A mixture of fog contains water and air in the ratio 5 ∶ 3. On adding 8 liters of air, the ratio of water and air becomes 1 ∶ 1. The quantity of water in the original fog is:
20 litres
Understanding the Initial Fog Mixture
The problem describes a fog mixture containing water and air. The initial ratio of water to air is given as 5 ∶ 3. This means that for every 5 parts of water, there are 3 parts of air.
Let the initial quantity of water be $5x$ liters and the initial quantity of air be $3x$ liters, where $x$ is a common multiplier.
Calculating Changes in the Mixture
Next, 8 liters of air are added to this mixture. The quantity of water remains unchanged, but the quantity of air increases.
- Initial quantity of water = $5x$ liters
- Initial quantity of air = $3x$ liters
- Quantity of air added = 8 liters
- New quantity of air = $(3x + 8)$ liters
- Quantity of water after adding air = $5x$ liters
Determining the New Ratio
After adding the air, the ratio of water to air becomes 1 ∶ 1. This implies that the quantities of water and air are now equal in the mixture.
We can set up an equation based on this new ratio:
$$ \frac{\text{Quantity of Water}}{\text{New Quantity of Air}} = \frac{1}{1} $$
Substituting the expressions for water and air quantities:
$$ \frac{5x}{3x + 8} = \frac{1}{1} $$
Solving for the Unknown Multiplier
To find the value of $x$, we can cross-multiply the equation:
$$ 5x \times 1 = 1 \times (3x + 8) $$
$$ 5x = 3x + 8 $$
Now, we need to isolate $x$. Subtract $3x$ from both sides of the equation:
$$ 5x - 3x = 8 $$
$$ 2x = 8 $$
Divide both sides by 2 to find the value of $x$:
$$ x = \frac{8}{2} $$
$$ x = 4 $$
Finding the Original Quantity of Water
The question asks for the quantity of water in the original fog. We established that the original quantity of water is $5x$ liters.
Substitute the value of $x = 4$ into the expression for the original quantity of water:
Original quantity of water = $5 \times x$
Original quantity of water = $5 \times 4$
Original quantity of water = $20$ liters
Verification
Let's check if this result satisfies the conditions:
- Original water = 20 liters
- Original air = $3x = 3 \times 4 = 12$ liters
- Initial ratio = 20:12 = 5:3 (Correct)
- Add 8 liters of air: New air = $12 + 8 = 20$ liters
- New ratio = Water:Air = 20:20 = 1:1 (Correct)
Therefore, the original quantity of water in the fog was 20 liters.
Important Questions from To Make a Mixture from Two Mixtures
In a 90 litre solution, acid and water are in the ratio 2 ∶ 1, To make the ratio of acid and water as 1 ∶ 2, how many litre of water should be added to the solution?
In a milk and water solution, the ratio of milk to water is 1 : 4. M litres of milk is added in it and ratio become 1 : 3. Again N litres of water is added in it and ratio become 2 : 7. If N – M = 4, then what is the initial quantity of solution?
A mixture contains acid and alcohol in the ratio of 3 : 2. On adding 10 litres of alcohol in mixture, the ratio of acid to alcohol becomes 3 : 5. The quantity of acid (in litres) in the original mixture was:
How many litres of acid are there in 12 litres of a 20% solution?
The ratio of number of male and female employees in a company was 5 ∶ 4. Some new males and females joined the company in the ratio 2 ∶ 3. The total number of employees become 1620 and the ratio of males to females become 68 ∶ 67. Initially what was the total number of employees in the company?
