The number of times one quantity contains another quantity of the same kind is called the ratio of the two quantities .

Clearly, a ratio may be expressed in the form of a fraction. A ratio, for example, of Rs.5/- and Rs. 7/- may be expressed either in the form  5 : 7 or in the form 5 / 7.

Proportion is an equality of ratios . For example, 15 : 45 = 1 : 3 or 15/45 = 1/3 .

Each of these expressions indicates a proportional.

The terms 15 , 45, 1, 3 are distinguished as the 1st , 2nd , 3rd and 4th proportional respectively.

The Ist and 4th proportional are called ‘Extremes’ . The 2nd and 3rd proportional are called the ‘Means’ .

In the above proportional, it is clear that  15 x 3 = 45 x 1

i.e.    Product of the extremes  =  Product of the means   

Examples :

Q.1    Find a fourth proportional to the numbers 6, 8, 9 .

Solution :   If X be the fourth proportional, then 6 : 8 = 9 : X  

Or 6X = 8 x 9 ( By Cross multiplication ) X = 8 x 9/6        Or     X = 12

Q.2 If 3 : 8 = 6 : X, then X = ?

Solution : X = 8 x 6 / 3 = 16

Q.3       Find the fourth proportional to 6, 18.

Solution : Here, we would  have to find out a fourth proportional to 6,18,18.

So, if X be the fourth proportional ,  then ,   6 : 18 = 18 : X

Hence ,  X =  18 x 18 / 6 =  54 .

Q.4    Find the mean proportional between 8 and 2.

Solution : If X be the required mean proportional , then 8 : X : : X : 2 ( Multiply Means & Extremes )

 Hence , X =  4 

Q.5   If 12 : X = X : 3, then X = ?

Solution :  X . X = 12×3 = 36   ,      Hence,  X   =  6 

Q.6    If 15 boys can do a piece of work in 120 days, in how many days will 18 boys  do it ?

Solution ( SHORT CUT ) :  Since the number of “boys days” required to complete any piece of work is always the same, so :

The number of boys    x The number of days in Ist case 

= The number of boys x The number of days in 2nd case

Hence, 15  x 120   =  18 x X       or        X    =  100 days

Q.7       If 12 men working 4 hours a day for 120 days can complete a piece of work, how many men working 6 hours a day for 12 days will be needed ?

Solution ( SHORT – CUT ) : Since the number of “ man day hours”  need to complete the work in both the cases is the same so, 12 X 4 X 120  = X x 6 X 12      or     X  =  80 men

Q.8       If 8 men reap 40 acres in 12 days, how many acres

will 6 men reap in 10 days.

Solution ( SHORT – CUT ) : Since the “man days” required per unit of work is always the same, so :           

Men 1 x Time / Work 1  = Men 2 x Time / Work 2

So, 8 x 12 / 40 = 6 x 10 / X   ( Cross Multiplication  )  Or     X = 25 acres

Q.9 If Rs. 16,000 maintain a family of 4 persons for 60 days, how long will  Rs. 21,000 maintain a family of 7 persons ?

Solution ( SHORT – CUT ) : 4 x 60 / 16,000 = 7x X/21,000     Or     X  =  45 days .

Q.10     If 9 men or 15 women can do a piece of work in 30 days, in how many days can the work be done by 6 men and 20 women working together ?

Solution  :        Since , 9 M = 15 W

M = 15 W /9 ;    6M = 15W/9 x  6 10 W  .    Now, 6 M + 20 W  = 10 W + 20 W = 30 Women

Now, we have to find out that if 15 women can do a work in 30 days, then 30 women can do  the same job in how many days ?

The proportion becomes , 30 : 15  =   30 : X          Or    X = 30 x 15 / 30 = 15 days

Q.11     If 1/18 : X = 9 : 4 , find the value of X .

Solution  :      1/18  :  X  =  9 : 4        or       9X  =  1/18 x 4      Or     X= 4 / 18 x 9 = 2 / 81