

Direct Formula / Rule 19 :
Theorem:
If x glasses of different sizes, say S_{1}, S_{2}, S_{3}...S_{x}, are filled with a mixture of spirit and water. The ratio
of spirit and water in each glass are as follows, a_{1} : b_{1},
a_{2} : b_{2}, a_{3} : b_{3},...., ax : bx. If the contents of all the glasses
are emptied into a single vessel, then proportion of spirit and water in it is given by
= 
( 
a_{1}S_{1} 
+ 
a_{2}S_{2} 
+ ... + 
a_{x}S_{x} 
) 
: 
( 
b_{1}S_{1} 
+ 
b_{2}S_{2} 
+ ... + 
b_{x}S_{x} 
) 
a_{1} + b_{1}  a_{2} + b_{2}  a_{x} + b_{x} 
a_{1} + b_{1}  a_{2} + b_{2}  a_{x} + b_{x} 
Example :
Three glasses of sizes 3 litres, 4 litres and 5 litres
contain mixture of spirit and water in the ratio 2 : 3, 3 : 7 and 4 : 11 respectively. The contents of all the three
glasses are poured into single vessel. Find the ratio
of spirit to water in the resultant mixture
Quicker Method : Here you can use direct formula :
= 
( 
2 X 3 
+ 
3 X 4 
+ 
4 X 5 
) 
: 
( 
3 X 3 
+ 
7 X 4 
+ 
11 X 5 
) 
2 + 3  3 + 7  4 + 11 
2 + 3  3 + 7  4 + 11 
= 
( 
6 
+ 
12 
+ 
20 
) 
: 
( 
9 
+ 
28 
+ 
55 
) 
5  10  15 
5  10  15 
= 
56 
: 
124 
= 56 : 124 
15  15 
or, Spirit : Water = 14 : 31.
Exercise :

Three glasses of capacity 2 litres, 5 litres and 9 litres
contain mixture of milk and water with milk concentrations
90%, 80% and 70% respectively. The contents of
three glasses are emptied into a large vessel. Find the
milk concentration and ratio of milk to water in the resultant
mixture.

Four glasses of sizes 3 litres, 4 litres, 6 litres and 7 litres
contain mixture of milk and water in the ratio 2 : 1, 5 : 3, 6 : 3
and 9 : 5 respectively. Find the ratio milk to water if the
contents of all the four glasses are poured into one large
vessel.

Two casks of 48 and 42 litres are filled with mixtures of
wine and water, the proportions in the two casks being
respectively 13 : 7 and 18 : 17. If the contents of the two
casks be mixed, and 20 litres of water added to the whole
what will be the proportion of wine to water in the result?
Answers :
1 = 7 : 3, 2 = 13 : 7, 3 = 12 : 13



