Rotate Your Phone!
For the best experience reading the mathematical steps, please rotate your device to landscape mode.
class 8 ganita prakash chapter 1
📖 Textbook Theory: The Golden 1% Rule
When estimating percentages of large numbers (like global populations), the smartest trick is to find exactly what 1% equals.
Unit Conversion: 1 Billion = 1,000 Million. Therefore, 8.2 Billion = 8,200 Million.
To find 1%, simply divide by 100. $8200 \div 100 = 82$. Therefore, 1% of the world is exactly 82 Million people. You can use this benchmark to estimate any country instantly!
2. The population of the world in 2025 is about 8.2 billion. The populations of some countries in 2025 are given. Match them with their approximate percentage share of the worldwide population.
(Options to choose from: 13%, 8%, 18%, 10%, 1%, 35%, 2%, 0.1%)
(Options to choose from: 13%, 8%, 18%, 10%, 1%, 35%, 2%, 0.1%)
1
Convert all numbers to the same unit. Let's use Millions. (1 Billion = 1000 Million).
2
World = $8200$ Million. To estimate quickly, establish that $1\% = 82$ Million. Use this "block" of 82 to guess the others.
1
India: $1.46 \text{ B} = 1460 \text{ M}$. Calculate $\frac{1460}{8200} \times 100$. (Or estimate: $10\% = 820$. $20\% = 1640$. 1460 is between them).
2
Bangladesh: $175 \text{ M}$. If $1\% = 82$, what is $2\%$?
3
USA: $347 \text{ M}$. If $1\% = 82$, how many times does 82 fit into 347? (Try $82 \times 4$).
!
Ignoring Units: If you divide India's 1.46 directly by 8200, you will get a wildly incorrect decimal. You MUST convert 1.46 Billion into 1460 Million before dividing by 8200 Million.
Estimation & Matching
Base (100%) = $8.2 \text{ Billion} = 8200 \text{ Million}$.
Therefore, $1\% \approx 82 \text{ Million}$.
Germany: 83 M. Very close to 82. $\rightarrow$ Approx 1%
Bangladesh: 175 M. Since $2\% = 164$, it is slightly more than 2%. Looking at the options... $\rightarrow$ Approx 2%
USA: 347 M. Since $82 \times 4 = 328$, it is slightly over 4%. The textbook options might be missing "4%", but mathematically $\frac{347}{8200} \approx 4.2\%$. Let's label it correctly: $\rightarrow$ Approx 4.2%
India: 1.46 B = 1460 M.
$\frac{1460}{8200} \times 100 \approx 17.8\%$. Looking at the options, the closest match is 18%. $\rightarrow$ Approx 18%
class 8 ganita prakash chapter 1
📖 Textbook Theory: The Math of Taxes
Taxes like GST (Goods and Services Tax) are added on top of the original price of an item. This makes GST mathematically identical to a "Percentage Increase".
Two Ways to Calculate:
1. Find the tax amount ($18\%$ of Base), and ADD it to the Base.
2. Recognize that the Final Price is $100\% + 18\% = 118\%$ of the Base, and multiply directly by $1.18$.
3. The price of a mobile phone is ₹8,250. A GST of 18% is added to the price. Which of the following gives the final price of the phone including the GST?
1
A GST of 18% means the final price is $100\% \text{ (Base)} + 18\% \text{ (Tax)} = 118\%$ of the Base Price.
2
We must scan the options for mathematical statements that accurately calculate either $(118\% \times 8250)$ or $(8250 + \text{Tax Amount})$.
1
How do you write 118% as a decimal multiplier? (Look at option v).
2
How do you write the "Tax Amount" mathematically? It is $18\% \text{ of } 8250$, which is $8250 \times 0.18$. (Look at option vi).
!
Adding raw numbers: Options (i) and (iii) try to add raw numbers ($+18$ and $+\frac{18}{100}$). You cannot add a raw number to a currency amount unless you have calculated the percentage OF the base first! 18% is not ₹18.
Evaluating the Options
Base Price = 8250. Tax = 18% of 8250.
Logic 1: Expansion
$\text{Final Price} = \text{Base} + \text{Tax Amount}$
$\text{Final Price} = 8250 + (8250 \times 0.18)$. This matches Option (vi).
Logic 2: Direct Multiplier
$\text{Final Price} = 118\% \text{ of } 8250$.
$\text{Final Price} = 1.18 \times 8250$. This matches Option (v).
Ans: Options (v) and (vi) are correct.
A restaurant bill for a family dinner comes to ₹2,500 (before tax).
A service charge of 10% is added. Then, a GST of 5% is added on top of that new total.
Which single mathematical expression calculates the final amount paid?
A service charge of 10% is added. Then, a GST of 5% is added on top of that new total.
Which single mathematical expression calculates the final amount paid?
1
This combines the GST logic from Q3 with the "Successive Percentage" logic from previous lessons.
2
You must apply the first multiplier (110%) to the base, and then apply the second multiplier (105%) to that result.
1
A 10% increase makes the multiplier $1.10$.
2
A 5% increase makes the multiplier $1.05$.
!
The Addition Trap: Do NOT say "$10\% + 5\% = 15\%$, so the multiplier is $1.15$." Because the 5% tax is calculated on the new total (which includes the service charge), it acts like compound interest. The true multiplier is slightly higher than 1.15.
Step-by-Step Translation
Base Bill = 2500.
Step 1: Add Service Charge (10%)
New Total = $2500 \times 1.10$.
Step 2: Add GST (5%) on the new total
Final Amount = $(2500 \times 1.10) \times 1.05$.
Ans: $2500 \times 1.10 \times 1.05$
(This evaluates to ₹2,887.50)
In Question 2, we found that 1.46 Billion is approx 18% of 8.2 Billion. If the world population drops to exactly 4.1 Billion, but India's population remains 1.46 Billion, what happens to India's percentage share?
A. It remains 18% because India's population didn't change.
B. It drops to 9% because the world population dropped by half.
C. It doubles to approx 36%. Because the 100% baseline shrank by half, India's absolute number now takes up twice as much "space" in the pie chart.
D. Percentages cannot be calculated if the world population drops.
1
A percentage is a relationship between a Part and a Whole.
1
$\text{Percentage} = \frac{\text{Part}}{\text{Whole}}$. What happens to a fraction when the denominator gets smaller?
!
Option A is a common misunderstanding. People think percentages are fixed properties of a number. They are not! They are purely relational.
Logical Explanation
The formula is $\frac{1.46}{\text{World Total}}$.
If the World Total gets cut in half (from 8.2 to 4.1), the denominator of our fraction is halved.
Mathematically, dividing by a number half the size results in an answer twice as big! $\frac{1.46}{4.1} \approx 35.6\%$.
Ans: Option (C)
In Question 3, Option (iv) suggests the final price is "$8250 \times 18$". If someone actually paid this amount at the store, what "GST percentage" did they effectively pay?
A. 18%
B. 1700% (A 17x markup).
C. 1800%
D. 1.8%
1
Remember how decimal multipliers work. A multiplier of $1.18$ means an $18\%$ increase. A multiplier of $2.00$ means a $100\%$ increase (doubling).
Reverse Engineering the Multiplier
The expression is $8250 \times 18$. The multiplier is $18.00$.
The formula for a multiplier is $(1 + \frac{R}{100})$.
So, $1 + \frac{R}{100} = 18$.
$\frac{R}{100} = 17 \implies R = 1700$.
If you multiply a price by 18, you are paying a 1700% tax! This is why it is critical to convert 18% into $0.18$ before doing math.
Ans: Option (B)
