Question 42·Hard·Percentages
In a laboratory experiment, a bacterial culture grows by 50% every 6 hours. The culture begins with bacteria at 8:00 a.m. on Day 1. After exactly 24 hours, an antiseptic treatment instantly destroys 30% of the bacteria that are present at that moment. The culture is then returned to the incubator, where the same 24-hour "grow-then-treat" cycle is repeated once more.
Which of the following expressions gives the number of bacteria in the culture immediately after the second antiseptic treatment, in terms of ?
For repeated-percent-change problems, convert each percentage to a multiplier (increase by 50% → multiply by 1.5; destroy 30% → multiply by 0.7), then count how many times each event happens and form a product. Group the process into one full cycle (here: 24 hours of growth, then one treatment), write an expression for that cycle, and then apply it as many times as the problem states, combining exponents for repeated factors to match the answer choices quickly.
Hints
Turn the percentages into multiplication factors
A 50% increase every 6 hours and a 30% decrease at treatment are both repeated multipliers. What single number do you multiply by for each 6-hour growth period, and what single number do you multiply by at each treatment?
Count how many growth periods happen before each treatment
Between treatments, the culture grows for 24 hours. How many 6-hour periods are there in 24 hours, and what overall factor does that give for one full day of growth?
Represent one full day as a product, then repeat
Write an expression for the bacteria after one full grow-then-treat cycle starting from (24 hours of growth, then one treatment). Then imagine doing that exact same multiply-by sequence one more time and combine the repeated factors.
Desmos Guide
Model one grow-then-treat cycle numerically
In Desmos, define a constant like N0 = 1 (any positive value works, since we care about the pattern). Then enter the expression N0*(1.5)^4*0.7 to represent one 24-hour growth (four 6-hour periods) followed by one treatment. Note this as the value after the first cycle.
Apply the same cycle a second time and compare to choices
In Desmos, define A = N0*(1.5)^4*0.7 and then B = A*(1.5)^4*0.7 to represent applying the same 24-hour grow-then-treat cycle again. Look at how Desmos rewrites B as a single product (its powers of 1.5 and 0.7) and choose the answer option whose exponents on 1.5 and 0.7 match that expression.
Step-by-step Explanation
Translate the percentage changes into multipliers
A 50% increase means the new amount is the old amount plus half of itself:
- New = old old old.
So every 6 hours, the culture is multiplied by .
Destroying 30% of the bacteria means 70% remain:
- Remaining fraction .
So each antiseptic treatment multiplies the current amount by .
Find the growth factor over one 24-hour period
Growth happens every 6 hours. In 24 hours, the number of 6-hour periods is
So in one day before any treatment, the culture is multiplied by four times:
Write the amount after the first 24-hour grow-then-treat cycle
Start with at 8:00 a.m. on Day 1.
- After 24 hours of growth (to 8:00 a.m. on Day 2):
- The antiseptic then destroys 30% of that amount, leaving 70%, so multiply by :
This is the number of bacteria immediately after the first treatment.
Repeat the same cycle for the second day and combine factors
The second 24-hour period works the same way, starting from :
- Another 24 hours of growth multiplies by again.
- The second antiseptic treatment multiplies by again.
So the amount immediately after the second treatment is
Combine like factors:
This matches answer choice C.