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class 9 ganita manjari chapter 10
📖 Textbook Reference: 10.1.2 Mixtures
When mixing completely different ingredients (like salt and sugar), you cannot average their percentages together. They must be calculated as independent active ingredients.
The Concept of Dilution: If you have a salt solution and you add pure water (or a sugar solution) to it, you are adding 0% salt to the mixture. This increases the total volume but keeps the amount of salt the same, naturally dropping the salt percentage.
A 600 mL solution with 5% salt is mixed with a 300 mL solution with 8% sugar. What are the concentrations of salt and sugar in the mixture?
Select the correct outcome:
(i) Salt: 5%, Sugar: 8%
(ii) Salt: 13%, Sugar: 3%
(iii) Salt: 6%, Sugar: 6%
(iv) Salt: 5.55%, Sugar: 8.88%
(v) Salt: 3.33%, Sugar: 2.67%
(vi) Salt: 4.1%, Sugar: 7.08%
1
Because salt and sugar are completely different ingredients, their percentages do not "mix" or average together. You must calculate their new concentrations independently.
2
Dilution: Adding the 300 mL sugar solution acts just like adding pure water to the salt. The volume increases, but no new salt is added, causing the salt percentage to drop.
1
What is the final total volume? $600\text{ mL} + 300\text{ mL} = 900\text{ mL}$.
2
For Salt: Calculate the exact mL of pure salt originally present ($5\%$ of $600\text{ mL}$). Then divide that amount by the new total volume ($900\text{ mL}$) and multiply by $100$.
3
For Sugar: Calculate the exact mL of pure sugar originally present ($8\%$ of $300\text{ mL}$). Then divide that amount by the new total volume ($900\text{ mL}$) and multiply by $100$.
!
Averaging the Percentages: Many students try to use the standard weighted average formula on the $5\%$ and $8\%$ to get something like $6\%$. You can only average percentages if they are measuring the exact same ingredient (e.g., mixing two salt solutions). Since they are different, they just dilute each other!
Independent Concentration Calculation
Total Volume of Mixture = $600 + 300 = 900\text{ mL}$.
Calculating New Salt Concentration:
Amount of pure salt = $5\%$ of $600\text{ mL} = \frac{5}{100} \times 600 = 30\text{ mL}$.
New Salt $\%$ = $\frac{30}{900} \times 100 = \frac{3000}{900} = \frac{10}{3} \approx 3.33\%$.
Calculating New Sugar Concentration:
Amount of pure sugar = $8\%$ of $300\text{ mL} = \frac{8}{100} \times 300 = 24\text{ mL}$.
New Sugar $\%$ = $\frac{24}{900} \times 100 = \frac{2400}{900} = \frac{8}{3} \approx 2.67\%$.
Ans: Option (v) Salt: 3.33%, Sugar: 2.67% is correct.
A 400 mL drink with 10% lemon juice is mixed with a 100 mL drink with 5% ginger juice. Find the new concentrations of lemon and ginger juice in the combined mixture.
1
Since lemon and ginger are different juices, they dilute independently when mixed. Calculate the final volume, then divide the raw amount of each juice by that new total volume.
1
Total volume = $400 + 100 = 500\text{ mL}$.
2
Find pure lemon juice: $10\%$ of $400\text{ mL}$. Divide that by $500$.
3
Find pure ginger juice: $5\%$ of $100\text{ mL}$. Divide that by $500$.
!
Do not add $10\%$ and $5\%$ or try to average them. Lemon does not turn into ginger!
Step-by-Step Calculation
Total Volume = $400 + 100 = 500\text{ mL}$.
Lemon Juice:
Amount = $10\%$ of $400 = 40\text{ mL}$.
New \% = $\frac{40}{500} \times 100 = 8\%$.
Ginger Juice:
Amount = $5\%$ of $100 = 5\text{ mL}$.
New \% = $\frac{5}{500} \times 100 = 1\%$.
Ans: Lemon: 8%, Ginger: 1%
When calculating the new salt concentration in the main problem, what is the "salt percentage" of the 300 mL sugar solution being added?
A. 8%
B. 5%
C. 0%
D. 13%
1
The 300 mL solution is a sugar solution. Unless stated otherwise, it contains zero salt.
Logical Explanation
When evaluating the salt concentration, the sugar solution acts exactly like adding pure water. It increases the volume by 300 mL, but contributes 0 mL of salt.
Therefore, from the perspective of the salt calculation, the sugar solution has a $0\%$ salt concentration.
Ans: Option (C)
Why does the concentration of salt decrease from 5% to 3.33% when the two liquids are mixed?
A. Because salt evaporates when exposed to sugar.
B. Because adding the sugar solution increases the total volume without adding any new salt, diluting it.
C. Because sugar chemically reacts with salt to reduce its mass.
D. Because 300 mL is exactly half of 600 mL.
1
This is the fundamental principle of dilution. If you increase the denominator (total volume) while keeping the numerator (amount of solute) the same, the overall fraction/percentage must decrease.
Logical Breakdown
Before mixing, you have $30\text{ mL}$ of salt sitting in $600\text{ mL}$ of liquid.
After mixing, you still have exactly $30\text{ mL}$ of salt, but it is now swimming in $900\text{ mL}$ of liquid!
Because the same amount of salt is spread out over a much larger volume, the concentration (percentage) drops. This is called dilution.
Ans: Option (B)
