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class 8 ganita prakash chapter 1
📖 Textbook Reference: Logical Comparison
When comparing two percentage quantities, you can often use logical estimation instead of exact calculation. Visualising or estimating can help. Compute only if necessary.
Textbook Tip: Look for patterns! If the percentages are identical, the side with the larger base value wins. If the base values are identical, the side with the larger percentage wins.
Q3. Surya made 60 ml of deep orange paint, how much red paint did he use if red paint made up 3/4 of the deep orange paint?
Q4. Identify and write appropriate symbols '>', '<', '=' in the boxes. Visualising or estimating can help.
- (i) 50% of 510 50% of 515
- (ii) 37% of 148 73% of 148
- (iii) 29% of 43 92% of 110
- (iv) 30% of 40 40% of 50
- (v) 45% of 200 10% of 490
- (vi) 30% of 80 24% of 64
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For Q3: This is a straightforward fraction multiplication problem. Just multiply the total volume by the fraction given.
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For Q4: Use logical deduction to save time. If the percentage is identical, the side with the larger base value is greater. If the base is identical, the side with the larger percentage is greater.
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Q3: Find $\frac{3}{4}$ of $60$. Divide 60 by 4, then multiply by 3.
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Q4 (i) & (ii): Apply the logical rules! No math needed.
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Q4 (iv): Find $30\%$ of $40$, and $40\%$ of $50$. ($3 \times 4$ vs $4 \times 5$).
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Q4 (vi): $30\%$ of $80 = 24$. For $24\%$ of $64$, just estimate. $25\%$ ($\frac{1}{4}$) of 64 is 16. Since $24 > 16$, the left side wins.
!
Brute Forcing Everything: You can multiply out $0.37 \times 148$ and $0.73 \times 148$ in Q4(ii), but you will waste valuable time! Trust the logical properties of multiplication.
Q3: Paint Calculation
$\text{Red Paint} = \frac{3}{4} \times 60$
$= 3 \times 15 = 45\text{ ml}$
Q4: Comparisons
(i) $<$ (Same %, $510 < 515$)
(ii) $<$ (Same base, $37\% < 73\%$)
(iii) $<$ (Left side has smaller % AND smaller base)
(iv) $<$ ($12 < 20$)
(v) $>$ ($90 > 49$)
(vi) $>$ ($24 > 15.36$)
Identify the appropriate symbol '>', '<', '=' for the following pairs without using a calculator.
(a) 15% of 400 15% of 395
(b) 60% of 150 150% of 60
(a) 15% of 400 15% of 395
(b) 60% of 150 150% of 60
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For (a), use the "Same Percentage, Different Base" rule.
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For (b), remember the mathematical property from Part 14: $x\% \text{ of } y = y\% \text{ of } x$.
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In (a), which base number is larger? 400 or 395?
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In (b), calculate the left side ($6 \times 15 = 90$) and the right side ($1.5 \times 60 = 90$).
!
Don't get tricked by the large percentage (150%) in part (b)! A larger percentage does not guarantee a larger result if its base value is significantly smaller.
Logical Breakdown
Part (a): Because the percentages are exactly the same (15%), the side with the larger base number yields a higher result. Since $400 > 395$, the left side is greater.
➔ >
Part (b): Due to the commutative property of multiplication, swapping the percentage and the base always yields the exact same answer. Both sides equal 90.
➔ =
Without calculating, why can we confidently state that $29\% \text{ of } 43 < 92\% \text{ of } 110$?
A. Because 43 is an odd number and 110 is an even number.
B. Because both the percentage and the base value on the left side are smaller than the corresponding values on the right side.
C. Because $29 \times 43$ does not end in a zero.
D. Because 92% is nearly 100%.
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Visualising or estimating can help avoid unnecessary computations.
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Compare the percentages: $29 < 92$. Compare the bases: $43 < 110$.
!
Option D is a true statement, but it is not the complete mathematical reason. The base value (110) being larger is equally important.
Estimation Logic
When multiplying two terms (Percentage $\times$ Base), if both terms on the left side are smaller than both terms on the right side, the product on the left side is guaranteed to be smaller.
Ans: Option (B)
In Q4(vi), you must evaluate $30\% \text{ of } 80$ against $24\% \text{ of } 64$. What is the most efficient mental path to prove the left side is larger?
A. $3 \times 8 = 24$. Then estimate that 25% (1/4) of 64 is 16. Since 24 is much larger than 16, the left side wins.
B. Set up long multiplication for $0.24 \times 64$ to find the exact decimal.
C. Guess that because 80 is larger than 64, it must win.
D. Add the percentages together.
1
Use "benchmark" percentages like 25% ($\frac{1}{4}$) to estimate difficult numbers quickly.
Mental Math Efficiency
The left side is easy: $3 \times 8 = 24$.
The right side ($0.24 \times 64$) is tedious. However, we know that $24\%$ is slightly less than $25\%$.
$25\%$ is $\frac{1}{4}$. And $\frac{1}{4}$ of 64 is exactly 16.
Since $24$ is significantly larger than $16$, we know the left side is greater without needing the exact decimal (15.36).
Ans: Option (A)
