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class 8 ganita prakash chapter 1
📖 Textbook Reference: Modeling Percentage Problems
Bar models visually break down the relationship between percentages and actual values. The 100% mark always represents the "base" or total original value.
If a bar extends past the 100% mark, the corresponding percentage is greater than 100%.
Textbook Tip: Estimate first before making any computations to solve the following questions. Try different methods including mental computations.
Find the missing numbers using mental computation and proportional reasoning.
(i) 100% = 60. Find the value for 20%.
(ii) 100% = 75. The value 90 corresponds to what percentage?
(iii) If 140 represents 140%, what value corresponds to 100%?
(i) 100% = 60. Find the value for 20%.
(ii) 100% = 75. The value 90 corresponds to what percentage?
(iii) If 140 represents 140%, what value corresponds to 100%?
1
These problems test the three core variations of percentage math: finding the part (i), finding the rate or percentage (ii), and finding the base or whole (iii).
2
Proportions are your best friend here. Set up the equation $\frac{\text{Percentage}}{100} = \frac{\text{Part}}{\text{Base}}$.
1
For (i): $20\%$ is exactly $\frac{1}{5}$ of $100\%$. Divide 60 by 5.
2
For (ii): The base is 75. The part is 90. Calculate $\frac{90}{75} \times 100$.
3
For (iii): If $140\%$ equals the number $140$, what does $100\%$ equal? (Divide by 1.4).
!
Assuming 100% is always the largest number: In part (ii), the value 90 is greater than the base of 75. Do not try to flip the fraction to $\frac{75}{90}$. If the value exceeds the base, the percentage must be over 100%.
Step-by-Step Calculation
Problem (i): Find the part
$20\%$ of $60 = \frac{1}{5} \times 60 = 12$
Problem (ii): Find the percentage
$\text{Percentage} = \frac{90}{75} \times 100$
$= \frac{6}{5} \times 100 = 120\%$
Problem (iii): Find the base
$140\% \text{ of Base} = 140$
$1.4 \times \text{Base} = 140$
$\text{Base} = \frac{140}{1.4} = 100$
Ans: (i) 12 (ii) 120% (iii) 100
A local school currently has 500 students, representing 100% capacity.
(A) If 10% of the students are absent today, how many students are absent?
(B) If the school expands and accepts 600 students next year, what percentage of the original capacity is the new population?
(A) If 10% of the students are absent today, how many students are absent?
(B) If the school expands and accepts 600 students next year, what percentage of the original capacity is the new population?
1
Part A requires finding the "part" from the "whole". Multiply the decimal percentage by the base.
2
Part B requires finding the "percentage". Create the fraction $\frac{\text{New}}{\text{Old}}$ and multiply by 100.
1
(A) $10\%$ is $\frac{1}{10}$. What is $\frac{1}{10}$ of $500$?
2
(B) The base is 500. The new value is 600. Calculate $\frac{600}{500} \times 100$.
!
In part B, do not calculate the percentage of the extra students ($\frac{100}{500} = 20\%$) unless the question asks for the "percentage increase". It asks for the percentage of the new population compared to the old, so the answer must be over 100%.
Step-by-Step Calculation
Part (A):
Absent = $10\%$ of $500 = 0.1 \times 500 = 50$ students.
Part (B):
Percentage = $\frac{600}{500} \times 100$
$= \frac{6}{5} \times 100 = 120\%$
Ans: (A) 50 absent (B) 120% capacity
In a bar model, if the 100% mark corresponds to the number 75, how do you mathematically find the percentage for a value of 90?
A. Subtract 75 from 90 and append a % sign (15%).
B. Set up the fraction $\frac{90}{75}$ and multiply by 100.
C. Set up the fraction $\frac{75}{90}$ and multiply by 100.
D. Add 15 to 100%.
1
To find a percentage, you place the "target" or "new value" over the "base" value (which is 100%), and multiply by 100.
1
The base is 75. The target value is 90.
!
Option A is a dangerous trap! The difference is indeed 15, but 15 is not 15%. 15 is $20\%$ of the base 75! Option D is mathematically a coincidence that works here, but is not the proper formula for finding a percentage.
Logical Explanation
The formula is $\frac{\text{Value}}{\text{Base}} \times 100$.
Since the base (100% mark) is 75, it goes in the denominator: $\frac{90}{75} \times 100 = 120\%$.
Ans: Option (B)
In a bar model, what does a coloured section extending past the 100% mark signify?
A. An error in the data collection.
B. A value that is mathematically greater than the original base quantity, representing a percentage > 100%.
C. A value that must be subtracted from the total.
D. A separate group of data that has no relation to the first group.
1
A bar model represents the base "whole" as exactly 100%.
Visual Interpretation
If a value exceeds the original whole (for instance, a harvest of 650kg compared to last year's 260kg), the bar must visually stretch beyond the 100% line.
This visually confirms that the new percentage is greater than 100%.
Ans: Option (B)
