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 Reference: 1.3 To Compare Proportions
When comparing the ingredients of two different products, looking at the raw weight (grams) can be very misleading if the total packet sizes are different.
The Golden Rule: It is easier to compare the proportions of the ingredients if we convert them into percentages. This mathematically standardizes both products to an imaginary "100 gram" serving size, allowing a perfect 1-to-1 comparison.
Madhu and Madhav recently learnt about the importance of reading labels on processed food before purchase. They are at a shop to buy badam drink mix. They are looking at two products and wondering which has a larger share of badam. Can you figure it out? Which product uses a smaller proportion of food chemicals?
| Ingredient | Product ABC (Total 200g) | Product DEF (Total 150g) |
|---|---|---|
| Sugar | 120 g | 99 g |
| Badam | 16 g | 15 g |
| Chemicals | 4 g | 6 g |
Example calculation: DEF’s sugar content = $\frac{99}{150} \times 100 = 66\%$
1
Comparing raw ingredients across different packet sizes is a logical fallacy.
2
To find the true nutritional composition, calculate the percentage by dividing the ingredient weight by the total packet weight, and multiplying by 100.
1
Badam: For ABC, calculate $\frac{16}{200} \times 100$. For DEF, calculate $\frac{15}{150} \times 100$.
2
Chemicals: For ABC, calculate $\frac{4}{200} \times 100$. For DEF, calculate $\frac{6}{150} \times 100$.
3
Shortcut for DEF: Any fraction over 150 can be converted to a percentage by multiplying the top number by $\frac{2}{3}$. For example, $99 \times \frac{2}{3} = 66\%$.
!
Falling for the Marketing Trap: If you only look at the grams, you will mistakenly conclude Product ABC is better because "16 is bigger than 15". But Product ABC forces you to consume 50 extra grams of filler/sugar just to get that 1 extra gram of Badam!
Step 1: Badam Comparison
Product ABC (Total 200g):
Badam % = $\frac{16}{200} \times 100 = \frac{16}{2} = 8\%$
Product DEF (Total 150g):
Badam % = $\frac{15}{150} \times 100 = \frac{1}{10} \times 100 = 10\%$
Result: $10\% > 8\%$. DEF has a larger share.
Step 2: Chemical Comparison
Product ABC:
Chemicals % = $\frac{4}{200} \times 100 = \frac{4}{2} = 2\%$
Product DEF:
Chemicals % = $\frac{6}{150} \times 100 = 6 \times \frac{2}{3} = 4\%$
Result: $2\% < 4\%$. ABC uses a smaller proportion.
Ans: DEF has more badam (10%). ABC has less chemicals (2%).
Rahul is comparing two protein bars.
Bar X weighs 50g and contains 15g of protein.
Bar Y weighs 80g and contains 20g of protein.
Which bar has a higher concentration (percentage) of protein?
Bar X weighs 50g and contains 15g of protein.
Bar Y weighs 80g and contains 20g of protein.
Which bar has a higher concentration (percentage) of protein?
1
Calculate the protein percentage for both bars: $\frac{\text{Protein}}{\text{Total Weight}} \times 100$.
1
Bar X: Calculate $\frac{15}{50} \times 100$. (Multiply 15 by 2).
2
Bar Y: Calculate $\frac{20}{80} \times 100$. ($\frac{20}{80}$ simplifies nicely to $\frac{1}{4}$).
!
Bar Y has 20g, which is 5g more than Bar X. If you don't calculate the percentage, you will incorrectly assume Bar Y is the better choice for high-density protein.
Step-by-Step Calculation
Bar X:
$\frac{15}{50} \times 100 = 15 \times 2 = 30\%$ protein.
Bar Y:
$\frac{20}{80} \times 100 = \frac{1}{4} \times 100 = 25\%$ protein.
$30\% > 25\%$.
Ans: Bar X has a higher concentration.
Why is it mathematically necessary to convert the badam weights into percentages rather than comparing 16g and 15g directly?
A. Because food labels are legally required to use percentages.
B. Because the base total weights (200g and 150g) are different, making a direct comparison of raw quantities disproportionate.
C. Because 16 and 15 are too close together to compare easily.
D. Because calculating percentages removes the food chemicals from the equation.
1
A direct comparison is only valid if the items are on the exact same scale.
1
Think about it: 16g of Badam in a 2kg sack is terrible. 15g of Badam in a 20g spoon is incredible. The total size dictates the value.
!
Option A is irrelevant to the math. Option C is false; 16 is easily compared to 15. The core issue is the unequal "whole".
Logical Explanation
Percentages standardize different quantities to an imaginary "out of 100" scale.
Because Product ABC forces you to buy 200g of material just to get 16g of Badam, its density is lower than Product DEF, which gives you 15g in only a 150g serving. Standardizing to percentages reveals this truth.
Ans: Option (B)
In the textbook's example, the calculation is given as $\frac{99}{150} \times 100 = 66\%$. What is the fastest mental-math shortcut to convert ANY fraction with a denominator of 150 into a percentage?
A. Divide the numerator by 2.
B. Multiply the numerator by 1.5.
C. Multiply the numerator by $\frac{2}{3}$.
D. Subtract 50 from the numerator.
1
Look at the constant part of the equation: $\frac{100}{150}$. Simplify this fraction.
Efficiency Breakdown
The formula is $\text{Numerator} \times \frac{100}{150}$.
The fraction $\frac{100}{150}$ simplifies to exactly $\frac{2}{3}$.
Therefore, to convert any amount out of 150 into a percentage, just multiply it by 2, then divide by 3.
Example: $99 \times \frac{2}{3} = \frac{198}{3} = 66\%$.
Ans: Option (C)
