Test: Ratio And Proportion, Indices, Logarithms - 1

The inverse ratio of 11:15 is 15:11

Let the numbers be 3x , 4x
Given ; the antecedent is 15

⇒ 3x = 15
⇒  x = 5

Therefore; 4x = 4*5 = 20  

The consequent is 20. 

The ratio of two quantities is 5:7
The inverse ratio is 7:5
The antecedent is 7

2×3×9×4×5×6×8×10 = 720:720
which gives 1:1

Given ratio is 3:4

since duplicate ratio is by a^2 :b^2 

then we have 9:16 

Given is 25: 36
The sub- duplicate ratio is given by √a :√b
Hence we have √25 : √36 is 5 : 6

The duplicate ratio of 3:4 is 3²/4² = 9/16.
The triplicate ratio of 2:3 is 2³/3³ = 8/27.

Now we have 4 ratios: 4:9, 9:16, 8:27 and 9:7.

We have to calculate the ratio compounded of all these ratios.

That will be (4 x 9 x 8 x 9)/(9 x 16 x 27 x 7) = 2/21.

Sub triplicate ratio of 8:27 is 3√8:3√27=2:3.

Dupliate ratio of 3:4, will be the square of the ratio
So, this will be 9:16
And the compounded ratio will be product of two ratios:
4:9 x 9:16 = 1:4 ( 9 will be cancelled and 16 by 4 )

The duplicate ratio of 4 : 5 is 4^2 : 5^2 i.e. 16 : 25

The triplicate ratio of 1 : 3 is 1^3 : 3^3 = 1 : 27

Sub duplicate ratio of 81 : 256 is (81)^1/2 : (256)^1/2 i.e. 9 : 16

Sub triplicate ratio of 125 : 512 is (125)^1/3: (512)^1/3 i.e. 5 : 8

Therefore the compounded ratio of (16:25) ; (1:27); (9:16) and (5:8) is:

16/25 x 1/27 x 9/16 x 5/8 = 9 x 5/25 x 27 x 8 = 1/(5 x 3 x 8) = 1/120.

A/b=3/4
So,
[(2 x 3)+(3 x 4)]/[(3 x 3)+(4 x 4)]
(6+12)/(9+16)
18/25

Let no. 2x and 3x then
(2x-4) / (3x-4) = 3 / 5
10x-20 = 9x-12
x=8
Now, numbers are: 2×8 = 16 & 3×8 = 24

Sum of angle of a triangle is 180 so,
2x + 7x + 11x = 180
20x = 180
x = 9

now put the value of x than we will get the angles

2x = 2*9 = 18
7x = 7*9 = 63
11x = 11*9 = 99

to verify add them if it is 180 then we are right

99 + 18 + 63 = 180.

Here is ratio of division is 11 : 7 so,
X share = 11a and Y is 7a
total 11a + 7a = 18a
18a = 324
 a = 18 
X share = 11a = 198 rs
Y share = 7a = 126 rs

Here find both persons earning in 1 hour.
As Anand earn Rs 80 in 7 hours , so he earn 80 / 7 in one hour
And Pramod earn Rs 90 in 12 hours , so he earn 90 / 12 in one hour .
So the ratio of their earning is Anand : pramod = 80/7 / 90/12  

= 80 x 12 / 90 x 7

= 32/21

Since the difference between the two numbers is 105, they are x and x + 105. The ratio of the two numbers is 7/10. So this may be written as

7 / 10 = x / (x + 105)

7(x + 105) = 10x  [Multiply both sides by 10(x + 105) to eliminate the denominators]

7x + 735 = 10x

735 = 3x  [Subtract 7x from both sides]

245 = x  [Divide both sides by 3]

350 = x + 105

So the two numbers are 245 and 350.

Verification

245 / 350 = 49 / 70 = 7 / 10 [Divide the numerator by 5 then by 7]

From the given data, we have x = (3/4) y or 4x = 3y

(x²y + xy²) :: (x² + y²) = xy(x + y) / {(x + y)(x² - xy + y²) = (xy) / (x² - xy + y²) 

Substituting x = (3/4)y from the above, 

(x²y + xy²) :: (x² + y²) = (3y²/4) / (9y² / 16 - 3y² / 4 + y²) = 12 / (9 - 12 + 16) = 12/13 

√(p-x²) /(q-x²)  = p/q --(1) 

[Sub- duplicate ratios of, 
  a/b= √a/b] 

Squaring both sides eq. -(1) ; 

(p-x²) /(q-x²) = p²/ q²

q²( p-x²)  = p²(q-x²) 

q² p - q² x2  = p² q - p² x²

X² ( p²-q²) = pq (p-q) 

x² =  pq (p-q) / (p+q) (p-q) 

x² = pq/p+q

Therefore, option (d) none of these is the correct answer. 

If given ratio x : y then it's duplicate

ratio = x^2 : y^2

According to the problem given,

2s / 3t = ( 2s -p)^2 / (3t -p)^2

2s( 3t -p )^2 = 3t( 2s - p )^2

2s( 9t^2 - 6tp + p^2 ) = 3t(4s^2 -4sp+ p^2)

18st^2-12spt+2p^2s= 12s^2t-12spt+3p^2t

18st^2-12s^2t= 3p^2t - 2p^2s

6st(3t -2s) = p^2(3t - 2s)

6st = p^2

Therefore ,

p^2 = 6st

The ratio are as follows

p : q = 2 : 3

Let The value of p = 2 a ,

Let The value of q = 3 a

And

x : y = 4 : 5

Let The value of x = 4 b

Let The value of y = 5 b

Now, According to question

Let 5 p x + 3 q y : 10 p x + 4 q y  = m :n

19-x / 31-x =1 / 4
76 - 4x =31 - x
-3x = -45
x=45 / 3
x÷15
so that number is 15.

Let the monthly income of one person be 4x and that of the other be 5x

Let the monthly expenses of one person be 7y and that of other be 9y

ATQ

4x-7y=50-(1)-*5=20x-35y=250

5x-9y=50-(2)-*4=20x-36y=200

=y=50

from 1

4x-7*50=50

4x=50+350

4x=400

x=100

Monthly income of one person=4x=4*100=400

Monthly income of the other person=5x=5*100=500

Speed of second train = Distance travelled / time = 400 /5 = 80 km/hr
Ratio of speeds = 7 :8 = (7/8) :1
If speed of second train is x, then the speed of first train is 7x/8
7x/8 = 7 x 80/8 = 70
So, speed of first train is 70 km/hr

For proportions; the product of the internal terms equals the product of the externals.

Therefore,

=> 4 : 8 = 6: x
=> 4x = 48
=> x = 12.

So, the answer is 12.

Let the third proportion be x

12 : 18 = 18 : x

x = 18*(18/12)

x = 18*(3/2)

x = 54/2

x = 27

The ratio of the number similar to 6∶13 is 12∶26

Given:

One ratio of the number is 6 : 13 which becomes 6/13

To find:

The another ratio of number similar to 6 : 13

Solution:

Let the unknown number be x

Given that another ratio of the number is x: 26 which becomes x/26

Given that the two ratios are similar, equating the two terms, we get

Hence the ratio of the number similar to 6∶13 is 12∶26

Let father age be X and son age be Y. Then
X/Y = 7/2
2X = 7Y ..... (1)
After ten years ,
(X+10)/(Y+10) = 9/4
4(X+10) = 9(Y+10)
4X + 40 = 9Y + 90
4X = 9Y + 50
2×2X = 9Y + 50

putting the value of 2X in equation 1

2×7Y = 9Y + 50
14Y = 9Y + 50
5Y = 50
Y = 10
Son age is 10 years .

putting the value of Y in equation 1

2X = 7Y
X = 7/2Y
X = 7/2×10
X = 70/2
X = 35
Father age is 35.

I think you take all in respect to A and cut it out
∴ a= a
b = 2a
c= 5a
a:b:c = 1:2:5

p/q = r/s = 2.5/1.5 = 5/3

Now, ps:qr = ps/qr = 5 x 3/3 x 5 = 1:1

x/y = 5/3

x = 5/3 y

z/w = 5/3

z = 5/3 w

x + z = 5/3 y + 5/3 w

= 5/3( y + w)

(x + z)/(y + w) = 5/3

( 5x - 3y ) / ( 5y - 3x ) = 3/4
Cross multiplying the numbers in the left and right ,
 4 (5x - 3y) = 3 (5y - 3x)
  Opening the brackets ,
  20x - 12y = 15y - 9x
  Grouping like terms to one side ,
 20x + 9x = 15y + 12y
  29x = 27y

→ 29 * x = 27 * y
→ x / y = 27 / 29
→ x:y = 27:29

The obtained ratio for A: B: C is 9 : 6 : 10
Given : A: B = 3: 2 and B: C = 3: 5

To compare two ratios, we need to make one of the ratios equal. Here we had B term in both the ratios, so we are making two terms equal by making B equal.

Multiplying A: B i.e. 3: 2 by 3  

and B: C i.e. 3: 5 by 2 to bring the value of B at par, we get

A: B = 9: 6 & B: C = 6: 10

Hence, the value of A: B: C is obtained to be 9 : 6 : 10

Making y same, we get

x:y = 8 :12 (multiplied by 4)

y:z = 12:9  (multiplied by 3)

x:y:z = 8:12:9