Fractions in Disguise · Lesson 5 of 12
Using Percentages to Compare Proportions
“Compare proportions with different totals and interpret ingredient labels and population shares.”
• Compare shares when their whole quantities are different. • Interpret an ingredient label using a common measurement basis. • Distinguish a larger proportion from a larger absolute amount. • Explain which conclusions a percentage comparison does and does not support.
Two groups organise a neighbourhood clean-up. One group collects 42 recyclable items out of 50 items in total. Another collects 70 recyclable items out of 80. The second group collects more recyclable items, but it also collects more items altogether. If your question is about which group has the greater recyclable share, comparing 42 and 70 alone is not enough. You must compare each recyclable count with its own total and then place both shares on a common scale.
This is one of the main reasons percentages are useful. They allow a fair comparison of proportions even when group sizes differ. They do not erase those differences in size; they answer a particular question about share rather than total amount.
As you work, distinguish the questions “How many?” and “What fraction of the whole?” They can have different answers, and both may be useful.
To Compare Proportions
For the first clean-up group, the fraction of items that are recyclable is 42/50. Doubling numerator and denominator gives 84/100, or 84%. For the second group, 70/80 simplifies to 7/8, which is 87.5%. The second group therefore has the greater recyclable proportion.
Why would comparing the items that were not recyclable be unreliable? The first group has eight such items and the second has ten. Eight is fewer than ten, but eight out of fifty is 16%, while ten out of eighty is 12.5%. The first group actually has the larger non-recyclable share.
The denominator gives the missing context. A difference of a few objects means more in a small group than in a large group. Converting to percentages puts both comparisons in terms of a hypothetical hundred items with the same proportions.
Problem
Compare the recyclable shares 42 out of 50 and 70 out of 80.
- 1.First share: (42/50) × 100 = 84%.
- 2.Second share: (70/80) × 100 = 87.5%.
- 3.Since 87.5 > 84, the second group has the greater recyclable proportion.
- 4.The first group’s non-recyclable share is 100% − 84% = 16%; the second’s is 12.5%. This gives the same conclusion from the remaining parts.
- 5.This calculation compares proportions. It does not explain why the groups collected different kinds of items.
A clear comparison has three stages. First identify the part and whole in each situation. Next convert both ratios to percentages using consistent units. Finally answer the precise question asked: greater proportion, greater count, or perhaps both.
For example, a small nursery might have a larger survival percentage than a large nursery even though fewer plants survive there in total. This is not a contradiction. One statement compares survival rates; the other compares numbers of plants.
Problem
Nursery A has 72 surviving plants out of 80 planted. Nursery B has 190 surviving plants out of 250. Compare both counts and proportions.
- 1.Nursery A survival percentage = (72/80) × 100 = 90%.
- 2.Nursery B survival percentage = (190/250) × 100 = 76%.
- 3.Nursery A has the greater survival proportion because 90% > 76%.
- 4.Nursery B has more surviving plants because 190 > 72.
- 5.Both conclusions are correct. State which quantity you are comparing instead of using the vague claim “A did more”.
Know Your Contents (KYC)
Food labels often describe ingredient masses or percentages. These can help you compare how much of an ingredient is present relative to the total product. If packet sizes differ, comparing only the grams of an ingredient can be misleading.
Consider two invented drink mixes. Packet A has total mass 150 g and contains 99 g sugar, 30 g milk solids, 15 g almond powder and 6 g other ingredients. Packet B has total mass 200 g and contains 120 g sugar, 40 g milk solids, 30 g almond powder and 10 g other ingredients. The ingredient masses in each packet add to that packet’s total.
Packet B has more grams of sugar, but that does not immediately tell us whether it has a greater sugar proportion. Dividing each ingredient mass by its own packet mass is essential. All quantities here are measured by mass, so the comparison uses one consistent basis.
Problem
Use the two drink mixes to compare their sugar and almond proportions.
- 1.Packet A sugar percentage = (99/150) × 100 = 66%. Packet B sugar percentage = (120/200) × 100 = 60%.
- 2.Packet A almond percentage = (15/150) × 100 = 10%. Packet B almond percentage = (30/200) × 100 = 15%.
- 3.Packet B has the greater almond proportion and the smaller sugar proportion.
- 4.If you take equal 20 g portions, A supplies 0.66 × 20 = 13.2 g sugar and B supplies 0.60 × 20 = 12 g sugar.
- 5.The equal-portion comparison agrees with the percentages. The full packets still have different total masses.
| Ingredient | Packet A: percentage of 150 g | Packet B: percentage of 200 g |
|---|---|---|
| Sugar | 66% | 60% |
| Milk solids | 20% | 20% |
| Almond powder | 10% | 15% |
| Other ingredients | 4% | 5% |
| Total | 100% | 100% |
The equal 20% milk-solids shares do not mean equal milk-solids masses in the original packets. Twenty per cent of 150 g is 30 g, while twenty per cent of 200 g is 40 g. Equal percentages become equal amounts only when the wholes are also equal.
If a complete ingredient list divides the whole into non-overlapping ingredients, its exact percentages total 100%. A rounded display might total 99.9% or 100.1%. A large mismatch calls for checking the arithmetic, missing ingredients or the measurement basis.
Do not assume that every list of percentages must add to one hundred. A survey might report overlapping categories, such as people who play cricket and people who play football. One person can belong to both. Ingredient shares of a complete mixture are different because each portion of the mass is assigned only once.
Problem
A 65 g snack packet contains 60% potato, 30% oil and 10% seasoning. Find each mass.
- 1.The whole packet is 65 g. Calculate each share using that same whole.
- 2.Potato mass = 0.60 × 65 = 39 g.
- 3.Oil mass = 0.30 × 65 = 19.5 g.
- 4.Seasoning mass = 0.10 × 65 = 6.5 g.
- 5.Check: 39 + 19.5 + 6.5 = 65 g. This is consistent with a complete ingredient breakdown.
Percentage comparisons also occur in population data. If a country’s population is 175 million and a stated world total is 8.2 billion for the same year, first use the same units: 8.2 billion is 8200 million. Then the country’s share is (175/8200) × 100, approximately 2.13%.
The country count and the world total must refer to compatible dates and definitions. For a learning problem, use the stated figures consistently. Do not treat an illustrative population estimate as a timeless fact or mix a recent numerator with a much older denominator.
Problem
In a simple election with one vote per voter, 1600 votes are cast. The winner receives 500 votes. Find the winner’s share and the smallest possible number of candidates if the winner has strictly more votes than every other candidate.
- 1.Winner’s share = (500/1600) × 100 = 31.25%.
- 2.The remaining candidates share 1600 − 500 = 1100 votes.
- 3.If there were only two other candidates, each would need fewer than 500 votes, so together they could receive at most 998 votes. That is insufficient.
- 4.Three other candidates could receive 400, 400 and 300 votes. These total 1100 and are all below 500.
- 5.Therefore at least four candidates are needed under the stated conditions. A winning share need not exceed half when several candidates compete.
Choose a packet with a clearly stated total mass and an ingredient percentage. Calculate the corresponding mass and explain what quantity represents 100%. If comparing two packets, calculate percentages or use equal serving masses. Report only conclusions supported by the numbers; an ingredient percentage alone does not establish every aspect of a food’s quality.
Quiz
Group A has 18 successful attempts out of 20 and group B has 40 out of 50. Which has the greater success proportion?
Two packets each contain 20% of an ingredient. What follows?
A 150 g mix contains 99 g sugar. What is the sugar percentage?
When should exact ingredient percentages total 100%?
What must you do before comparing 175 million with 8.2 billion as a percentage?
Practice Problems
- Compare the proportions 21 out of 30 and 36 out of 45. Solution: 21/30 = 0.7 = 70%. 36/45 = 0.8 = 80%. The second share is greater.
- A 250 g mix contains 35 g nuts. Another 180 g mix contains 27 g nuts. Which has the greater nut proportion? Solution: First percentage = 35/250 × 100 = 14%. Second = 27/180 × 100 = 15%. The second has the greater proportion even though it contains fewer grams of nuts overall.
- A 120 g mixture contains 45 g ingredient A, 30 g ingredient B and the rest ingredient C. Find all percentages. Solution: C has mass 120 − 45 − 30 = 45 g. A and C each form 45/120 × 100 = 37.5%; B forms 30/120 × 100 = 25%. Their sum is 100%.
- A club has a participation rate of 80%, while another has 60%. Can you conclude the first has more participants? Give a numerical counterexample. Solution: No. If the first club has 20 members, 80% gives 16 participants. If the second has 100 members, 60% gives 60 participants. The first has the larger proportion but the smaller number.
- Using the stated figures 1.46 billion and 8.2 billion, estimate the first population’s percentage share, then calculate to one decimal place. Solution: Both values are in billions, so the common unit cancels. Since 1.46 is near one fifth of 8.2 but smaller than 1.64, estimate slightly below 20%. Calculate (1.46/8.2) × 100 ≈ 17.8049%, which rounds to 17.8%.
Key Takeaways
• Compare each part with its own whole before comparing proportions. • A greater percentage does not always mean a greater number or mass. • Use consistent units and a consistent measurement basis. • Equal ingredient percentages imply equal ingredient amounts only for equal whole amounts. • A complete non-overlapping set of exact shares totals 100%. • State only conclusions that the available proportions and totals support.