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class 9 ganita manjari chapter 10
📖 Textbook Reference: What does the 100% stacked bar tell us?
A 100% stacked bar chart helps us compare fractions/percentages inside each stack, but it completely hides the actual numbers (absolute totals) of the items being measured.
The Golden Rule: You can confidently compare fractions within the *same* garden (e.g., Summer vs Monsoon in Fatima's garden). However, you cannot compare fractions *across* different gardens (e.g., Fatima's Summer vs Naveen's Summer) without knowing their absolute total counts!
Example 8: The 100% stacked bar below compares the proportion of flowers blooming in Fatima's garden and Naveen's garden across seasons.
Which of the following statements can be logically inferred directly from this chart?
(i) In Fatima's garden, there were more blooms in the summer than in the monsoon.
(ii) In summer, Fatima's garden had more blooms than Naveen's garden.
(iii) In winter, Fatima's garden had fewer blooms than Naveen's garden.
(iv) The total number of flowers, across seasons, is the same in both gardens.
Which of the following statements can be logically inferred directly from this chart?
(i) In Fatima's garden, there were more blooms in the summer than in the monsoon.
(ii) In summer, Fatima's garden had more blooms than Naveen's garden.
(iii) In winter, Fatima's garden had fewer blooms than Naveen's garden.
(iv) The total number of flowers, across seasons, is the same in both gardens.
Which statement(s) are mathematically valid to infer?
(i) True
(ii) True
(iii) True
(iv) True
1
A 100% stacked bar chart only shows fractions of a whole. It tells you "what piece of the pie" a category is, but it tells you nothing about the "size of the pie".
2
You can safely compare pieces of the *same* pie (e.g., inside Fatima's garden only).
3
You cannot compare the absolute size of a piece from one pie to a piece from another pie (e.g., Fatima vs Naveen) without knowing their total flower counts!
1
Statement (i): Compares Summer vs Monsoon *within* Fatima's garden. Since it's the same garden, is 50% larger than 30%? Yes.
2
Statement (ii): Compares Fatima's Summer (50%) to Naveen's Summer (40%). But what if Naveen has 10,000 flowers and Fatima only has 10 flowers? 40% of 10,000 is much bigger than 50% of 10!
3
Statement (iv): Does the chart prove that the total physical number of flowers is the same? No, the chart artificially stretches both to *look* like they are exactly 100% long.
!
The "Bigger Percentage = More Stuff" Fallacy: This is the most common mistake when interpreting 100% stacked charts. A large percentage of a tiny number is still a tiny number. A small percentage of a massive number is still a massive number. You cannot compare absolute quantities across different groups using only percentages.
Detailed Evaluation of Statements
Statement (i) is True.
It compares seasons within the *same* garden (Fatima's). Within her garden, the summer fraction ($50\%$) is larger than the monsoon fraction ($30\%$). Therefore, she absolutely had more absolute flowers in summer than monsoon.
Statements (ii) and (iii) Cannot Be Inferred.
These compare across different gardens. Fatima has a higher percentage in summer ($50\%$) than Naveen ($40\%$). However, the chart does not tell us how many flowers they each had in total. Naveen's total could be massive. $40\%$ of Naveen's massive total might be more actual flowers than $50\%$ of Fatima's small total. We cannot be sure.
Statement (iv) Cannot Be Inferred.
A 100% stacked chart forces all bars to an equal visual width representing $100\%$. It intentionally erases the absolute total, meaning we have no idea if their total flower counts are the same or different.
Ans: Only statement (i) can be correctly inferred.
Suppose we peek behind the percentages and find the absolute data: Fatima's garden had exactly 100 total flowers this year, and Naveen's garden had exactly 200 total flowers this year.
Using the percentages in the chart above, calculate the absolute number of Summer flowers for each person. Does this prove Statement (ii) True or False?
Using the percentages in the chart above, calculate the absolute number of Summer flowers for each person. Does this prove Statement (ii) True or False?
1
To find absolute numbers from a 100% chart, multiply the percentage (as a decimal) by the newly revealed total.
1
Fatima Summer: Calculate $50\%$ of $100$.
2
Naveen Summer: Calculate $40\%$ of $200$.
3
Compare the two results. Did Fatima have more?
!
Assuming that a higher percentage automatically means a higher headcount. $50\%$ is larger than $40\%$, but watch what happens to the math when the baseline totals are so different!
Logical Breakdown
Fatima's Summer Blooms:
$50\%$ of $100$ total flowers $= 50$ flowers.
Naveen's Summer Blooms:
$40\%$ of $200$ total flowers $= 80$ flowers.
Comparison:
Naveen ($80$) clearly has more summer blooms than Fatima ($50$), despite having a lower percentage on the chart! This proves exactly why we cannot infer across gardens without the absolute totals.
Ans: Statement (ii) is False in this scenario.
Under what specific condition would statement (ii) ("In summer, Fatima's garden had more blooms than Naveen's garden") actually be guaranteed to be True?
A. If Fatima's bar was colored a brighter shade of yellow.
B. If we were given extra data proving that both gardens had the exact same total number of flowers for the year.
C. If the chart was converted into a pie chart.
D. It is mathematically impossible for it to ever be true.
1
Percentages can only be directly compared as absolute values if their "base" (the total amount they are calculated from) is identical.
Logical Explanation
The only reason we cannot trust the percentages across groups is because we don't know the totals.
If someone explicitly told us "Fatima has 100 total flowers and Naveen has 100 total flowers", then the totals are strictly equal.
In that identical-base scenario, Fatima's $50\%$ ($50$ flowers) is undeniably greater than Naveen's $40\%$ ($40$ flowers). If the bases are equal, the percentages perfectly reflect absolute reality.
Ans: Option (B)
Look at Naveen's bar in the chart. Which statement is mathematically valid to infer?
A. Naveen had exactly 40 flowers bloom in the Summer.
B. Within Naveen's garden, more flowers bloomed in the Summer (40%) than in the Monsoon (25%).
C. Naveen's Winter blooms (35%) were greater than Fatima's Winter blooms (20%).
D. Naveen's total garden is smaller than Fatima's garden.
1
You can only safely make absolute size comparisons when looking at fractions *within the exact same stack*.
Logical Breakdown
Option A is false because $40\%$ does not mean $40$ absolute flowers.
Option C is false because you cannot compare percentages across two different people without knowing their totals.
Option D is false because the 100% chart completely hides the total sizes.
Option B is perfectly true! Because $40\%$ and $25\%$ are both fractions of Naveen's exact same personal total, the $40\%$ piece is mathematically guaranteed to be an absolutely larger number than the $25\%$ piece.
Ans: Option (B)
