A bowl of 500 mL lemonade has 10% jaggery. A glass of 200 mL lemonade has 20% jaggery. If we mix the lemonade from the bowl and the glass, what is the concentration of jaggery in the mixture?
Can you estimate what part of the mixture is jaggery? Is it 15%, or is it more or less? Why do you think so?
Step-by-step solution
To find: An estimate, then the exact concentration of jaggery in the mixture
Idea: Each concentration pulls the mixture’s concentration towards itself in proportion to the quantity it brings. The larger 10% bowl pulls harder, so the answer is below the midway value 15%.
- Estimate: if the quantities were equal, the answer would be 15% (midway). Here the bowl (10%) gives 500 mL and the glass (20%) only 200 mL, so the mixture is closer to 10%: less than 15%.½ mark
- Jaggery from the bowl = 500 × 0.1 = 50 mL; jaggery from the glass = 200 × 0.2 = 40 mL. Total jaggery = 90 mL.1 mark
- Total mixture = 500 + 200 = 700 mL.½ mark
- Concentration = quantity of jaggery in the mixturequantity of the mixture = 90700 ≈ 0.13, that is about 13% (more precisely 12.86%). This is less than 15%, as estimated.1 mark
Check: 13% lies between 10% and 20% and is nearer to 10% ✓. Distances: 13 − 10 = 3 and 20 − 13 = 7, roughly in the ratio 200 : 500 ✓.
Answer to write in the exam
Estimate: less than 15%, since more lemonade (500 mL) has 10% jaggery
Jaggery = 500 × 0.1 + 200 × 0.2 = 50 + 40 = 90 mL
Mixture = 500 + 200 = 700 mL
Concentration = 90700 ≈ 0.13
∴ Concentration of jaggery in the mixture ≈ 13%
Common mistakes that cost marks
- Answering 15% by averaging 10% and 20% without looking at the quantities.
- Estimating ‘more than 15%’ because 20% is the bigger number. It is the bigger quantity, not the bigger percentage, that pulls the answer towards it.
- Dividing 90 by 500 or by 200 instead of by the whole mixture, 700 mL.
How this can come in the exam
300 mL of juice with 10% sugar is mixed with 100 mL of juice with 30% sugar. The sugar concentration of the mixture is
- 20%
- 15%
- 25%
- 17.5%
Show answer
(B) 15%
300 × 0.1 + 100 × 0.3400 = 30 + 30400 = 60400 = 0.15 = 15%.
1 litre of 5% salt solution is mixed with 4 litres of 15% salt solution. Without calculating, say whether the mixture is above or below 10% salt. Then find its concentration.
Show answer
Above 10%, because most of the mixture (4 L) is the 15% solution (1 mark). 1 × 0.05 + 4 × 0.155 = 0.655 = 0.13 = 13% (1 mark).Try one yourself
400 mL of lemonade with 25% jaggery is mixed with 100 mL of lemonade with 5% jaggery. Estimate first, then find the concentration.
Show answer
Estimate: closer to 25%. Exactly: 400 × 0.25 + 100 × 0.05500 = 100 + 5500 = 0.21 = 21%.
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