Growing by a percentage is multiplying, over and over. The exponent counts how many times.
The problem
A town of 20 000 grows 5% a year. What is its population after 8 years?
P(1 + r)ⁿ = 20,000(1.05)⁸ = 20,000 × 1.4775 ≈ 29,549. B is simple growth — eight lots of 5% is 40%, giving 28,000. Compounding beats it because each year's growth is calculated on the new, larger total.
Growing by 5% means keeping 100% and adding 5% — a multiplier of 1.05.
20 000 × 1.05 = 21 000
after 1 year
21 000 × 1.05 = 22 050
after 2 — the rise is bigger already
22 050 × 1.05 = 23 152.5
after 3
The second year's growth is 1050, not 1000, because the 5% is taken on the new, larger figure. That compounding is the entire difference between this and straight-line growth.
A = A₀(1 + r)t
A₀
20 000
r
0.05
t
8
A = 20 000(1.05)8
eight years of multiplying
1.058 ≈ 1.4775
square it three times: 1.1025, 1.2155, 1.4775
A ≈ 29 549
people, so round sensibly
Writing 20 000(0.05)⁸ shrinks the town to almost nothing, and 20 000(5)⁸ turns it into a galaxy. Convert the percent to a decimal, then add it to 1.
(1 + r)
5% rise → 1.05
(1 − r)
12% fall → 0.88
A car worth ₱800 000 losing 12% a year is worth 800 000(0.88)³ ≈ ₱545 000 after three years. The multiplier is what you keep, never what you lose.
A sample holds 80 mg of a substance whose half-life is 6 hours. How much is left after 18 hours?
A = A₀(½)t/h
t ÷ h = 18 ÷ 6 = 3
three half-lives have passed
80 → 40 → 20 → 10
halve it three times
A = 80(½)³ = 10 mg
same answer, one line
18 in the exponent would mean eighteen halvings. Dividing by the half-life first is what converts the clock into the number of times the multiplier is applied — the same reason a rate “per quarter” needs the years multiplied by four.
(a) A culture triples every 4 hours. By what factor does it grow in 10 hours? (b) 400 g of a substance falls to 25 g in 20 days. What is its half-life?
(a) periods = 10 ÷ 4 = 52
not a whole number — that is fine
35/2 = 3² × 31/2 = 9√3
split off the whole part; a half power is a square root
(b) 400 → 200 → 100 → 50 → 25
count the halvings: 4
20 days ÷ 4 = 5 days
each halving takes the same time
In (a) the factor is 3 raised to 52, about 15.6 — not 2.5, and not 3 × 2.5. In (b), 400 ÷ 25 = 16 = 2⁴: the 16 is the factor, the 4 is the number of halvings.
| Linear | Exponential | |
|---|---|---|
| each period | add the same amount | multiply by the same factor |
| the model | A₀ + dt | A₀(1 + r)t |
| graph | straight line | a curve that steepens |
| after 8 years | 28 000 | 29 549 |
“Grows by 1000 a year” and “grows by 5% a year” start out almost identical here and end up thousands apart. Read which one the question said.
Find the rate. A fund grows from ₱2,500 to ₱3,600 in 2 years at a fixed yearly percent. What is the rate?
2500(1 + r)² = 3600
the same formula, r unknown
(1 + r)² = 1.44
divide by 2500
1 + r = 1.2 → r = 20%
square root, take the positive one
What is different: the unknown is inside the bracket, so we root instead of logging.
The total rise is 44%, but compounding means each year's rate is less than half of it: 2500 × 1.2 = 3000, then 3000 × 1.2 = 3600 ✓.
Find the start. A machine loses 10% of its value a year. After 3 years it is worth ₱72,900. What did it cost?
A₀(0.9)³ = 72 900
decay multiplier 1 − 0.10
0.9³ = 0.729
0.81 × 0.9
A₀ = 72 900 ÷ 0.729 = ₱100,000
divide; do not multiply by 1.1³
What is different: we undo the decay by dividing by 0.9³. Multiplying by 1.1³ = 1.331 gives about ₱97,030, which is wrong, because losing 10% and gaining 10% do not cancel.
Units that do not match. 48 g of a substance has a half-life of 30 minutes. How much is left after 2 hours?
2 hours = 120 minutes
one unit for both
120 ÷ 30 = 4 halvings
periods, as before
48(½)⁴ = 48 ÷ 16 = 3 g
48 → 24 → 12 → 6 → 3
What is different: 2 ÷ 30 mixes hours with minutes. Put both in one unit (120 min and 30 min, or 2 h and 0.5 h), then count periods.
“How long until it doubles?” puts t upstairs, which is what logarithms are for.
20 000(1.05)t = 40 000
double the town
1.05t = 2
divide by 20 000 — the start cancels
t = log 2log 1.05 ≈ 14.2 years
take logs of both sides
Notice the starting size never mattered: any town growing at 5% doubles in about 14 years. Doubling time depends on the rate alone, and so does half-life.
Most items are built so the numbers are whole doublings or halvings: divide the target by the start, write it as a power of 2 (or ½), and the exponent counts the periods. For “first exceeds” questions, multiply 1.1, 1.21, 1.331 … until you pass the target — faster than logs and safe from rounding.
The problem
A colony of 500 bacteria doubles every 3 hours. How many after 12 hours?
Count the doublings, not the hours: 12 ÷ 3 = 4. So 500 × 2⁴ = 500 × 16 = 8,000. C uses only three doublings. The exponent is always time divided by the period.
