Before going to solve word problems in proportion we shold remember some property of proportion, let recollect it:
1. When two ratios are equal, then the four quantities composing them are said to be proportionals. if
a/b = c/d, then it is written as a : b = c : d or a : b :: c: d
Hence "a and d" are known as extremes and "b and d" are known as means .
2. Continued proportion : If a,b,c,d,.... are some quantities then the proportion
a/b = b/c = c/d = ....... are said to be in continued proportion.
3. Product of extremes = product of means
a × d = b × c
4. If a , b and c are in contiued proportion then 'b' is said to the " mean" and 'c' is said to be the third proportion.
How to Calculate Proportion :
1.Nine litres are drawn from a vessel full of juice, it is then filled with water, then nine litres of the mixture are drawn , and the vessel is again filled with water . If the quantity of juice now in the vessels be to the quantity of water in it as 16 to 9, how much does the vessel hold ?
Let the vessel holds x litres of juice in the begning. The vessel, after taking 9 litres from it, is left with (x - 9) litres of juice, but since 9 litres of water have been added to it, the ratio of juice to water will be (x - 9) : 9.
Again from this mixture , 9 litres have been taken out
∴ Quantity of juice drawn in this mixture
= [( x - 9 ) / x ] × 9 = ( 9x - 81) / x
∴ Remaining quantity of juice in the vessel
= ( x - 9 ) - [ (9x - 81) / x ]
= ( x2 - 18x + 81 ) / x ....................(i)
Quantity of water drawn = ( 9 / x ) / x = 81 / x
∴ Remaining water in the vessel = 9 - (81 / x) = (9x - 81) / x
Since again 9 litres of water have been added.
∴ Total quantity of water = [ (9x - 81 ) / x ] + 9
= (18x - 81 ) / x .....................(ii)
Ratio of juice to water from (i) and (ii)
= ( x2 - 18x + 81 ) / x : (18x - 81 ) / x = 16 / 9
or 9x2 - 162x + 729 = 128x - 1296
⇒ x = 45.
∴ Total the vessel holds 45 litres.
2. In Canada the population increased by 15.9 percent between 2001 and 2011, if the town population increased by 18 percent and the country population 4 percent . Compare the town and country populations iin 2001 ?
Let the population of the country in 2001 be y and that of the town be x.
The increase in the population of the town is 18% and in that of the country is 4%.
∴ Increased population of the town = 118x / 100 = 59x / 50
and population of the country = 104y / 100 = 52y / 50.
∴ Total population of the town and country
= (52y / 50 ) + (59x / 50)
But given that total population has increased by 15.9%
∴ Total population = [ (x + y) (1159 / 1000) ] .
Hence ( 59x / 50 ) + (52y / 50 ) = (1159 / 1000) (x + y)
= 21x / 1000 = 119y / 1000
⇒ x / y = 17 / 3 i.e. x : y : : 17 : 3.