1. Ratio:
The ratio of two quantities x and y in the same units, is the fractionx/yand we write it as x : y.
In the ratio x : y, we call x as the first term or antecedent and y, the second term or consequent.
Eg. The ratio 2 : 3 represents 2/3 with antecedent = 2, consequent = 3.
Rule: The multiplication or division of each term of a ratio by the same nonzero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. 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. Fourth Proportional:
If a : b = c : d, then d is called the fourth proportional to a, b, c.
Third Proportional:
a : b = c : d, then c is called the third proportion to a and b.
Mean Proportional:
Mean proportional between a and b is √ab.
4. Comparison of Ratios:
We say that (a : b) > (c : d) ⇔ a/b > c/d.
Compounded Ratio:
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
5. Duplicate Ratios:
Duplicate ratio of (a : b) is (a^{2} : b^{2}).
Subduplicate ratio of (a : b) is (√a : √b).
Triplicate ratio of (a : b) is (a^{3} : b^{3}).
Subtriplicate ratio of (a : b) is (a^{1/3} : b^{1/3}).
If a/b = c/d , then a + b/a  b = c + d/c  d [componendo and dividendo]
6. Variations:
We say that x is directly proportional to y, if x = ky for some constant k and we write, x ∝ y.
We say that x is inversely proportional to y, if xy = k for some constant k and
we write, x ∝ 1/y.
1. 

The required expression (E) = (2a+3b)/(3a+4b)
= [2(a/b)+3]/[3(a/b)+4]
Given a/b = 4/3
So, E =[2(3/4)+3]/[3(3/4)+4]
= 18/25
2. 

Let the no. of 50 P, 25 P, and 20 P coins be 4X ,2X ,X Respectively
Total value of coins = 50(4X) + 25(2X) + 20(X) = 5400
270X = 5400
X = 20
So, number of 25 P coins = 2X = 2x20 = 40
3. 

Let Ajay's and Balu's marks be 4X and 5X respectively.
(4X+36)/(5X+36) = 7/8
8(4X+36) = 7(5X+36)
32X + 288 = 35X + 252
3X = 36
X =12
So, Balu's marks = 5X = 60
4. 

Let the amounts received by the old man's wife, the eldest son and youngest son be a, b and c respectively.
Then, according to the first condition
given a = 1/2(b+c)
2a = b + c
Adding 'a' on both sides,
a = (a + b + c)/3
According to the 2nd condition given,
c = (1/3)(a+b)
3c = a + b
Adding 'c' on both sides,
4c = a + b + c
c = (a + b + c)/4
So, (a+b+c)/3  (a+b+c)/4 = 60000
a + b + c = 720000
5. 

Let the two numbers be 4X and 5X
Let K be added in both of these for the ratio to become 5:6
(4X+K)/(5X+K) = 5/6
24X+6K = 25X+5K
K = X
As X is unknown, K cannot be found.
6. 

Let, the speed of bus be x kmph and the consumption of diesel per hour be y litres.
Given, y ∝ x^{2},
⇒ y = kx^{2}
Given, 1 = k(40)^{2}
K = 1/1600
∴ y = x^{2}/1600
Let the required speed be x kmph
Time taken to cover 400 km at c kmph = 400/x hours
∴ consumption of diesel in 400/x hours = 400/x litres
∴ cost of diesel = 400y/x(40) = ₹16000y/x
= (16000/x)(x^{2}/1600) = 10x [∵y= x^{2}/16000]
and other expenses for 400/x hours = 400/x (40) = ₹16000/x
7. 

Let the quantities of milk in the 1st to 5st vessels be 4x, 5x, 6x, 7x and 8x respectively.
Total quantity of milk in the vessels is 30x
Total capacity of the 5 vessels = 30x(100/75) = 40x
Capacity of each vessel is 8x
64% of 8x = 5.12x
The number of vessels of vessels which contain at least 5.12x of milk is three.
8. 

Let the truck is rented for 8 hours or less.
Then the number of hours it is rented for = 800/100 = 8 hours
Buts, if it charged ₹ 10 per km, the amount that should paid be = 120(8) = 960
But company paid only ₹800
So, The truck is rented for more than 8 hours.
So, number of hours = 800/80 = 10 hours
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