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
We also have the relationship,
Product of the MEANS = Product of the
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
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
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
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.
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. .
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
the fourth proportional to the numbers 0.8, 1.6 and 1.6. .
fourth proportional of a, b, c is given by = 3.2