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Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Convert a retained percentage into a multiplier.
- Write and evaluate an exponential-decay rule.
- Compare percentage losses with absolute losses.
Before you start
Convert percentages to decimals and evaluate positive integer powers.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
In an idealized classroom model, each identical filter passes 80% of the light reaching it. Starting with 100 intensity units, how many units remain after three filters?
Why this math matters
A repeated percentage acts on the current amount at every step. This creates a geometric sequence and an exponential model. The amount lost shrinks as the remaining quantity shrinks, even though the percentage loss remains constant.
Set up the model
A useful answer starts with clear assumptions:
- Each filter transmits exactly 80% of its incoming intensity in the model.
- Other optical effects are excluded; these numbers illustrate exponential arithmetic, not a real device specification.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Follow a quantity through repeated percentage reductions
PausedQuestion: Start with the question. Paused.
Question
Start with the question
In an idealized classroom model, each identical filter passes 80% of the light reaching it. Starting with 100 intensity units, how many units remain after three filters?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Record the retained fraction
80% = 0.8; Iₙ = 100(0.8)ⁿ
n counts filters. At n = 0, no reduction has occurred and the initial intensity is one hundred units.
Follow successive stages
100 → 80 → 64 → 51.2
Every arrow multiplies the current amount by 0.8. It does not subtract twenty units repeatedly.
Check the total reduction
100 − 51.2 = 48.8 units; overall reduction = 48.8%
The separate losses are 20, 16, and 12.8 units. They total 48.8, rather than three times twenty percent of the original amount.
The result
After three filters, 51.2 intensity units remain in the model.
Repeated twenty-percent losses leave 0.8³ = 0.512 of the starting amount. The model stays positive for every finite number of filters even though its values approach zero as the filter count grows.
Common mistakes to catch
- A twenty-percent loss uses multiplier 0.8, not 0.2.
- Adding three twenty-percent losses as sixty percent ignores the changing base.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
A quantity starts at 200 units and retains 90% at each step. Find the value after two steps.
Show a hint
Use 200(0.9)².
Reveal answer and explanation
162 units
The first step leaves 180 units and the second leaves 162. The total loss is thirty-eight units.
Practice 2
A stage starts with 80 units and loses 20%. How many units are lost, and how many remain?
Show a hint
The percentage now uses eighty as its base.
Reveal answer and explanation
16 lost; 64 remain
0.2 × 80 = 16, and 80 − 16 = 64. This is the second stage of the original sequence.
Take the idea with you
For each percentage change, write down its base. A repeated multiplier is the signal to consider an exponential model.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use a logarithm to count doubling cycles
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