The Question
The function f is defined by f(x) = 78(0.39)x. For any positive integer n, f(n) is p% less than f(n – 1). What is the value of p?
(A) 78
(B) 61 ✓
(C) 39
(D) 22
Why Most Students Get This Wrong
This is a classic SAT trap. You see the base 0.39 and assume the percent decrease is 39%. That feels right, but it is wrong.
An exponential function of the form f(x) = a(b)x has two key parts. The coefficient a is the starting value, and the base b controls how the function changes at each step. In this function, b is always the decay factor, meaning the proportion that remains, not the proportion that is lost.
The base does not tell you what is lost. It tells you what stays.
Step-by-Step Solution
Here is the full algebraic proof, useful if you want to show your work or understand why the shortcut works.
Step 1. Write out two back-to-back terms
The problem says f(n) is p% less than f(n – 1). Start by writing both terms explicitly.
f(n) = 78(0.39)n and f(n – 1) = 78(0.39)n-1
These are just two consecutive outputs of the same function, one step apart.
Step 2. Divide to find what fraction one term is of the previous
Form a ratio by dividing the newer term by the older one.
f(n) / f(n – 1) = [78(0.39)n] / [78(0.39)n-1]
The 78 cancels top and bottom. Then 0.39n / 0.39)n-1 = 0.391 = 0.39.
So f(n) / f(n – 1) = 0.39
Every term is exactly 0.39x the one before it.
Step 3. Translate “p% less” into a math equation
If something is p% less, it keeps (100 – p)% of what it was. In decimal form, that means multiplying by (1 – p/100).
f(n) = (1 – p/100) x f(n – 1)
Divide both sides by f(n – 1), f(n) / f(n – 1) = 1 – p/100
A 20% decrease means you keep 80%, so you multiply by 0.80 = 1 – 20/100. Same logic here.
Step 4. Set the two expressions equal and solve for p
From Step 2, the ratio = 0.39. From Step 3, the ratio = 1 – p/100. They describe the same thing, so set them equal.
- 0.39 = 1 – p/100
- p/100 = 1 – 0.39 = 0.61
- p = 0.61 x 100 = 61
- p = 61
Fast SAT Shortcut
You do not need the full ratio method once you recognize the pattern.
In a(b)x where 0 < b < 1, subtract the base from 1 and convert to a percent.
Base is 0.39, so 1 – 0.39 = 0.61, which means a 61% decrease.
Why Each Wrong Answer Looks Tempting
A is 78. This is the starting coefficient, not the rate of change. When you form the ratio f(n)/f(n-1), the 78 cancels out completely. It has no effect on p.
B is 61. Correct. The base 0.39 tells you 39% remains. The percent lost is 100 – 39 = 61.
C is 39. The most common mistake. Students read the base 0.39 and treat it as the percent decrease. But 0.39 is the percent that stays, not the percent that disappears.
D is 22. Comes from subtracting 39 from 61, or incorrectly mixing the coefficient and base. There is no clean mathematical path to 22 from this problem. If you got this, retrace your algebra from Step 2.
Build the Pattern With These Examples
| Base (b) | What it means | Percent change |
|---|---|---|
|
0.39 |
39% remains each step |
61% decrease ← this problem |
|
0.82 |
82% remains each step |
18% decrease |
|
0.50 |
50% remains each step |
50% decrease |
|
1.12 |
Grows by 12% each step |
12% increase |
Quick Transfer Check
If the base were 0.82, the percent decrease would be (1 – 0.82) × 100% = 18%. Same rule, different base.
Frequently Asked Questions
Because 0.39 is the fraction that remains after each step, not the fraction that disappears. The percent decrease is 1 – 0.39 = 0.61, or 61%.
The base represents the percent remaining. To find the percent decrease, subtract the base from 1 and multiply by 100.
If the base is b and 0 < b < 1, then percent decrease = (1 − b) × 100%.
Because when you compare consecutive terms using a ratio, the 78 cancels out. The percent change depends only on the base.
A 39% decrease would mean 61% remains. But here the base is 0.39, which means only 39% remains. So the decrease must be 61%.
In exponential growth, the base is greater than 1, so the values increase each step. In exponential decay, the base is between 0 and 1, so the values decrease.
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