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class 8 ganita prakash chapter 1
📖 Textbook Reference: Modeling Percentage Problems
When a specific percentage of a total quantity corresponds to a specific value, you can find the remaining quantity or the total quantity using proportional reasoning.
Textbook Tip: Drawing rough diagrams can help understand the given situation better and make it easier to think further about the problem.
There are multiple ways to solve these: The Unitary Method (scaling up or down), Proportions, or setting up an Algebraic Equation.
A cyclist cycles from Delhi to Agra and completes 40% of the journey. If he has covered 92 km, how many more kilometres does he have to travel to reach Agra?
1
If a certain percentage of a total quantity corresponds to a specific value, you can find the remaining quantity using proportional reasoning or the unitary method.
2
Read carefully: The question asks for the remaining distance, not the total distance between Delhi and Agra.
1
Method 1 (Scaling): If 40% is 92 km, what is 20%? (Divide by 2). Once you know 20%, multiply by 3 to find the remaining 60%.
2
Method 2 (Algebra): Set up the equation for the total distance $d$: $\frac{40}{100} = \frac{92}{d}$. Solve for $d$, then subtract 92 to find the remaining distance.
!
Stopping at the Total: A very common mistake is calculating the total distance (230 km) and writing that as the final answer. The question asks "how many more kilometres does he have to travel".
Method 1: Unitary Scaling
40% is 92 km.
Therefore, 20% is 46 km (dividing by 2).
He needs to travel the remaining 60%.
This makes 60% to be $92 + 46 = 138$ km.
Method 2: Algebraic Total
Let $d$ be the total distance.
$\frac{40}{100} = \frac{92}{d}$
$d = 92 \times \frac{100}{40} = 230$
Remaining distance = $230 - 92 = 138$ km.
Ans: He has to travel 138 more kilometres.
A runner completes 30% of a marathon. If she has run 12 km, how many more kilometres does she have left to run to finish the race?
1
Use the scaling method (Unitary method). Find what a smaller, friendlier percentage equals, and scale it up to the remaining percentage.
1
If 30% equals 12 km, what does 10% equal? (Divide by 3).
2
If she ran 30%, she has 70% left to run.
3
Multiply your 10% value by 7 to find the 70% value.
!
Do not find the total distance (40 km) and stop! The question asks how many more kilometres she has left. You must subtract the 12 km she already ran.
Step-by-Step Calculation
30% = 12 km.
Divide by 3: 10% = 4 km.
Remaining distance is 70% ($100\% - 30\%$).
Multiply 10% value by 7: $4 \times 7 = 28$ km.
Ans: She has 28 kilometres left to run.
Why does the "Scaling Method" (Method 1) work so efficiently for the Delhi to Agra problem?
A. Because 92 is an even number.
B. Because both 40% and 60% are perfect multiples of 20%.
C. Because the total distance is exactly 100 km.
D. Because scaling avoids the use of fractions entirely.
1
Scaling relies on finding a common factor between the percentage you have and the percentage you want.
1
The cyclist completed 40% and has 60% remaining.
2
What is the greatest common divisor of 40 and 60? It's 20.
!
Scaling doesn't "avoid fractions". Finding 20% by dividing 40% in half is essentially multiplying by the fraction 1/2.
Logical Explanation
Since the known value is 40% and the target value is 60%, calculating a common "stepping stone" like 20% makes the math mental and instantaneous.
If the percentages were 37% and 63%, scaling would be much harder, and algebra (Method 3) would be the better choice.
Ans: Option (B)
In Method 3, the total distance '$d$' is calculated as 230 km. Why is 230 km NOT the final answer to the problem?
A. Because 230 is not divisible by 40.
B. Because the cyclist traveled backwards.
C. Because the question specifically asks for the remaining distance, not the total distance.
D. Because 230 km represents 100% of the journey, which is impossible.
1
Always re-read the final sentence of a word problem to ensure you are answering the specific question being asked.
Reading Comprehension
The formula $\frac{40}{100} = \frac{92}{d}$ correctly solves for $d = 230$ km, which is the total distance from Delhi to Agra.
However, the question asks "how many *more* kilometres does he have to travel".
Therefore, you must subtract the 92 km he already traveled from the total 230 km to get the remaining 138 km.
Ans: Option (C)
