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class 8 ganita prakash chapter 1
📖 Textbook Reference: To Compare Proportions
If the maximum marks are the same, the comparison becomes easier, isn't it? For these kinds of comparisons, we need to convert both values to percentages.
The Golden Rule: Never compare raw "mistakes" or "marks lost" when the total exams are different sizes. Always convert the scores into a percentage (a common denominator of 100) to find out who actually performed better proportionally.
Eesha scored 42 marks out of 50 on an English test and 70 marks out of 80 in a Science test. Since she lost only 8 marks in English but 10 marks in Science, she thinks she has done better at English. Is Eesha correct? Which subject did she actually score better in?
1
When totals are different, comparing the "marks lost" is a logical fallacy.
2
To accurately compare performances across different scales, you must convert both scores into a percentage (a standardized score out of 100).
1
English: Calculate $\frac{42}{50} \times 100$. (Multiply the numerator by 2, since $50 \times 2 = 100$).
2
Science: Calculate $\frac{70}{80} \times 100$. Cancel the zeros to make it $\frac{7}{8} \times 100$.
3
Compare the two percentages. The higher percentage is the better performance.
!
The "Marks Lost" Trap: Losing 8 marks on a 50-mark test means you lost $\frac{8}{50}$, which is a $16\%$ loss. Losing 10 marks on an 80-mark test is $\frac{10}{80}$, which is only a $12.5\%$ loss! Eesha focused on the raw number instead of the proportion.
Step-by-Step Calculation
English Score:
$= \frac{42}{50} \times 100$
$= 42 \times 2 = 84\%$
Science Score:
$= \frac{70}{80} \times 100$
$= \frac{7}{8} \times 100 = 87.5\%$
Since $87.5\% > 84\%$, Eesha's logic was flawed.
Ans: Eesha is incorrect. She scored better in Science.
Two basketball players are comparing their season stats. Player A successfully made 15 out of 20 shots. Player B successfully made 24 out of 30 shots. Which player has the better shooting success rate?
1
Just like test scores, you cannot compare the raw number of shots made (15 vs 24) because they attempted a different number of total shots.
2
Convert both players' stats into percentages to find their true success rate.
1
Player A: $\frac{15}{20} \times 100$. ($100 \div 20 = 5$. Multiply $15 \times 5$).
2
Player B: $\frac{24}{30} \times 100$. (Cancel the zeros: $\frac{24}{3} \times 10 = 8 \times 10$).
!
Player B made 9 more shots than Player A, so it's tempting to immediately say Player B is better. But you must verify the math! Player B also *missed* more shots (6) than Player A (5). Only percentages reveal the truth.
Step-by-Step Calculation
Player A:
$= \frac{15}{20} \times 100 = 15 \times 5 = 75\%$
Player B:
$= \frac{24}{30} \times 100 = \frac{240}{3} = 80\%$
$80\% > 75\%$
Ans: Player B has the better success rate.
In the textbook example, Eesha thought she did better in English because she only lost 8 marks, compared to losing 10 marks in Science. Why is this logic mathematically flawed?
A. Because Science is generally a harder subject than English.
B. Because losing 8 marks out of 50 is a larger proportional penalty (16%) than losing 10 marks out of 80 (12.5%).
C. Because 42 is smaller than 70.
D. Because 8 and 10 are both even numbers.
1
Raw numbers cannot be compared directly if their "wholes" (denominators) are different sizes.
1
Losing 1 mark on a 2-mark quiz is a 50% failure. Losing 10 marks on a 1000-mark exam is a 1% failure. The size of the test dictates the impact of the loss.
!
Option C is true (42 is smaller than 70), but that doesn't prove anything. If the test was out of 45, scoring 42 would be incredible! The relationship between the score and the total is what matters.
Logical Explanation
When evaluating performance, the proportion (percentage) is what matters, not the raw count of errors.
Losing 8 out of 50 translates to a 16% deduction, leaving an 84% score. Losing 10 out of 80 is a 12.5% deduction, leaving an 87.5% score.
Ans: Option (B)
In the calculation $\frac{70}{80} \times 100$, what is the most efficient mental-math pathway to find the final percentage?
A. Multiply $70 \times 100$ to get $7000$, then do long division $7000 \div 80$.
B. Add 20 to the numerator and denominator to make it $\frac{90}{100}$.
C. Cancel the zeros to get $\frac{7}{8}$. Knowing that $\frac{1}{8} = 12.5\%$, multiply $7 \times 12.5$.
D. Subtract 10 from 100.
1
Efficiency in percentage math relies on simplifying fractions and memorizing common fractional equivalents (like $\frac{1}{8} = 12.5\%$).
Efficiency Breakdown
Option A works, but long division is slow and prone to errors.
Option B is a mathematical violation (you cannot add numbers to force an equivalent fraction).
Option C uses simplification. $\frac{70}{80}$ simplifies to $\frac{7}{8}$. If a student knows that $\frac{1}{8}$ is exactly half of $\frac{1}{4}$ (which is 25%), they know $\frac{1}{8} = 12.5\%$.
$7 \times 12.5 = 87.5\%$.
Ans: Option (C)
