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class 8 ganita prakash chapter 1
📖 Textbook Reference: Interpreting Growth Statements
When describing how a quantity has grown, the English language gives us two distinct ways to say the exact same mathematical thing.
We can either describe the final state relative to the original state ("The new value is 165% of the old value"), or we can describe the action of growth itself ("The old value increased by 65%").
The Golden Rule: An increase of $x\%$ always means the new total is $(100 + x)\%$ of the original base.
Do the following two statements mean the same thing?
- (i) The population of this state in 1991 is 165% of that in 1961.
- (ii) The population of this state has increased by 65% from 1961 to 1991.
1
To prove two English statements are equivalent, translate them both into mathematical equations using variables.
2
Let the original 1961 population be '$p$' and the final 1991 population be '$q$'. Simplify both equations and compare the final results.
1
Statement (i): Translate "165% of $p$". Convert 165% to a decimal ($1.65$).
2
Statement (ii): Translate "$p$ increased by 65% of $p$". This means $p + 0.65p$. Add the terms together.
!
Confusing the wording: "Increased to 165%" means the new total is 165%. "Increased by 165%" means the new total is $100\% + 165\% = 265\%$. Statement (ii) says increased by 65%, which correctly totals 165%.
Algebraic Proof
Suppose $p$ is the population in 1961 and $q$ is the population in 1991.
Statement (i) implies:
$q = 165\% \text{ of } p$
$q = \frac{165}{100} \times p = 1.65p$
Statement (ii) implies:
$q = p + 65\% \text{ of } p$
$q = p + 0.65 \times p = 1.65p$
Ans: Yes, both mean the same. The 1991 population is 1.65 times that of 1961.
A financial report states: "Our company's revenue in 2024 is 120% of its revenue in 2023."
If a news outlet wants to report this as a percentage increase, how should they phrase it? "The revenue has increased by ____% from 2023 to 2024."
If a news outlet wants to report this as a percentage increase, how should they phrase it? "The revenue has increased by ____% from 2023 to 2024."
1
Use the reverse logic of Example 3. If the final state is described as a total percentage, subtract the original 100% base to find the "increase" portion.
1
Total 2024 Revenue = 120%.
2
Base 2023 Revenue = 100%.
3
Calculate: $120\% - 100\%$.
!
Do not report that revenue "increased by 120%". An increase of 120% would mean the new revenue is 220% of the old revenue!
Mathematical Translation
Let $R$ be the revenue in 2023.
The 2024 revenue is $1.20R$.
We can split this mathematically: $1.20R = 1.00R + 0.20R$.
The $1.00R$ is the original base. The $0.20R$ is the extra amount added.
Therefore, the growth is exactly $20\%$.
Ans: "The revenue has increased by 20%..."
In the textbook's algebraic proof for Statement (ii), the equation is written as $q = p + 0.65 \times p$. What does the standalone "$p$" represent in this specific equation?
A. It represents the 100% baseline population from 1961 that the 65% growth is being added to.
B. It represents an unknown random variable used to balance the equation.
C. It represents the final population in 1991.
D. It represents the 65% increase.
1
When translating "increased by", you must retain the original amount and add the increase to it.
1
If your salary increased by ₹500, your new salary is not just ₹500. It is your (Old Salary) + ₹500.
!
If you forget the standalone "$p$", the equation becomes $q = 0.65p$. This would mean the population dropped to 65% of its original size!
Algebraic Breakdown
The phrase "increased by 65%" translates to taking the original amount and adding the 65% growth to it.
Therefore, the "$p$" serves as the anchor—the 100% original population from 1961.
By factoring out $p$, the equation becomes $p(1 + 0.65) = 1.65p$.
Ans: Option (A)
If a store owner says, "The price of this bag has increased by 100%," which of the following statements is mathematically equivalent?
A. The new price is 100% of the old price.
B. The new price is 1.0 times the old price.
C. The new price is exactly double the old price.
D. The bag is now free.
1
Use the textbook's formula: $\text{New Value} = \text{Original Base} + \text{Increase}$.
Mathematical Translation
Let the old price be $P$.
The equation is: $\text{New Price} = P + (100\% \text{ of } P)$.
Since $100\% \text{ of } P$ is just $P$, the equation becomes $\text{New Price} = P + P = 2P$.
Therefore, increasing by 100% mathematically means the price has doubled.
Ans: Option (C)
