Dorjee has collected 1 litre of water from the Dead Sea! Using the information in the table, answer the following questions. A calculator can be used if necessary.
- (i) What is the salinity of the mixture if he mixes 1 litre of water from the Dead Sea with 2 litres of purified drinking water?
- (ii) Is it possible to mix water from the Dead Sea and purified drinking water to get a mixture with the salinity of groundwater? Why/Why not? What quantity of purified drinking water should Dorjee mix with 1 litre of water from the Dead Sea to get a mixture having the salinity of groundwater?
- (iii) Is it possible to mix water from the Dead Sea and groundwater to get a mixture with the salinity of purified drinking water? Why/Why not? What quantity of groundwater should he mix with 1 litre of water from the Dead Sea to get a mixture having the salinity of purified drinking water?
Answer: (i) 0.34 + 2 × 0.000013 ≈ 11.33%. (ii) Yes, since 0.01% lies between 0.001% and 34%; he needs about 3777 litres (3776.67 L) of purified water. (iii) No: both waters are saltier than 0.001%, so any mixture is at least 0.01% salty.
Step-by-step solution
Given:
; Dorjee has 1 litre of Dead Sea water
| Water Source | Salinity |
|---|---|
| Dead Sea | ≈ 34% |
| Other seas and oceans | ≈ 3.5% |
| Ground water | ≈ 0.01% |
| Purified drinking water | ≈ 0.001% |
Idea: The salinity of a mixture is the weighted mean of the salinities, weighted by the volumes. So it always lies between the salinities of the waters being mixed, and closer to the one used in larger amount.
(i) What is the salinity of the mixture if he mixes 1 litre of water from the Dead Sea with 2 litres of purified drinking water?
- Salt in the mixture (in litres) = 1 × 0.34 + 2 × 0.00001 = 0.34002; volume = 1 + 2 = 3 L.½ mark
- Salinity = 0.340023 ≈ 0.1133 = 11.33%.½ mark
≈ 11.33%
(ii) Is it possible to mix water from the Dead Sea and purified drinking water to get a mixture with the salinity of groundwater? Why/Why not? What quantity of purified drinking water should Dorjee mix with 1 litre of water from the Dead Sea to get a mixture having the salinity of groundwater?
- Yes. A mixture’s salinity lies between the two salinities, 0.001% and 34%, and groundwater’s 0.01% is between them. Using a very large amount of purified water brings the mixture down close to 0.001%.½ mark
- Let x L of purified water be added: 34 × 1 + 0.001x1 + x = 0.01 ⇒ 34 + 0.001x = 0.01 + 0.01x.1 mark
- 33.99 = 0.009x ⇒ x = 33.990.009 ≈ 3776.67. He needs about 3777 litres of purified water.½ mark
Yes; about 3777 L (3776.67 L) of purified water
(iii) Is it possible to mix water from the Dead Sea and groundwater to get a mixture with the salinity of purified drinking water? Why/Why not? What quantity of groundwater should he mix with 1 litre of water from the Dead Sea to get a mixture having the salinity of purified drinking water?
- No. The mixture’s salinity must lie between 0.01% (groundwater) and 34% (Dead Sea). The target 0.001% is lower than both, so no amount of groundwater can reach it; the mixture is always at least 0.01% salty.1 mark
- Algebra agrees: 34 + 0.01x1 + x = 0.001 ⇒ 34 + 0.01x = 0.001 + 0.001x ⇒ 0.009x = −33.999 ⇒ x ≈ −3777.7, a negative quantity, which is impossible. No quantity of groundwater works.1 mark
Not possible; no quantity of groundwater works
(i) About 11.33%. (ii) Yes, because 0.01% lies between 0.001% and 34%; about 3777 litres of purified water are needed. (iii) No, because 0.001% is below both 0.01% and 34%; the equation gives a negative quantity.
Check: (ii): 34 + 0.001 × 3776.673777.67 = 37.783777.67 ≈ 0.0100% ✓.
Answer to write in the exam
(i)
Salinity = 1 × 34 + 2 × 0.0011 + 2 %
= 34.0023 % = 11.33%
∴ Salinity of the mixture ≈ 11.33%
(ii)
0.001% < 0.01% < 34%, so it is possible
Let purified water = x L: 34 × 1 + 0.001x1 + x = 0.01
34 + 0.001x = 0.01 + 0.01x ⇒ 0.009x = 33.99
x = 3776.67
∴ Yes; about 3776.67 L (≈ 3777 L) of purified water is needed
(iii)
Mixture salinity lies between 0.01% and 34%
0.001% < 0.01%, so the target is outside this range
Check: 34 + 0.01x1 + x = 0.001 ⇒ 0.009x = −33.999 ⇒ x < 0
∴ Not possible; no quantity of groundwater will do
Common mistakes that cost marks
- Forgetting that purified water also has a little salt (0.001%). Here it hardly changes the answers, but the method should include it.
- Mixing percentages and decimals in one equation, for example 34 with 0.0001.
- In (iii), reporting the negative value as an answer instead of concluding that it is impossible.
How this can come in the exam
MCQ (1 mark)
A 20% solution is mixed with a 5% solution in some ratio. The concentration of the mixture could be
- 3%
- 12%
- 22%
- 25%
Show answer
(B) 12%
A mixture’s concentration lies between 5% and 20%; only 12% does.
Try one yourself
How much 1% salt water must be mixed with 1 litre of 10% salt water to get 4% salt water?
Show answer
0.1 + 0.01x1 + x = 0.04 ⇒ 0.1 + 0.01x = 0.04 + 0.04x ⇒ 0.03x = 0.06 ⇒ x = 2 litres.
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