Mathematics • Year 9 • Unit 1 • Lesson 17
Scientific Notation at Work
Use $\times$, $\div$, $+$ and $-$ on scientific-notation numbers in real situations: sunlight travel time, humanity's daily breath of air, world-population water use, computer benchmarks, and a daily TikTok upload count. Then explain your method in your own words.
1. Word problems
Each problem needs at least one operation in scientific notation. Show all working — coefficients with arithmetic, indices with the product or quotient rule, and re-standardise the coefficient back into $[1, 10)$ when needed.
1.1 — How long does sunlight take? Light travels at about $3 \times 10^8$ m/s. The Sun is about $1.5 \times 10^{11}$ m from Earth.
(a) Use $t = \dfrac{\text{distance}}{\text{speed}}$ to find the time, in seconds, in scientific notation.
(b) Convert your answer to minutes, to one decimal place. 3 marks
1.2 — Humanity's daily breath. Earth has about $7.8 \times 10^9$ people. Each person breathes about $6 \times 10^3$ litres of air per day.
(a) Find the total litres of air breathed by humanity in one day, in scientific notation.
(b) Name the index law you used on the powers of $10$. 3 marks
1.3 — Drinking water for the planet. The recommended daily water intake is about $2 \times 10^0$ litres per person (i.e. $2$ L). Earth's population is about $8 \times 10^9$.
(a) Find the total recommended litres for one day, in scientific notation.
(b) For one year ($365$ days $\approx 3.65 \times 10^2$ days), find the total in scientific notation. 3 marks
1.4 — Computer benchmark. A laptop runs $4 \times 10^9$ operations per second. A short maths simulation takes $2.5 \times 10^3$ seconds to finish.
(a) How many operations does the laptop do during the simulation? Give your answer in scientific notation.
(b) If a phone runs at $5 \times 10^8$ operations per second, how many times longer would the same simulation take on the phone? 3 marks
1.5 — Adding two scales. A short video file is $4 \times 10^7$ bytes. A still photo is $3 \times 10^6$ bytes. You upload both at once.
(a) Find the total bytes uploaded, in scientific notation.
(b) The video uses how many times more bytes than the photo? Give the answer in scientific notation. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate writes "$3 \times 10^4 + 2 \times 10^3 = 5 \times 10^7$". In your own words, explain (i) which two rules they have confused, (ii) why addition of scientific-notation numbers needs the powers of $10$ to match first, and (iii) how to get the correct answer. Refer to "common power" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Sunlight to Earth
(a) $t = \dfrac{1.5 \times 10^{11}}{3 \times 10^8} = \dfrac{1.5}{3} \times 10^{11 - 8} = 0.5 \times 10^3 = \mathbf{5 \times 10^2}$ s (re-standardised).
(b) $5 \times 10^2$ s $= 500$ s, and $500 \div 60 \approx \mathbf{8.3}$ minutes.
Real-world check: it really does take sunlight about 8 minutes to reach Earth.
1.2 — Humanity's daily breath
(a) $(7.8 \times 10^9) \times (6 \times 10^3) = (7.8 \times 6) \times 10^{9 + 3} = 46.8 \times 10^{12} = \mathbf{4.68 \times 10^{13}}$ L per day.
(b) Product rule on powers of 10: $10^m \cdot 10^n = 10^{m+n}$ (add the indices).
That's about 47 trillion litres of air every day.
1.3 — Drinking water for the planet
(a) $(2 \times 10^0) \times (8 \times 10^9) = 16 \times 10^{0 + 9} = 16 \times 10^9 = \mathbf{1.6 \times 10^{10}}$ L per day.
(b) Per year: $(1.6 \times 10^{10}) \times (3.65 \times 10^2) = (1.6 \times 3.65) \times 10^{10 + 2} = 5.84 \times 10^{12}$ L. Answer: $\mathbf{5.84 \times 10^{12}}$ L per year.
Re-standardised both times: the $16$ became $1.6$ and bumped the index by 1; the $5.84$ stays put.
1.4 — Computer benchmark
(a) Operations $= (4 \times 10^9) \times (2.5 \times 10^3) = (4 \times 2.5) \times 10^{9 + 3} = 10 \times 10^{12} = \mathbf{1 \times 10^{13}}$ operations (re-standardised — $10$ became $1$ with index up by 1).
(b) Ratio of speeds $= \dfrac{4 \times 10^9}{5 \times 10^8} = 0.8 \times 10^{9 - 8} = 0.8 \times 10^1 = \mathbf{8}$. The phone would take $\mathbf{8}$ times longer.
1.5 — Adding two scales
(a) Common power $10^7$: $3 \times 10^6 = 0.3 \times 10^7$. Then $(4 + 0.3) \times 10^7 = \mathbf{4.3 \times 10^7}$ bytes total.
(b) Ratio $= \dfrac{4 \times 10^7}{3 \times 10^6} = \dfrac{4}{3} \times 10^{7 - 6} \approx 1.33 \times 10^1 \approx \mathbf{1.33 \times 10^1}$ (or $\approx 13.3$ times).
So the video uses about 13 times more bytes than the photo.
2.1 — Explain your thinking (sample response)
My classmate has confused the rule for multiplying scientific-notation numbers with the rule for adding them. They have added both the coefficients ($3 + 2 = 5$) and the indices ($4 + 3 = 7$), but adding indices is only valid when you are multiplying powers of $10$. When you are adding two scientific-notation terms, the powers of $10$ are not being multiplied at all — they have to match first, becoming a single common power that both terms share. To fix it, rewrite $2 \times 10^3$ as $0.2 \times 10^4$ so both terms share the common power $10^4$; now add the coefficients: $(3 + 0.2) \times 10^4 = \mathbf{3.2 \times 10^4}$. A quick check: $3 \times 10^4 = 30{,}000$ and $2 \times 10^3 = 2{,}000$, so the sum is $32{,}000 = 3.2 \times 10^4$ — definitely not $5 \times 10^7 = 50{,}000{,}000$.
Marking: 1 mark for naming the confused rules (add vs multiply); 1 for stating powers must match; 1 for the correct answer $3.2 \times 10^4$; 1 for a clear full-sentence explanation using "common power".