12. RATIO AND PROPORTION
IMPORTANT FACTS AND
FORMULAE
I. RATIO: The ratio of two
quantities a and b in the same units, is the
fraction a/b and we write it as a:b.
In the ratio a:b, we call a as the first term or
antecedent and b, the second
term or consequent.
Ex. The
ratio 5: 9 represents 5/9 with antecedent = 5, consequent = 9.
Rule: The
multiplication or division of each term of a ratio by the same non-zero number does not affect
the ratio.
Ex. 4: 5 = 8: 10 = 12: 15 etc. Also, 4: 6 = 2: 3.
2. PROPORTION: The equality of two ratios is called proportion.
If a: b = c: d, we write, a: b:: c : d and
we say that a, b, c, d are in proportion . Here a and d are called extremes, while b and c are called
mean terms.
Product of means
= Product of extremes.
Thus, a: b:: c : d <=> (b x c) = (a x d).
3. (i) Fourth Proportional: If a : b = c: d, then
d is called the fourth proportional
to
a, b, c.
(ii) Third Proportional: If a: b = b:
c, then c is called the third proportional to
a
and b.
(iii) Mean Proportional: Mean proportional between a and b is
square root of ab
4. (i) COMPARISON OF RATIOS:
We say that (a: b) > (c: d) <=> (a/b)>(c /d).
(ii)
COMPOUNDED RATIO:
The
compounded ratio of the ratios (a: b), (c: d), (e : f) is (ace:
bdf)
5. (i)
Duplicate ratio of (a : b) is (a2 : b2).
(ii) Sub-duplicate ratio of (a : b) is (√a : √b).
(iii)Triplicate ratio of (a : b) is (a3 : b3).
(iv) Sub-triplicate ratio of (a : b) is (a ⅓ : b ⅓ ).
(v) If (a/b)=(c/d), then
((a+b)/(a-b))=((c+d)/(c-d)) (Componendo
and dividendo)
6. VARIATION:
(i) We say that x is directly proportional to y, if
x = ky for some constant k and
we write, x µ y.
(ii) We say that x is
inversely proportional to y, if xy = k for some constant k and
we write, x∞(1/y)
SOLVED PROBLEMS
Ex. 1. If a : b = 5 : 9 and
b : c = 4: 7, find a : b : c.
Sol. a:b=5:9 and b:c=4:7= (4X9/4): (7x9/4) =
9:63/4
a:b:c = 5:9:63/4 =20:36:63.
Ex. 2. Find:
(i)
the fourth proportional to 4, 9, 12;
(ii)
the third proportional to 16 and 36;
iii)
the mean proportional between 0.08 and
0.18.
Sol.
i) Let
the fourth proportional to 4, 9, 12 be x.
Then, 4 : 9 : : 12 : x ó4
x x=9x12 ó
X=(9 x 12)/14=27;
Fourth proportional to 4, 9, 12 is 27.
(ii) Let
the third proportional to 16 and 36 be x.
Then, 16 : 36 : : 36 : x ó16 x x = 36 x 36 ó x=(36 x 36)/16 =81
Third proportional to 16 and 36 is 81.
(iii) Mean
proportional between 0.08 and 0.18
Ö0.08
x 0.18 =Ö8/100
x 18/100= Ö144/(100
x 100)=12/100=0.12
Ex. 3. If x : y = 3 : 4, find
(4x + 5y) : (5x - 2y).
Sol. X/Y=3/4 ó (4x+5y)/(5x+2y)= (4( x/y)+5)/(5 (x/y)-2)
=(4(3/4)+5)/(5(3/4)-2)
=(3+5)/(7/4)=32/7
Ex. 4. Divide Rs. 672 in the
ratio 5 : 3.
Sol.
Sum of ratio terms = (5 + 3) = 8.
First part = Rs. (672 x (5/8)) = Rs. 420; Second part = Rs. (672 x
(3/8)) = Rs. 252.
Ex. 5. Divide Rs. 1162 among A,
B, C in the ratio 35 : 28 : 20.
Sol.
Sum of ratio terms = (35 + 28 + 20) = 83.
A's share = Rs. (1162 x (35/83))= Rs. 490; B's share = Rs. (1162
x(28/83))= Rs. 392;
C's share = Rs. (1162 x (20/83))= Rs. 280.
Ex. 6. A bag contains 50 p, 25
P and 10 p coins in the ratio 5: 9: 4, amounting to
Rs. 206. Find the number of coins
of each type.
Sol.
Let the number of 50 p, 25 P and 10 p coins be 5x, 9x and 4x respectively.
(5x/2)+( 9x/ 4)+(4x/10)=206ó 50x
+ 45x + 8x = 4120ó1O3x = 4120 óx=40.
Number of 50 p coins = (5 x 40) = 200; Number of 25 p coins = (9 x 40) =
360;
Number of 10 p coins = (4 x 40) = 160.
Ex.
7. A mixture contains alcohol and water in the ratio 4
: 3. If 5 litres of water is added to the mixture, the ratio becomes 4: 5. Find
the quantity of alcohol in the given mixture
Sol.
Let the quantity of alcohol and water be 4x litres and 3x litres respectively
4x/(3x+5)=4/5 ó20x=4(3x+5)ó8x=20 óx=2.5
Quantity of alcohol = (4 x 2.5) litres = 10 litres.
