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class 8 ganita prakash chapter 1
📖 Textbook Theory: The Total Base
When a label gives the composition of ingredients as percentages, the sum of all those percentages must equal exactly 100%.
Finding Physical Weight: To find the actual physical weight of any ingredient, multiply its percentage (as a decimal) by the total physical weight of the entire item.
12. The percentage of ingredients in a 65 g chips packet is shown in the picture. Find out the weight each ingredient makes up in this packet.
1
The entire packet weighs 65 g. This is the 100% Base.
2
To find the weight of each ingredient, calculate its specific percentage out of the 65 g total.
1
Potato: Calculate $70\%$ of $65$.
2
Oil: Calculate $24\%$ of $65$.
3
Salt/Spices: Calculate $3\%$ of $65$. Convert $3\%$ to $0.03$ before multiplying!
!
Treating Percentages as Weights: A very common mistake is answering "Potato is 70 grams". 70% does not equal 70 grams unless the total packet weighs exactly 100 grams! Since the packet is only 65g, the potato weight must be less than 65g.
Calculating the Weights
Total Weight = 65 g.
1. Potato (70%)
$= \frac{70}{100} \times 65 = 0.7 \times 65 = 45.5$ g.
2. Vegetable Oil (24%)
$= \frac{24}{100} \times 65 = 0.24 \times 65 = 15.6$ g.
3. Salt (3%)
$= \frac{3}{100} \times 65 = 0.03 \times 65 = 1.95$ g.
4. Spices (3%)
$= \frac{3}{100} \times 65 = 0.03 \times 65 = 1.95$ g.
Check: $45.5 + 15.6 + 1.95 + 1.95 = 65.0$ g. Perfect!
class 8 ganita prakash chapter 1
📖 Textbook Theory: Calculating Real Discounts
When stores offer "Buy $X$, Get $Y$ Free" deals, the discount percentage is NOT based on the number of items you paid for. It is based on the Total number of items you carry out of the store.
The Golden Formula:
$\text{Discount } \% = \left( \frac{\text{Free Items}}{\text{Total Items Received}} \right) \times 100$.
13. Three shops sell the same items at the same price. The shops offer deals as follows:
• Shop A: “Buy 1 and get 1 free”
• Shop B: “Buy 2 and get 1 free”
• Shop C: “Buy 3 and get 1 free”
(i) If the price of one item is ₹100, what is the effective price per item in each shop? Arrange the shops from cheapest to costliest.
(ii) Calculate the percentage discount on the items.
(iii) Suppose you need 4 items. Which shop would you choose? Why?
• Shop A: “Buy 1 and get 1 free”
• Shop B: “Buy 2 and get 1 free”
• Shop C: “Buy 3 and get 1 free”
(i) If the price of one item is ₹100, what is the effective price per item in each shop? Arrange the shops from cheapest to costliest.
(ii) Calculate the percentage discount on the items.
(iii) Suppose you need 4 items. Which shop would you choose? Why?
1
Effective Price: Find the total money paid, and divide it by the total items received.
2
Discount Percentage: Find what fraction of your shopping cart was free. ($\frac{\text{Free}}{\text{Total}}$).
3
Choosing 4 Items: Run a specific simulation for each shop. How much total money leaves your pocket if you walk out with exactly 4 items?
1
Shop A: You pay ₹100, you get 2 items. $\frac{100}{2} = ?$
2
Shop B: You pay ₹200, you get 3 items. $\frac{200}{3} = ?$
3
For part (iii), Shop B's deal gives you 3 items for ₹200. To get a 4th item, you must pay full price (₹100). Total = ₹300. Test the other shops!
!
The Denominator Error: For Shop B (Buy 2 Get 1), many people calculate the discount as $\frac{1}{2} = 50\%$. This is mathematically false! The 100% Base is the total value of what you received. You received 3 items. Therefore the discount is $\frac{1}{3} = 33.33\%$.
Part (i) & (ii): Price and Discount
Shop A (Buy 1 Get 1):
Pay 100, Get 2. Effective Price = $\frac{100}{2} =$ ₹50/item.
Discount = $\frac{1}{2}$ items free = 50%.
Shop B (Buy 2 Get 1):
Pay 200, Get 3. Effective Price = $\frac{200}{3} \approx$ ₹66.67/item.
Discount = $\frac{1}{3}$ items free = 33.33%.
Shop C (Buy 3 Get 1):
Pay 300, Get 4. Effective Price = $\frac{300}{4} =$ ₹75/item.
Discount = $\frac{1}{4}$ items free = 25%.
Order (Cheapest to Costliest): Shop A, Shop B, Shop C.
Part (iii): Buying exactly 4 items
Shop A: Buy 2, get 2 free. Total paid = ₹200.
Shop B: Deal gives 3 items for ₹200. Need 1 more at full price (₹100). Total paid = ₹300.
Shop C: Deal gives 4 items. Total paid = ₹300.
Ans: I would choose Shop A. It only costs ₹200 for 4 items, making it the cheapest.
A local pizza place has a deal: "Buy 3 pizzas, get 2 free."
What is the exact percentage discount being offered by this deal?
What is the exact percentage discount being offered by this deal?
1
Use the formula: $\text{Discount } \% = \left( \frac{\text{Free Items}}{\text{Total Items Received}} \right) \times 100$.
1
Free Items = 2.
2
Total Items Received = $3 \text{ Paid} + 2 \text{ Free} = 5$.
3
Calculate $\frac{2}{5} \times 100$.
!
If you calculate $\frac{2}{3} = 66.6\%$, you have used the Paid items as the base. You must use the Total items as the 100% base!
Step-by-Step Translation
Total Pizzas Received = 5.
Free Pizzas = 2.
Discount % = $\left( \frac{2}{5} \right) \times 100$.
$\frac{2}{5} = 0.40 \implies 40\%$.
Ans: The discount is exactly 40%.
In Question 13, Shop C's deal is "Buy 3, Get 1 Free". Why is this a 25% discount and not a 33.3% discount?
A. Because 3 + 1 = 4, and 100 divided by 3 is 33.3.
B. Because Shop B is already 33.3%.
C. Because the "100% Base" is the total retail value of what you put in your bag (4 items). Getting 1 item free out of 4 total items means you saved $\frac{1}{4}$ of the total cost, which is 25%.
D. It depends on the price of the item.
1
A percentage discount measures how much of the Total Value you did NOT have to pay.
1
Assume each item is ₹10. Total value of 4 items is ₹40. You only paid for 3 (₹30). You saved ₹10. What is 10 out of 40?
!
Dividing by the "Paid" amount (1/3) calculates your return on investment, but it does NOT calculate the discount percentage off the retail price. Retail discounts must be measured against the total retail value.
Logical Explanation
When you walk out of Shop C, you have 4 items in your bag. This represents the total 100% value of the goods.
You only paid for 3 of them. Therefore, 1 of them was free.
The discount is the fraction of the total value that was free: $\frac{1 \text{ free}}{4 \text{ total}} = 25\%$.
Ans: Option (C)
In Question 12, salt made up 3% of the 65g chips packet, which equalled 1.95 grams. If the company created a new, massive "Family Size" packet that weighed 200 grams, but kept the recipe exactly the same, how much salt would be in the new packet?
A. 1.95 grams
B. 6 grams
C. 3 grams
D. 60 grams
1
A percentage is a relative ratio. If the recipe is the same, the ratio (3%) remains identical, but the physical weight it represents changes because the Base weight changed.
Recalculating from a New Base
The new Base is 200 g.
The salt percentage remains 3%.
Salt Weight = $3\%$ of $200$.
$= \frac{3}{100} \times 200 = 3 \times 2 = 6$ grams.
Because the Base increased, the physical weight of the 3% slice also increased proportionally.
Ans: Option (B)
