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The ratio of milk and water in the mixture of 90 litre is 7 : 2. If some amount of mixture is replaced by water, then the ratio of milk to water becomes 5 : 2. Find the quantity of water added to the mixture.
- a)6 litre
- b)8 litre
- c)12 litre
- d)10 litre
- e)None of these
Correct answer is option 'E'. Can you explain this answer?
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The ratio of milk and water in the mixture of 90 litre is 7 : 2. If so...
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The ratio of milk and water in the mixture of 90 litre is 7 : 2. If so...
Understanding the Initial Mixture
The initial mixture consists of 90 liters with a milk to water ratio of 7:2.
- Total parts = 7 + 2 = 9 parts
- Milk = (7/9) * 90 = 70 liters
- Water = (2/9) * 90 = 20 liters
Replacing Mixture with Water
Let \( x \) be the quantity of the mixture replaced with water. After replacing \( x \) liters, the new composition will be:
- Milk remaining = \( 70 - \frac{7}{9}x \)
- Water remaining = \( 20 - \frac{2}{9}x + x = 20 - \frac{2}{9}x + \frac{9}{9}x = 20 + \frac{7}{9}x \)
Now, the new ratio of milk to water is given as 5:2.
Setting Up the Equation
This gives us the equation:
\[
\frac{70 - \frac{7}{9}x}{20 + \frac{7}{9}x} = \frac{5}{2}
\]
Cross-multiplying gives:
\[
2(70 - \frac{7}{9}x) = 5(20 + \frac{7}{9}x)
\]
Expanding both sides:
\[
140 - \frac{14}{9}x = 100 + \frac{35}{9}x
\]
Solving for \( x \)
Rearranging terms:
\[
140 - 100 = \frac{35}{9}x + \frac{14}{9}x
\]
\[
40 = \frac{49}{9}x
\]
Multiplying by 9:
\[
360 = 49x
\]
\[
x = \frac{360}{49} \approx 7.35 \text{ liters}
\]
Conclusion
Since the question asks for the quantity of water added and \( x \) is approximately 7.35 liters, the closest answer choice is not present, thus the correct option is "None of these".
The initial mixture consists of 90 liters with a milk to water ratio of 7:2.
- Total parts = 7 + 2 = 9 parts
- Milk = (7/9) * 90 = 70 liters
- Water = (2/9) * 90 = 20 liters
Replacing Mixture with Water
Let \( x \) be the quantity of the mixture replaced with water. After replacing \( x \) liters, the new composition will be:
- Milk remaining = \( 70 - \frac{7}{9}x \)
- Water remaining = \( 20 - \frac{2}{9}x + x = 20 - \frac{2}{9}x + \frac{9}{9}x = 20 + \frac{7}{9}x \)
Now, the new ratio of milk to water is given as 5:2.
Setting Up the Equation
This gives us the equation:
\[
\frac{70 - \frac{7}{9}x}{20 + \frac{7}{9}x} = \frac{5}{2}
\]
Cross-multiplying gives:
\[
2(70 - \frac{7}{9}x) = 5(20 + \frac{7}{9}x)
\]
Expanding both sides:
\[
140 - \frac{14}{9}x = 100 + \frac{35}{9}x
\]
Solving for \( x \)
Rearranging terms:
\[
140 - 100 = \frac{35}{9}x + \frac{14}{9}x
\]
\[
40 = \frac{49}{9}x
\]
Multiplying by 9:
\[
360 = 49x
\]
\[
x = \frac{360}{49} \approx 7.35 \text{ liters}
\]
Conclusion
Since the question asks for the quantity of water added and \( x \) is approximately 7.35 liters, the closest answer choice is not present, thus the correct option is "None of these".
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