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class 8 ganita prakash chapter 1
📖 Textbook Reference: 1.2 Ratios and Percentages
A given ratio can be converted to a fraction, and then to a percentage.
When you are given a ratio like $2:7$, these numbers represent "parts". The first step is always to find the total number of parts to form the denominator of your fraction.
Estimation First: The practice of estimating first before calculating can improve number sense and help reduce mistakes. Often, we may not need exact values.
To prepare a particular millet kanji (porridge), suppose the ratio of millet to water to be mixed for boiling is 2:7. What percentage does the millet constitute in this mixture? If 500 ml of the mixture is to be made, how much millet should be used?
1
A ratio describes the "parts". Before you can find a percentage, you must find the total number of parts to form your fraction's denominator.
2
Once you have the percentage, you multiply it by the physical volume (500 ml) to find the exact amount of that ingredient.
1
Total parts = $2 \text{ (millet)} + 7 \text{ (water)} = 9 \text{ parts}$.
2
The fraction for millet is $\frac{2}{9}$. Multiply by $100$ to get the percentage.
3
Multiply that percentage by $500\text{ ml}$ to find the final volume of millet.
!
The "2/7" Trap: A very common mistake is using the other side of the ratio as the denominator, creating the fraction $\frac{2}{7}$. This is completely wrong! The denominator must be the total mixture (9).
Step 1: Convert Ratio to Percentage
Total parts = $2 + 7 = 9$.
Millet fraction = $\frac{2}{9}$.
Percentage = $\frac{2}{9} \times 100$
$= \frac{200}{9} = 22.22\%$
Step 2: Calculate Volume in 500 ml
Volume = $22.22\%$ of $500\text{ ml}$
$= \frac{22.22}{100} \times 500$
$= 22.22 \times 5$
$= 111.1\text{ ml}$
Ans: Millet is 22.22%, requiring 111.1 ml.
To make a specific concrete mixture, the ratio of cement to sand is 1:4. What percentage of the mixture is cement? If a builder needs to make 300 kg of this concrete, how many kilograms of cement are needed?
1
Use the exact same logic as Example 4. First, determine the total parts in the mixture.
1
Total parts = $1 \text{ (cement)} + 4 \text{ (sand)} = 5 \text{ parts}$.
2
Fraction of cement = $\frac{1}{5}$. Convert to percentage.
3
Find that percentage of $300\text{ kg}$.
!
Do not use $\frac{1}{4}$! The fraction $\frac{1}{4}$ represents the relationship of cement *to* sand, not cement to the *whole* mixture. $\frac{1}{4}$ would lead to 25%, which is incorrect.
Step-by-Step Calculation
Total parts = $1 + 4 = 5$.
Cement percentage = $\frac{1}{5} \times 100 = 20\%$
Amount needed = $20\%$ of $300\text{ kg}$
$= \frac{20}{100} \times 300 = 20 \times 3 = 60\text{ kg}$
Ans: Cement is 20%, requiring 60 kg.
When converting the ratio $2:7$ (millet to water) into a fraction to find the percentage of millet, why must the denominator be $9$?
A. Because 9 is the highest single-digit number.
B. Because you multiply $2 \times 4.5$ to get 9.
C. Because a fraction for percentage must be $\frac{\text{Part}}{\text{Whole}}$, and the whole mixture is $2 + 7 = 9$ parts.
D. Because 9 is the next odd number after 7.
1
A ratio compares two parts to each other. A percentage compares one part to the *total whole*.
1
If I have 2 parts millet and 7 parts water, how many parts are in the bowl altogether?
!
Using $\frac{2}{7}$ tells you that the millet is $\approx 28\%$ of the *water's* volume. But we want to know what percentage it is of the *entire mixture's* volume!
Logical Explanation
In the ratio 2:7, in one unit of the mixture, millet occupies 2 units and water occupies 7 units.
Therefore, the total volume of the mixture is $2 + 7 = 9$ units. Since percentage is out of the total, the denominator must be 9.
Ans: Option (C)
Using the textbook's estimation method, how can we logically deduce that $\frac{2}{9}$ is slightly less than $25\%$ before calculating?
A. Because 2 is an even number and 9 is an odd number.
B. Because $25\%$ is exactly $\frac{1}{4}$ (or $\frac{2}{8}$). Since a denominator of 9 is larger than 8, the fraction $\frac{2}{9}$ must be smaller than $\frac{2}{8}$.
C. Because $9 \times 2.5 = 22.5$.
D. Because 20% of 9 is 1.8.
1
Estimation builds strong number sense. You can compare an unknown fraction to a known benchmark fraction (like $25\% = \frac{1}{4}$).
Estimation Logic
We know that $25\% = \frac{1}{4}$.
If we make the numerator 2, the equivalent fraction is $\frac{2}{8}$.
When comparing $\frac{2}{9}$ and $\frac{2}{8}$, the fraction with the larger denominator is actually the smaller number.
Therefore, $\frac{2}{9}$ is less than $\frac{2}{8}$ ($25\%$).
Ans: Option (B)
