Question : A mixture of milk and water in three bottles of equal capacity is in the ratio 4 : 1, 4 : 3, and 1 : 2, respectively. These three bottles are emptied into a large bottle. What will be the ratio of milk and water respectively in the large bottle?
Option 1: 179 : 136
Option 2: 152 : 165
Option 3: 198 : 175
Option 4: 133 : 145
Correct Answer: 179 : 136
Solution :
Let the capacity of each bottle be $x$.
In the first bottle, quantity of milk $\frac{4}{5}x$ and quantity of water is $\frac{1}{5}x$.
In the second bottle, quantity of milk $\frac{4}{7}x$ and quantity of water is $\frac{3}{7}x$.
In the third bottle, quantity of milk $\frac{1}{3}x$ and quantity of water is $\frac{2}{3}x$.
⇒ Total quantity of milk = $\frac{4}{5}x$ + $\frac{4}{7}x$ + $\frac{1}{3}x$ = $\frac{84}{105}x$ + $\frac{60}{105}x$ + $\frac{35}{105}x$ = $\frac{179}{105}x$
⇒ Total quantity of water = $\frac{1}{5}x$ + $\frac{3}{7}x$ + $\frac{2}{3}x$ = $\frac{21}{105}x$ + $\frac{45}{105}x$ + $\frac{70}{105}x$ = $\frac{136}{105}x$
So, the require ratio = $\frac{\text{Total quantity of milk}}{\text{Total quantity of water}}$ = $\frac{ \frac{179}{105}x}{ \frac{136}{105}x}$ = $\frac{179}{136}$
Hence, the correct answer is 179 : 136.
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