Proportion
When two ratios are equal, the four
quantities involved in the two ratios are said to be proportional i.e., if a/b
= c/d, then a, b, c and d are proportional. This is represented as a : b :: c :
d and is read as “a is to b (is) as c is to d”. .
When a, b, c and d are in proportion, then
the items a and d are called the EXTREMES and the items b and c are called the
MEANS. .
We also have the relationship,
Product of the MEANS = Product of the
EXTREMES
i.e., bc = ad .
If a : b = c : d then
b : a = d : c → (1)
a : c = b : d
a + b : b = c + d : d → (2) obtained by adding 1 to both sides of the given relationship
a – b : b = c – d : d → (3) obtained by subtracting 1 from both sides of the given
relationship
a + b : a – b = c + d : c – d → (4) obtained by dividing relationship (2) above by (3). .
Relationship (1) above is called INVERTENDO;
(2) is called COMPONENDO; (3) is called DIVIDENDO and (4) is called
COMPONENDO-DIVIDENDO. The last relationship, i.e., COMPONENDO-DIVIDENDO is very
helpful in simplifying problems. .
Whenever we know a/b = c/d, then we can write
(a + b)/ (a - b) = (c + d)/(c - d) by this rule. The converse of this is also
true — wherever we know that (a + b)/(a - b) = (c + d)/(c - d), then wean
conclude that a/b = c/d. .
If three quantities a, b and c are such that
a : b : : b : c, then we say that they are in CONTINUED PROPORTION. We also
get, b2 = ac. .
Worked out examples
EXAMPLE 1
There are three, numbers, 6 times the first
and seven times the second are equal. 5 times the second and 6 times the third
are also equal. If the first number is 20 more than the third, find the third
number. .
Solution
Let the first number, second number and third
number be denoted as a, b and c respectively. .
6a = 7b
&⇒ a = 
b
5b = 6c
&⇒ c = 
b
a – c = 20
&⇒ b – b = 20
&⇒ b = 60. .
c = 
= 50.
EXAMPLE 2
A bag has coins of denominations of
one-rupee, two-rupees and five-rupees in the ratio 9 : 6 : 4. If the total
value of five-rupee coins is Rs.32 more than the total value of two-rupee
coins, find the total value of the coins in the bag. .
Solution
Let the number of coins of denominations of
one-rupee, two-rupee and five-rupee be 9x, 6x and 4x respectively. Total value
of coins 4(5x) - 6(2x) = 32 .
x = 4
Total value of the coins in the bag
9x + 12x + 20x
= 41x = Rs.164 .
x = 4
EXAMPLE 3
Calculate
the fourth proportional to the numbers 0.8, 1.6 and 1.6. .
Solution
The
fourth proportional of a, b, c is given by 
= 3.2
