

Direct Formula / Rule 17 :
Theorem:
If x glasses of equal size 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_{x}, : b_{x}. If the contents
of all the x glasses are emptied into a single vessel,
ti en proportion of spirit and water in it is given by
= 
( 
a_{1} 
+ 
a_{2} 
+ ... + 
a_{x} 
) 
: 
( 
b_{1} 
+ 
b_{2} 
+ ... + 
b_{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 :
In three vessels each of 10 litres capacity, mixture of
milk and water is filled. The ratios of milk and water
are 2 : 1, 3 : 1 and 3 : 2 in the three respective vessels.
If all the three vessels are emptied into a single large
vessel, find the proportion of milk and water in the
mixture.
Quicker Method : Here you can use direct formula :
required ratio is
= 
( 
2 
+ 
3 
+ 
3 
) 
: 
( 
1 
+ 
1 
+ 
2 
) 
2 + 1  3 + 1  3 + 2  2 + 1  3 + 1  3 + 2 
= 
( 
2 
+ 
3 
+ 
3 
) 
: 
( 
1 
+ 
1 
+ 
2 
) 
3  4  5  3  4  5 
= 
40 + 45 + 36 
= 
20 + 15 + 24 
= 26 
3 X 4 X 5  3 X 4 X 5 
=121 : 59
Exercise :

Three equal glasses are filled with mixtures of milk and
water. The proportion of milk and water in each glass is
as follows. In the first glass as 3 : 1, in the second glassas
5 : 3 and in the third as 9 : 7. The contents of the three
glasses are emptied into a single vessel. What is the
proportion of milk and water in it?

Four vessels of equal sizes contain mixture of spirit and
water. The concentration of spirit in 4 vessels are 60%,
70%, 75% and 80% respectively. If all the four mixtures
are mixed, find in the resultant mixture the ratio of spirit
to water.

Two equal glasses filled with mixtures of alcohol and
water in the proportions of 2 : 1 and 1 : 1 respectively
were emptied into a third glass. What is the proportion
of alcohol and water in the third glass?
Answers :
1 = 17 : 31, 2 = 57 : 23, 3 = 7 : 5



