MYP 3 · Maths

PERCENTAGE

Expressing one quantity as a percentage of another

QUESTION 1 [6 marks] — Criterion B Medium
Investigate what happens to the percentage when you SWAP which quantity is treated as the 'part' and which is the 'whole'.
a. Find what percentage 15 is of 60, then find what percentage 60 is of 15.
[3]
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15 as a % of 60: $\frac{15}{60}\times100=25\%$. 60 as a % of 15: $\frac{60}{15}\times100=400\%$.
b. These two answers are very different. Explain why swapping the 'part' and 'whole' changes the result so dramatically, and what a percentage over 100% actually means.
[3]
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Swapping part and whole essentially flips the fraction (from $\frac{15}{60}$ to $\frac{60}{15}$), giving a completely different value — division is not commutative. A percentage over 100% simply means the 'part' is actually LARGER than the 'whole' being compared to (here, 60 is 4 times as large as 15).
QUESTION 2 [4 marks] — Criterion C Medium
A student calculates 'what percentage is 40 of 25' by computing $\frac{25}{40}\times100$, getting $62.5\%$.
a. Identify the student's error, and give the correct calculation and answer.
[2]
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The student divided the wrong way around — to find what percentage 40 IS OF 25, the 'part' (40) must go on top: $\frac{40}{25}\times100=160\%$, not $\frac{25}{40}\times100$.
b. Explain, using the phrase itself ('what percentage is 40 OF 25'), a reliable way to remember which number goes on top of the fraction.
[2]
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The number right after 'is' (or being described) goes on TOP (the part), and the number right after 'of' goes on the BOTTOM (the whole) — so 'what percentage IS 40 OF 25' becomes $\frac{40}{25}$.
QUESTION 3 [5 marks] — Criterion D Medium
In a basketball season, Player A scored 84 points out of 120 shot attempts. Player B scored 105 points out of 150 shot attempts.
a. Find each player's shooting success rate as a percentage.
[3]
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Player A: $\frac{84}{120}\times100=70\%$. Player B: $\frac{105}{150}\times100=70\%$.
b. A commentator claims 'Player B is clearly the better shooter since they scored more total points (105 vs 84)'. Evaluate this claim using your percentages.
[2]
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The claim is misleading — both players have exactly the same shooting SUCCESS RATE (70%). Player B simply took more shots overall (150 vs 120), which is why they scored more total points, not because they were more accurate.
QUESTION 4 [5 marks] — Criterion A Medium
A water conservation project reduced a city's daily water usage from 240 million litres to 156 million litres over 5 years.
a. Find the NEW usage as a percentage of the ORIGINAL usage.
[2]
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$$\frac{156}{240}\times100=65\%$$
b. Express the REDUCTION itself (84 million litres) as a percentage of the original usage, and verify your two percentages (from this part and part a) sum to 100%.
[3]
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$\frac{84}{240}\times100=35\%$. Check: $65\%+35\%=100\%$ ? (makes sense, since 'new usage %' and 'reduction %' together must account for the whole original amount).
QUESTION 5 [7 marks] — Criterion B Medium
Investigate what happens to 'what percentage is $a$ of $b$' as $a$ gets closer and closer to $b$, and as $a$ gets closer to 0.
a. Calculate what percentage 45 is of 50, then what percentage 49 is of 50, then what percentage 50 is of 50.
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$\frac{45}{50}\times100=90\%$. $\frac{49}{50}\times100=98\%$. $\frac{50}{50}\times100=100\%$.
b. Calculate what percentage 5 is of 50, then what percentage 1 is of 50, then what percentage 0 is of 50.
[2]
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$\frac{5}{50}\times100=10\%$. $\frac{1}{50}\times100=2\%$. $\frac{0}{50}\times100=0\%$.
c. Based on both patterns, state (without further calculation) what percentage would result if $a$ EQUALS $b$ exactly, and explain why this makes intuitive sense.
[2]
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When $a=b$, the percentage is always $100\%$ — this makes sense because 'what percentage is $a$ of $b$' is really asking 'how much of the whole ($b$) does $a$ represent', and when $a$ equals the whole amount exactly, it represents ALL of it, i.e. 100%.
QUESTION 6 [6 marks] — Criterion B Hard
Investigate whether 'what percentage is $a$ of $b$' plus 'what percentage is $b$ of $a$' always sums to 100%, using $a=30, b=70$.
a. Calculate both percentages: what percentage is 30 of 70, and what percentage is 70 of 30.
[3]
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$\frac{30}{70}\times100\approx42.9\%$. $\frac{70}{30}\times100\approx233.3\%$.
b. Do these two percentages sum to 100%? Explain why the '$a$ of $b$' plus '$b$ of $a$' pattern does NOT generally work the same way as the 'part + remainder = 100%' pattern investigated earlier (which specifically involved a part and the REMAINDER of the same whole).
[3]
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No — they sum to $42.9+233.3=276.2\%$, nowhere near 100%. This is different from the earlier pattern because here, $a$ and $b$ are each being treated as the 'whole' in turn (two DIFFERENT wholes), rather than $a$ being a part of ONE fixed whole $b$ with the remainder being $b-a$ — these are fundamentally different mathematical relationships, so there's no reason to expect them to sum to 100%.
QUESTION 7 [3 marks] — Criterion C Medium
A student calculating 'what percentage is 18 of 45' writes $\frac{45}{18}\times100\approx250\%$, then says 'that seems too big, so I think I need to just flip it', getting the right final answer by trial and error rather than by understanding WHY they needed to flip it.
a. Calculate the CORRECT answer, and explain — using the actual SIZES of 18 and 45 — why an answer over 100% should have immediately signalled an error, without needing trial and error.
[3]
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Correct: $\frac{18}{45}\times100=40\%$. Since 18 is SMALLER than 45, the percentage 18 represents OF 45 must be under 100% (a part can't exceed 100% of a whole that's bigger than it) — recognizing this BEFORE calculating would have immediately flagged the first (250%) attempt as using the numbers the wrong way round, without needing to guess-and-check.