Quants Basic Concepts Ratio and Proportion Day – 1

Ratio is a comparison of two quantities by division. Ratio represents the relation that one quantity bears to the other. It is represented as a:b. In any ratio a:b, a is called Antecedent and B is called Consequent. It is an abstract (without units) quantity.

A ratio remains unaltered if its numerator and denominator are multiplied or divided by the same number, e.g. 4:3 is the same as the (4 x 10) : (3 x 10) ie 40:30.

Duplicate Ratio:a²: b² is called duplicate ratio of a : b.

Triplicate Ratio:a³: b³ is called triplicate ratio of a : b.

Sub – Duplicate Ratio: √a :√b is called sub-duplicate ratio of a : b.

Sub – Triplicate Ratio:³√a :³√b is called sub-triplicate ratio of a : b

Compound Ratio:ab : cd is the compound ratio of a : c and b : d. It is the ratio of the product of the antecedents to that of the consequents of two or more given ratios.

Inverse Ratio:1/a : 1/b is the inverse ratio of a : b.

Componendo and Dividendo: If a/b = c/d, then (a + b)/(a – b) = (c + d)/(c – d)

Proportion is a statement that two ratios are similar. When two ratios are equal, they make a proportion, i.e. if a/b = c/d, then a, b, c and d are in proportion. This is represented as a:b :: c:d. When a, b, c and d are in proportion, then a and d are called the Extremes and b and c are called the Means, also Product of the Means = Product of the Extremes i.e. bc = ad.

Continued Proportion:If these quantities a, b and c are such that a:b :: b:c, then b²= ac and a, b and c are in continued proportion. Also the quantity c is called the third proportion of a and b.

Fourth Proportion:If four quantities a, b, c and x are such that a:b :: c:x, then ax = bc and x is called the fourth proportion of a, b and c.

Mean or second Proportion: If three quantities a, b and x are such that a:x ::x:b, then x²= ab and x is called the mean of a and b. Also, if a:b = c:d, then the following properties holds good.

i)b:a = d:c (Invertendo)

ii)a:c = b:d (Alternendo)

iii)(a + b) : b = (c + d) : d (Componendo)

iv)(a – b) : b = (c – d) : d (Divendendo)

v)(a + b)/(a – b) = (c + d)/(c – d) (Componendo – Divendendo)

If two quantities x and y are related in such a way that as the quantity x changes it also brings a change in the second quantity y, then the two quantities are in variation.

Direct Variation:The quantity x is in direct variation to y, if an increase in x makes y to increase proportionally. Also decrease in x males y to decrease proportionally it can be expressed as x = ky, where k is called the constant of proportionality.

Eg: Cost is directly proportional to the number of articles brought.

Inverse Variation:The quantity x is in inverse variation to y, if an increase in x makes y o decrease proportionally. Also a decrease in x makes y to increase proportionally. It can be expressed as x = k/y, where k is a constant of proportionality.

Eg: The time taken by a vehicle in covering a certain distance is inversely proportional to the speed of the vehicle.

Joint Variation:If there are more than two quantities x, y and z and x varies with both y and z, then x is in joint variation to y and z. It can be expressed as kyz, where k is constant of proportionality.

Eg: Men doing a work in some number of days working certain hours a day.

Distribution of Amount:If an amount A is distributed in the ratio a:b, then

First part =a/(a+b) x A;Second Part =b/(a + b) x A

1) If a:b :: b:c, then a/b = b/c => c = b²/a

2) If a:b :: c:d, then a/b = c/d => d = bc/a

3)If a:x :: x:b, then x = √ab (x is mean proportional)

4)If x/y = 1, then (x + a)/(y + a) = 1 and (x – a)/(y – a) = 1

5)If x/y> 1, then (x + a)/(y + a)< x/y and (x –a)/(y – a)> x/y

6)If x/y< 1, then  (x + a)/(y + a)> x/y and (x –a)/(y – a)< x/y

7)If a/b = c/d = e/f = …………… = k (constant), then (a + c + e + ……..)/(b + d + f + ……..) = k

Let the third number be x.

Then first number is 125% of x = 125x/100 = 5x/4 and second number is 160% of x = 160x/100 = 8x/5.

Now the ratio of first to second number is 5x/4 : 8x/5 = 5x x 5 : 8x x 4 = 25 : 32.

4x/(9x + 32) = 4/17 =>68x = 36x + 128 => x = 4.

So the number of boys in the school is (4 x 4) = 16.

Let B gets Rs.x. Then we can say A gets Rs.(x + 20) and C gets Rs.(x + 35)

x + 20 + x + x + 35 = 385 => 3x + 55 = 385 => 3x = 330 => x = 110.

C’s share = Rs.(110 + 35) = Rs.145.

We can assume ratio as 5x, 6x, 8x

Now, the increment of new salary of Vinod is 10% = 110/100, Kamal is 20% = 120/100 and Krishna is 25% = 125/100

Vinod’s new salary = 110 x 5x/100 = 55x/10; Kamal’s new salary = 36x/5 and Krishna’s new salary = 10x.

New ratio = 55x/10 : 36x/5 : 10x = 11/5 : 36/5 : 50/5 = 11 : 36 : 50

Let the number of 50p, 25p, 10p coins be 5x, 9x, 4x respectively

We can say two 50 paise, four 25 paise, ten 10 Paise make one rupee respectively.

Then 5x/2 + 9x/4 + 4x/10 = 206 => x = 40

So Number of 50 paise coins = 5x = 5 x 40 = 200 coins

Number of 25 paise coins = 9x = 9 x 40 = 360 coins

Number of 10 paise coins = 4x = 4 x 40 = 160 coins