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.
