Mathematics • Year 9 • Unit 1 • Lesson 4
The Quotient Rule in the Real World
Use $\dfrac{a^m}{a^n} = a^{m-n}$ in everyday contexts: knockout tournaments, scaling down a population, file compression, dividing inheritance, and decoding ladder distances. Then explain your method in your own words.
1. Word problems
Each problem uses the quotient rule from Lesson 4: $\dfrac{a^m}{a^n} = a^{m-n}$. Show your working — a single final answer with no working only earns half marks.
1.1 — Knockout tournament. A basketball knockout tournament starts with $2^7$ teams. Each round, exactly half the remaining teams are eliminated. After $3$ rounds, the remaining teams are $\dfrac{2^7}{2^3}$.
(a) Apply the quotient rule to express the number of remaining teams as a single power of $2$.
(b) Evaluate it as a normal number. 3 marks
1.2 — Scaling down a colony. A community of $5^6$ ants in a forest is studied. A nearby outpost colony has $5^2$ times fewer ants than the main colony.
(a) Write the outpost population as a quotient of two powers of $5$.
(b) Apply the quotient rule and evaluate as a normal number. 3 marks
1.3 — File compression. A photo is $2^{20}$ bytes (one megabyte). A compression app shrinks it by a factor of $2^4$.
(a) Write the new file size as a quotient of two powers of $2$.
(b) Apply the quotient rule. What is the new file size as a power of $2$? 3 marks
1.4 — Inheritance shared among siblings. A family inherits $10^6$ dollars. There are $10^2$ family members entitled to a share.
(a) Express each person's share as a quotient of powers of $10$.
(b) Apply the quotient rule, evaluate, and write the amount in dollars. 3 marks
1.5 — Doubling ladder. A "doubling ladder" in a video game has rungs at heights $2^1, 2^2, 2^3, \ldots$ metres. A player jumps from the rung at height $2^9$ down to the rung at height $2^5$.
(a) How many times taller is the upper rung than the lower rung? Express this as a single power of $2$ using the quotient rule.
(b) Evaluate that power of $2$ as a normal number. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes "$\dfrac{2^{10}}{2^2} = 2^5$ because $10 \div 2 = 5$". They're confident they're right. In your own words, explain (i) what mistake they have made, (ii) what the quotient rule actually says, and (iii) the correct simplification of $\dfrac{2^{10}}{2^2}$. Use the words "subtract" and "top minus bottom" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Knockout tournament
(a) $\dfrac{2^7}{2^3} = 2^{7-3} = \mathbf{2^4}$ teams.
(b) $2^4 = \mathbf{16}$ teams remain after 3 rounds.
Quick check: $128$ starting $\div\ 2$ three times $= 128 / 8 = 16$ ✓.
1.2 — Scaling down a colony
(a) Outpost $= \dfrac{5^6}{5^2}$ ants.
(b) $5^{6-2} = 5^4 = \mathbf{625}$ ants in the outpost.
So the main colony with $5^6 = 15{,}625$ ants is $25$ times bigger.
1.3 — File compression
(a) New size $= \dfrac{2^{20}}{2^4}$ bytes.
(b) $2^{20-4} = \mathbf{2^{16}}$ bytes ($= 65{,}536$ bytes, about $64$ KB).
That's a $16{\times}$ size reduction — typical for moderate JPEG compression.
1.4 — Inheritance shared among siblings
(a) Each share $= \dfrac{10^6}{10^2}$ dollars.
(b) $10^{6-2} = 10^4 = \mathbf{\$10{,}000}$ each.
One million dollars split between one hundred people gives ten thousand dollars each.
1.5 — Doubling ladder
(a) $\dfrac{2^9}{2^5} = 2^{9-5} = \mathbf{2^4}$ times taller.
(b) $2^4 = \mathbf{16}$ times taller.
The upper rung at $2^9 = 512$ m is $16$ times higher than the lower at $2^5 = 32$ m. Check: $512 \div 32 = 16$ ✓.
2.1 — Explain your thinking (sample response)
My classmate has divided the indices ($10 \div 2 = 5$) when they should have subtracted them. The quotient rule for same-base powers is $\dfrac{a^m}{a^n} = a^{m-n}$, which says: take the top minus bottom index, not the top divided by the bottom. The reason is that $\dfrac{2^{10}}{2^2}$ has $10$ copies of $2$ on top and $2$ copies on the bottom; the $2$ on the bottom cancels against $2$ of the copies on top, leaving $10 - 2 = 8$ copies on top — that's $\mathbf{2^8}$, not $2^5$. A quick check: $2^{10} = 1024$ and $2^2 = 4$, so $1024 \div 4 = 256 = 2^8$ ✓ (and $2^5 = 32$, which is clearly not $256$).
Marking: 1 mark for naming the mistake (divided instead of subtracted); 1 for stating the correct rule with "top minus bottom"; 1 for the correct answer $2^8$; 1 for a clear, full-sentence explanation.