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class 8 ganita prakash chapter 1
📖 Textbook Reference: 1.2 Percentage of Some Quantity
Before starting this section, the textbook discusses the need for percentages. It uses the example of a biscuit factory comparing two varieties with sugar fractions $\frac{9}{34}$ and $\frac{13}{45}$. It's hard to compare them directly. But converting them to 26.47% and 28.88% standardizes the base to 100, making comparison instant!
However, percentages represent proportions. It would be inappropriate to compare just the percentages when they are referring to different quantities or wholes.
The Golden Rule: To compare amounts accurately across different wholes, you must apply the percentage to the total weight to find the exact numerical value of the part.
Madhu and Madhav each ate biscuits of a different variety. Madhu's biscuits had 25% sugar, while Madhav's had 35% sugar. Suppose Madhu ate 120 g of biscuits and Madhav ate 95 g of biscuits. Who consumed more sugar?
1
Percentages cannot be compared directly if they are referring to different quantities (different wholes).
2
To find out who consumed more sugar, you must calculate the exact grams of sugar each person ate by applying their percentage to the total weight of their biscuits.
1
Find Madhu's sugar: Calculate $25\%$ of $120\text{ g}$. You can do this by converting $25\%$ to $\frac{25}{100}$ (or $\frac{1}{4}$) and multiplying by $120$.
2
Find Madhav's sugar: Calculate $35\%$ of $95\text{ g}$. Multiply $95$ by $\frac{35}{100}$.
3
Compare the two final gram amounts.
!
The "Bigger Percentage Wins" Trap: Just because 35% > 25% doesn't automatically mean Madhav ate more sugar! If Madhu had eaten 200g of biscuits, 25% of 200g is 50g, making her the winner. You *must* calculate the exact values to be sure.
Step 1: Calculate Madhu's Sugar
Madhu ate 120 g of biscuits having 25% sugar.
$\text{Sugar} = \frac{25}{100} \times 120$
$= \frac{1}{4} \times 120$
$= 30\text{ g}$
Step 2: Calculate Madhav's Sugar
Madhav ate 95 g of biscuits having 35% sugar.
$\text{Sugar} = \frac{35}{100} \times 95$
$= 0.35 \times 95$
$= 33.25\text{ g}$
Step 3: Compare
$33.25\text{ g} > 30\text{ g}$.
Ans: Madhav consumed more sugar.
In a school election, Candidate A gets 60% of the votes from Class X (which has 40 students). Candidate B gets 45% of the votes from Class Y (which has 60 students). Who received more actual votes?
1
This uses the exact same logic as Example 1. Because the classes are different sizes, a lower percentage in a larger class might equal more votes than a higher percentage in a smaller class.
1
Find Candidate A's votes: Calculate $60\%$ of $40$.
2
Find Candidate B's votes: Calculate $45\%$ of $60$.
3
Compare the two whole numbers.
!
Don't assume Candidate A won just because 60% is a landslide majority! It's a majority of a small pond. Calculate the real numbers.
Step-by-Step Calculation
Candidate A votes = $60\%$ of $40$
$= \frac{60}{100} \times 40 = 24\text{ votes}$
Candidate B votes = $45\%$ of $60$
$= \frac{45}{100} \times 60 = 27\text{ votes}$
$27 > 24$.
Ans: Candidate B received more actual votes.
According to the textbook, why is it mathematically inappropriate to conclude Madhav ate more sugar just by looking at the percentages (35% vs 25%)?
A. Because the percentages refer to different total quantities (wholes).
B. Because 35% is not a multiple of 25%.
C. Because sugar content cannot be accurately measured in grams.
D. Because Madhu's biscuits might weigh less than 100 grams.
1
Percentages represent fractional quantities or proportions. It would be inappropriate to compare just the percentages when they are referring to different quantities or wholes.
1
Think about it: 100% of a grape is still smaller than 10% of a watermelon! The "whole" matters.
!
Option D is a true statement about a hypothetical scenario, but it doesn't explain the underlying mathematical principle of why direct percentage comparison fails.
Logical Explanation
A percentage is a ratio. To compare amounts accurately, you must apply the percentage to the total weight (the "whole") to find the exact numerical value of the part.
Ans: Option (A)
In the proportional reasoning method (Method 2) used to solve Madhu's sugar intake, what does the equation $\frac{25}{100} = \frac{s}{120}$ represent?
A. 25 grams of sugar divided by 120 grams of biscuits.
B. The ratio of 25 parts per 100 is equivalent to '$s$' parts per 120.
C. 100 grams of biscuits contains '$s$' grams of sugar.
D. 25% of 100 is equal to '$s$'% of 120.
1
A proportion sets two ratios equal to each other. The left side is the percentage ratio ($\frac{25}{100}$), and the right side is the actual weight ratio ($\frac{\text{sugar}}{\text{total weight}}$).
Equation Breakdown
The left side ($\frac{25}{100}$) states that there are 25 grams of sugar for every 100 grams of biscuits.
The right side ($\frac{s}{120}$) scales this ratio up to the actual total weight of 120 grams, where '$s$' is the unknown amount of sugar.
Ans: Option (B)
