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class 9 ganita manjari chapter 10
📖 Textbook Reference: 10.1.2 Mixtures
Let us consider what happens when things are mixed.
When combining mixtures, the concentration of the final mixture is determined by calculating the total quantity of the active ingredient (e.g., jaggery) and dividing it by the total quantity of the mixture.
Equal Quantities Rule: If the quantities mixed are exactly equal, the final concentration will be exactly midway between the individual concentrations (i.e., a simple average).
Two glasses of equal quantities of lemonade are prepared. One glass has 10% jaggery and the other has 20% jaggery. If we mix the lemonade from both glasses, what is the concentration of jaggery in the mixture?
1
When mixing two solutions, the final concentration is calculated by finding the total amount of the active ingredient (jaggery) and dividing it by the total volume of the mixture.
2
However, just like we saw in the previous example with equal days, when the quantities of the mixtures being combined are exactly equal, the final concentration lands exactly halfway between the two starting concentrations.
1
Let the volume of each glass be $y$.
2
Amount of jaggery in glass 1 = $10\%$ of $y = 0.1y$.
3
Amount of jaggery in glass 2 = $20\%$ of $y = 0.2y$.
4
Total jaggery = $0.1y + 0.2y$. Total liquid = $y + y = 2y$.
!
Adding Percentages Directly: Do not simply add $10\% + 20\% = 30\%$. You cannot add concentrations together. You must add the quantities of the ingredient, then divide by the total volume.
Mathematical Proof
Suppose $y$ is the quantity of lemonade in each glass.
Total Jaggery = $\frac{10}{100}y + \frac{20}{100}y$
Total Mixture Volume = $y + y = 2y$
$\text{Concentration} = \frac{\text{Total Jaggery}}{\text{Total Mixture}}$
$= \frac{0.1y + 0.2y}{2y}$
$= \frac{0.3y}{2y}$
$= 0.15 = 15\%$
Ans: The concentration of jaggery in the mixture is 15%.
Two identical jars of saline solution are mixed together. The first jar has a 12% salt concentration and the second has an 18% salt concentration. What is the salt concentration of the combined mixture?
1
Because the problem explicitly states the jars are "identical", we know the volumes are equal.
2
When mixing equal volumes, the final concentration is exactly halfway between the two starting concentrations (the simple average).
1
You don't need a complex formula here. Just find the simple average of $12\%$ and $18\%$.
!
Adding Percentages: Don't say the answer is $30\%$. You cannot create a saltier solution just by pouring two less salty solutions together.
Direct Calculation
Since the quantities (jars) are identical, we can take the simple average of the concentrations.
$\text{Final Concentration} = \frac{12\% + 18\%}{2}$
$= \frac{30\%}{2}$
$= 15\%$
Ans: The salt concentration is 15%.
When combining two liquid mixtures of EQUAL quantities, how is the final concentration calculated?
A. By adding the two percentages together.
B. By taking the simple average of the two percentages.
C. By multiplying the two percentages.
D. It cannot be determined without knowing the exact volume in liters.
1
The general weighted average formula is $\frac{w_1 x_1 + w_2 x_2}{w_1 + w_2}$.
2
When quantities are equal, $w_1 = w_2$. This simplifies the formula to $\frac{x_1 + x_2}{2}$, which is the simple average.
1
Think about what happened in Example 3. The 10% and 20% mixtures combined exactly at the halfway point (15%).
!
Option D Trap: You do not need the exact volume in liters. As long as you know the ratio of the volumes (in this case, 1:1 since they are equal), you can find the final concentration percentage.
Logical Explanation
When glasses have equal quantities, they carry equal weight in the mixture equation.
Therefore, the final concentration will always rest exactly midway between the individual concentrations, which is the definition of a simple average.
Ans: Option (B)
If we mix $y$ ml of a 10% jaggery solution and $y$ ml of a 20% jaggery solution, the total amount of jaggery in the final mixture is mathematically represented as:
A.
$$10y + 20y$$
B.
$$0.1y + 0.2y$$
C.
$$\frac{0.1y + 0.2y}{2y}$$
D.
$$0.3y^2$$
1
To find the total amount of an ingredient in a mixture, you must convert the percentage concentration into a decimal, and multiply it by the volume of that specific mixture.
1
$10\%$ is equal to $0.1$. So, the jaggery in the first glass is $0.1 \times y$.
2
$20\%$ is equal to $0.2$. So, the jaggery in the second glass is $0.2 \times y$.
!
Option C Trap: Option C represents the final concentration percentage of the mixture. The question specifically asked for the "total amount of jaggery" (the numerator), not the concentration.
Derivation
Jaggery in Glass 1 = $\frac{10}{100} \times y = 0.1y$
Jaggery in Glass 2 = $\frac{20}{100} \times y = 0.2y$
Total Jaggery = $0.1y + 0.2y$
Ans: Option (B)
