Ratio Word Problems — a little musical tasting menu
Drawing upon what you learned in the lesson, complete and then check the worked answers below. Read each solution slowly — like savouring a small, perfect bite.
Problems (student printable)
- A musician is playing notes with the frequencies 264.94 Hz and 529.88 Hz together. Will you like how these notes sound together? Justify your answer with mathematics then explain in words.
- A composer knows they find the sound of notes with the frequencies 220 Hz and 330 Hz pleasing to their ear. Find another note that, when paired with 330 Hz, the composer will find equally pleasing. (Give at least one frequency and explain why.)
- Tamya is playing a song on the piano that has 32 evenly spaced notes. How many evenly spaced notes would Luis need to play on the drums in order to make a 4:3 rhythm?
Extra clapping question
Amorell and Erik are clapping two different beats at the same time. Amorell claps five times in the measure while Erik claps three times. What is the ratio of beats Amorell claps to beats Erik claps?
Worked solutions (step-by-step, student-facing)
1. 264.94 Hz and 529.88 Hz — do these sound pleasing together?
Mathematics first: form the ratio of the higher frequency to the lower frequency.
529.88 ÷ 264.94 = 2.000000 (exactly 2 to the precision given).
The ratio is 2 : 1. In musical terms, a 2:1 frequency ratio is an octave. Notes an octave apart share the same note name and sound very consonant together — they blend beautifully, like the same colour in two different intensities.
Conclusion: Yes — these two notes form an octave (2:1), which is normally very pleasing to the ear because the sound waves align so regularly.
2. 220 Hz and 330 Hz — find another note to pair with 330 Hz
First check the ratio between 330 and 220:
330 ÷ 220 = 1.5 = 3/2.
This 3:2 ratio is the perfect fifth, a very pleasing musical interval.
We want a frequency that, when paired with 330 Hz, gives the same pleasing interval (3:2). There are two natural places to look: below 330 (so 330 is the higher of the pair) or above 330 (so 330 is the lower of the pair).
- If 330 is the higher note: lower = 330 × (2/3) = 220 Hz (the original note).
- If 330 is the lower note: higher = 330 × (3/2) = 495 Hz.
So, another note that will sound equally pleasing with 330 Hz is 495 Hz (a perfect fifth above). 220 Hz also works (a perfect fifth below), but 495 Hz is a different, new note that keeps the same 3:2 relationship.
3. Tamya: 32 evenly spaced piano notes; Luis wants a 4:3 rhythm — how many drum hits?
A 4:3 rhythm means Tamya's count : Luis's count = 4 : 3. Tamya has 32 notes, so set up the proportion:
32 : x = 4 : 3
Solve for x: x = 32 × (3/4) = 32 × 0.75 = 24.
Answer: Luis should play 24 evenly spaced drum hits to form a 4:3 ratio with Tamya's 32 piano notes.
Extra: Amorell and Erik clapping
Amorell claps 5 times while Erik claps 3 times in the same measure. The ratio of Amorell’s claps to Erik’s is simply 5 : 3. That is already given in simplest whole-number form.
Short explanations — in words (to reinforce understanding)
Think of frequency ratios like mixing colours or spices. Simple whole-number ratios (like 2:1 or 3:2) produce sounds whose wave cycles line up regularly — our ears interpret that regularity as pleasant. More complicated ratios are less regular and can sound more jarring.
ACARA v9 mapping (teacher-facing)
Mapped to Australian Curriculum v9 (Mathematics — Number and Algebra): this activity develops students' ability to work with ratios and rates, interpret multiplicative relationships, and solve problems using ratio reasoning. It is suitable for Years 7–8 (age 13–14), linking mathematical ratio concepts to real-world contexts (music and rhythm).
Teacher comments (approx. 100 words each, unique) — give to students after they complete each task
Comment for Task 1 (264.94 Hz & 529.88 Hz)
You approached this classic musical ratio beautifully. You showed the calculation of 529.88 ÷ 264.94 clearly, and you recognised the result as 2:1 — an octave — which you then explained in musical terms. Your justification linked the arithmetic to the sound: how regular wave alignment creates consonance. For improvement, you might add a quick sketch or small diagram showing two wave cycles aligned (one wave completing two cycles where the other completes one) — that visual can make the concept stick. Excellent clarity in both computation and explanation; keep blending maths with real-world examples.
Comment for Task 2 (220 Hz & 330 Hz; find another pleasing note)
You identified the 3:2 ratio correctly and named it as a perfect fifth — well done. I liked that you calculated both directions (finding a lower partner and a higher partner) and gave 495 Hz as an alternative pleasing note above 330 Hz. That shows strong flexible thinking about intervals. To deepen your response, note how octaves and fifths relate on a piano keyboard (e.g., counting keys or semitones) or consider why simple integer ratios are so musically important. Your solution shows mathematical precision and musical insight — a delightful combination.
Comment for Task 3 (Tamya 32 notes, 4:3 rhythm)
Your proportional setup was correct: 32 : x = 4 : 3, and you solved quickly to get x = 24. You also explained the reasoning clearly, not just the arithmetic — demonstrating that you understand what a 4:3 rhythm means in practice. For extension, try sketching both sequences of beats across the same measure to show visually how 32 and 24 align at the start and end (they will meet again every full measure). Great job applying ratio reasoning to rhythm — you translated abstract numbers into a musical pattern.
Comment for the clapping question (Amorell & Erik)
You recognised the ratio immediately as 5:3 and left it in simplest whole-number form — exactly what we need. You might extend your answer by explaining how to find a common pulse so both clappers can follow the same measure: divide the measure into 15 equal parts (LCM of 5 and 3), then Amorell claps every 3 parts and Erik every 5 parts. This gives a clear picture of how two different rhythms can coexist neatly. Nice concise reasoning and an awareness of simplest-ratio form.
Extended rubric — what to aim for
Task 1 rubric (264.94 Hz & 529.88 Hz)
Exemplary: Correctly computes the ratio 2:1, names the interval (octave), and gives a clear physical or visual justification (wave-cycle diagram or explanation of harmonic alignment). Connects result to musical perception (why an octave sounds consonant). Uses precise language and checks arithmetic.
Proficient: Correctly computes the ratio and identifies the interval as very consonant (octave). Provides a clear sentence linking the ratio to why it sounds pleasing. Arithmetic and explanation are accurate.
Task 2 rubric (220 Hz & 330 Hz)
Exemplary: Correctly identifies 330/220 = 3/2, names the perfect fifth, and finds at least one other frequency (495 Hz or 220 Hz) that forms the same interval with 330. Explains why 3:2 is musically pleasing and optionally relates to keyboard semitones or octave equivalence.
Proficient: Shows the 3:2 ratio, names the interval, and gives one correct frequency that keeps the same interval. Explanation links simple ratio to consonance.
Task 3 rubric (32 notes, 4:3 rhythm)
Exemplary: Correct proportion set-up and solution x = 24; includes a supporting visual (timeline or beats) showing alignment and describing least common multiple if needed. Explains why the counts produce a 4:3 relationship in words.
Proficient: Correctly computes x = 24 using a proportion and explains in one or two sentences that 24 drum hits produce a 4:3 ratio with 32 piano notes.
Clapping rubric (5 vs 3)
Exemplary: Gives the ratio 5:3 and describes how to find a common subdivision (e.g., divide measure into 15 parts) or draws a time-line showing the alignment of claps.
Proficient: Correctly states the ratio 5:3 and keeps it in simplest form.
If you want, I can produce a one-page printable PDF layout of this worksheet (student-facing) or create a small diagram showing wave alignment and timelines for the rhythms. Would you like that?