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class 8 ganita prakash chapter 1
📖 Textbook Theory: Proportions & Logic
This question combines standard percentage calculation with a brilliant logical puzzle.
To find the minimum number of candidates in an election where the "winner" got a specific number of votes, we must assume that the remaining votes are distributed as evenly as possible among the losers. If an even distribution results in a loser getting MORE votes than the winner, the scenario is impossible, and we must add another candidate!
4. 1600 people voted in an election and the winner got 500 votes. What percent of the total votes did the winner get? Can you guess the minimum number of candidates who stood for the election?
1
Percentage: The total base is 1600. Divide the winner's amount (500) by the total (1600) and multiply by 100.
2
Logic: If the winner got 500, it means no other single candidate can get more than 499 votes.
1
Calculate $\frac{500}{1600} \times 100$. Cancel the zeros first!
2
There are $1600 - 500 = 1100$ remaining votes. If there were only 2 losing candidates, they would share the 1100 votes. What is $1100 \div 2$? Does that person beat the winner? Keep adding candidates until the losers' share drops below 500!
!
Assuming 3 Candidates: Many students assume a 3-person race. But if there are 3 candidates total (1 winner, 2 losers), the 2 losers share 1100 votes. Even if they split it perfectly, they get 550 each. 550 > 500, which means the "winner" wouldn't have won! The math proves this is impossible.
Part 1: Percentage Calculation
Total Votes = 1600. Winner's Votes = 500.
Winner \% = $\left(\frac{500}{1600}\right) \times 100$
$= \frac{5}{16} \times 100 = \frac{500}{16} = 31.25\%$.
Part 2: Logical Deduction
Remaining votes for all other candidates = $1600 - 500 = 1100$.
Since the winner got 500, no other candidate can get 500 or more.
If there were 2 other candidates (3 total):
They would average $1100 \div 2 = 550$ votes each. (Impossible, 550 > 500).
If there were 3 other candidates (4 total):
They would average $1100 \div 3 = 366.6$ votes each. (Possible! 366 < 500).
Ans: Winner got 31.25%. Minimum candidates = 4.
class 8 ganita prakash chapter 1
📖 Textbook Theory: Rate of Inflation
Inflation is the rate at which prices increase over time. Mathematically, it is identical to Percentage Increase.
The Golden Rule: The chronological starting price (the older price) is ALWAYS the 100% denominator.
5. The price of 1 kg of rice was ₹38 in 2024. It is ₹42 in 2025. What is the rate of inflation? (Inflation is the percentage increase in prices.)
1
Calculate the raw difference (the increase in price) from 2024 to 2025.
2
Divide the raw increase by the Original Starting Price (the 2024 price) and multiply by 100.
1
Increase = $42 - 38 = 4$.
2
Set up the fraction: $\frac{4}{38}$. Simplify to $\frac{2}{19}$ before multiplying by 100.
!
The Wrong Base: Dividing 4 by 42 will give you roughly 9.5%. This is wrong! You cannot divide by the *new* price. The growth happened *on top of* the old price (38), so 38 is the denominator.
Inflation Calculation
Price in 2024 (Original Base) = ₹38.
Price in 2025 = ₹42.
Increase = $42 - 38 =$ ₹4.
Rate of Inflation = $\left( \frac{\text{Increase}}{\text{Original Base}} \right) \times 100$
$= \frac{4}{38} \times 100$
Simplify fraction: $= \frac{2}{19} \times 100 = \frac{200}{19}$
$= 10.526...\%$
Ans: The rate of inflation is 10.53% (approx).
2000 students voted for the school captain. The winning candidate received 800 votes.
(A) What percentage of the votes did the winner get?
(B) What is the MINIMUM number of candidates who stood in this election, assuming the winner truly got the most votes?
(A) What percentage of the votes did the winner get?
(B) What is the MINIMUM number of candidates who stood in this election, assuming the winner truly got the most votes?
1
This mirrors the textbook election logic perfectly. Apply the same reasoning!
1
Part (A): Divide 800 by 2000 and multiply by 100.
2
Part (B): Remaining votes = $2000 - 800 = 1200$. Start dividing 1200 by 2, then by 3, etc., until the average for the losers falls below 800.
!
If there was only 1 losing candidate (2 candidates total), the loser would get all 1200 remaining votes. That means the "loser" beat the winner (1200 > 800)!
Step-by-Step Logic
Part (A):
Winner \% = $\frac{800}{2000} \times 100 = \frac{8}{20} \times 100 = 40\%$.
Part (B):
Remaining votes for losers = $2000 - 800 = 1200$.
If 1 loser (2 total candidates): Loser gets 1200. (1200 > 800, impossible).
If 2 losers (3 total candidates): Average loser gets $1200 \div 2 = 600$. (600 < 800, possible!).
Therefore, minimum 3 candidates are needed (e.g., Winner 800, Loser1 600, Loser2 600).
Ans: (A) 40% (B) 3 candidates
In the textbook election question, there were 1100 remaining votes. Why does dividing 1100 by 2 mathematically prove that there MUST be at least 4 total candidates?
A. Because 1100 is an odd number and cannot be divided easily.
B. Because elections usually have 4 candidates.
C. If there were only 3 candidates, the 2 losers would share 1100 votes. Even an equal split means one gets 550. Since 550 > 500, a "loser" would have beaten the winner, which contradicts the premise.
D. Because 1600 divided by 4 is 400.
1
This is a classic "Proof by Contradiction."
1
Assume there are only 3 candidates. See what happens to the math.
!
If you don't divide the remaining votes among the hypothetical losers, you miss the logic puzzle entirely.
Logical Explanation
The problem states the winner got exactly 500 votes. Therefore, no other single person can have 500 or more.
If the 1100 remaining votes were split between only 2 other people, the lowest maximum score one of them could get is 550 (a 550-550 tie).
Since 550 is greater than the winner's 500, the 3-candidate scenario breaks the rules of reality. Therefore, we must add a 4th candidate to dilute the losers' votes.
Ans: Option (C)
If the price of rice fell from ₹42 back down to ₹38, would the "percentage decrease" be exactly the same as the 10.53% inflation rate calculated in Question 5?
A. Yes, because the difference in price (₹4) is the same.
B. No. The percentage decrease would be calculated using the larger ₹42 as the base denominator, resulting in a smaller percentage (roughly 9.5%).
C. Yes, percentages always reverse symmetrically.
D. No, it would be exactly double.
1
The Golden Rule: Percentage change is always calculated on the starting base value.
The Asymmetry of Percentages
When price goes from ₹38 to ₹42, the starting base is ₹38. Calculation: $\frac{4}{38} \approx 10.53\%$.
When price goes from ₹42 to ₹38, the starting base is now ₹42!
Calculation: $\frac{4}{42} = \frac{2}{21} \approx 9.52\%$.
Because the denominator changed, the percentage must change. Percentages do not reverse symmetrically!
Ans: Option (B)
