Multiply
the
starting
amount
by
g
once
for
each
period.
A
=
18,000
×
0.85
5
A
=
18,000
×
0.4437
A
=
about
$7,986.70
Check
Put the numbers back in. Both sides should match.
0.85
=
1
−
15
100
✓
7,986.70
≈
18,000
×
0.85
5
✓
Topic check
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