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class 8 ganita prakash chapter 1
📖 Textbook Reference: Comparing Bases & Patterns
When comparing percentages from different bases (like days vs. weeks), you must convert them to the same unit before comparing the final values.
Additionally, real-world math often requires assumptions. For instance, if workers complete 20% of a job in 18 days, predicting the end date relies on the assumption that their working speed remains perfectly constant.
- 7. Is 10% of a day longer than 1% of a week? Create such questions and challenge your peers.
- 8. Mariam's farm has a peculiar bull. One day she gave the bull 2 units of fodder and it ate 1. Next day, 3 units and it ate 2. This continued, and on the 99th day she gave 100 units and it ate 99. Represent these quantities as percentages. What do you observe?
- 9. Workers in a coffee plantation take 18 days to pick berries in 20% of the plantation. How many days will they take to complete the entire plantation? Why is this assumption necessary?
1
Q7: To compare periods of time, standardize them into the same unit (hours) before calculating the percentage.
2
Q8: This explores the mathematical sequence of fractions $\frac{n}{n+1}$. As the numbers get larger, the value approaches 1 (or 100%).
3
Q9: Use proportionality ($\frac{20}{100} = \frac{18}{\text{Total}}$) to scale the days.
1
Q7: 1 day = 24 hours. 1 week = $24 \times 7 = 168$ hours.
2
Q8: Calculate $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{99}{100}$ as percentages.
3
Q9: If 20% takes 18 days, then 100% (which is $5 \times 20\%$) will take $5 \times 18$ days.
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Comparing Raw Numbers: In Q7, do not say "10% is larger than 1%, so the day is longer." The base values are different! You must calculate the actual hours to prove it.
Q7 Solution
1 Day = 24h. $\rightarrow 10\% \text{ of } 24 = 2.4\text{ hours}$.
1 Week = 168h. $\rightarrow 1\% \text{ of } 168 = 1.68\text{ hours}$.
Yes, 10% of a day is longer. *(Peer Challenge Idea: Is 50% of a minute longer than 1% of an hour?)*
Q8 Solution
Day 1: $\frac{1}{2} = 50\%$. Day 2: $\frac{2}{3} = 66.6\%$.
Day 3: $\frac{3}{4} = 75\%$. Day 99: $\frac{99}{100} = 99\%$.
Observation: As the quantities increase, the percentage eaten strictly increases, getting closer and closer to 100%.
Q9 Solution
$20\% \rightarrow 18\text{ days}$.
$100\% = 5 \times 20\% \rightarrow 5 \times 18 = 90\text{ days}$.
Assumption: We assume the workers maintain the exact same speed/rate of picking every day. If they slow down, it will take longer.
(A) A construction crew paves 25% of a road in 12 days. Assuming a constant rate of work, how long will it take to pave the entire road?
(B) Is 5% of a standard month (30 days) longer than 20% of a week (7 days)?
(B) Is 5% of a standard month (30 days) longer than 20% of a week (7 days)?
1
For (A), scale the percentage. 100% is four times larger than 25%.
2
For (B), convert both time periods into the same unit (days) before calculating the percentages.
1
(A) Multiply 12 days by 4.
2
(B) Find 5% of 30 days. Then find 20% of 7 days.
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In part A, do not say 48 days more. The question asks how long it will take to pave the entire road, which is 48 days total. (If it asked for the remaining time, it would be 36 days).
Step-by-Step Calculation
Part (A):
If $25\% = 12$ days, then $100\% = 12 \times 4 = 48$ days.
Part (B):
Month: $5\% \text{ of } 30 = 0.05 \times 30 = 1.5$ days.
Week: $20\% \text{ of } 7 = 0.20 \times 7 = 1.4$ days.
$1.5 \text{ days} > 1.4 \text{ days}$.
Ans: (A) 48 days (B) Yes, 5% of a month is longer.
In Q9 regarding the coffee plantation workers, why is the mathematical assumption (that the rate of work stays the same) necessary to solve the problem?
A. Because 100% cannot be calculated without it.
B. Because real-world work rates can fluctuate due to weather or fatigue, meaning a linear multiplication ($18 \times 5$) would be inaccurate without assuming constant speed.
C. Because coffee berries grow at a constant rate.
D. Because 20% is exactly one-fifth of the plantation.
1
Mathematics creates perfect, linear models, but the real world is messy.
1
If workers get tired on day 19, will the next 20% of the plantation still take exactly 18 days? Probably not!
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Option D is a true mathematical statement, but it does not explain why an assumption about human workers is necessary.
Logical Explanation
To scale 18 days up by a factor of 5 to get 90 days, the math assumes that every 20% chunk of work takes exactly 18 days. This is a linear rate of work.
In reality, workers might slow down, so explicitly stating the assumption justifies the simple mathematical calculation.
Ans: Option (B)
In Q8 (the peculiar bull), the fractions follow the pattern $\frac{1}{2}, \frac{2}{3}, \frac{3}{4} ... \frac{n}{n+1}$. As the values get larger, what mathematical limit is the percentage approaching?
A. 0%
B. 50%
C. 99%
D. 100%
1
This problem explores limits in fractions. As the numerator and denominator grow larger with a constant difference of 1, the ratio gets closer to 1.
Pattern Recognition
Let's look at the percentages: 50%, 66.6%, 75%, 80%, 83.3%... 99%.
If she fed the bull 1000 units and it ate 999, the percentage is 99.9%.
The larger the numbers, the closer the fraction $\frac{n}{n+1}$ gets to a full 1, which represents $100\%$.
Ans: Option (D)
