Ratio And Proportion, Indices, Logarithms - 1 - CA Foundation Quant Aptitude

The inverse ratio of 11:15 is 15:11

The correct answer is Option d: 20
3/4 = 15/x
Now, Cross Multiplying 
3 X x= 4 X 15
3x= 60
x= 20

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

To find the ratio compounded of the given ratios 2 : 3, 9 : 4, 5 : 6, and 8 : 10, we need to multiply the numerators together and the denominators together:

Given ratios:
2 : 3, 9 : 4, 5 : 6, 8 : 10

Compounding these ratios involves multiplying the numerators together and the denominators together:

Numerators:
2 × 9 × 5 × 8

Denominators:
3 × 4 × 6 × 10

Let’s calculate each step by step.

Multiplying the numerators:
2 × 9 = 18
18 × 5 = 90
90 × 8 = 720

Multiplying the denominators:
3 × 4 = 12
12 × 6 = 72
72 × 10 = 720

So, the compounded ratio is:
720 : 720

That simplifies to 1 : 1.

Therefore, the ratio compounded of 2 : 3, 9 : 4, 5 : 6, and 8 : 10 is 1 : 1.

Given ratio is 3:4

The duplicate ratio of a : b is a² : b²

Therefore, the answer is 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.

The sub triplicate ratio involves taking the cube roots of each term in the original ratio. For the given ratio 8 : 27, the process is as follows:

• Calculate the cube root of 8: ∛8 = 2
• Calculate the cube root of 27: ∛27 = 3

Thus, the sub triplicate ratio is 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 )

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

Substituting the value of x, we get the angles.

2x = 2X9 = 18
7x = 7X9 = 63
11x = 11X9 = 99

To verify, add the angles. 

99 + 18 + 63 = 180.

This confirms the answer.

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

Given:

Ratio of two numbers = 3 ∶ 2

Sum of their squares = 468

Calculation:

Let the numbers be 3x and 2x

Sum of their squares = (3x)2 + (2x)2

⇒ Sum of their squares = 9x2 + 4x2 = 13x2

Now, 13x2 = 468

⇒ x2 = 36

⇒ x = 6

Larger number = 3x 

⇒ 3 × 6 = 18 

∴ The larger number is 18

Here find both persons earning in 1 hour.
As Anand earns Rs. 80 in 7 hours, his earning per hour is 80/7
And Promod earns Rs 90 in 12 hours, his earning per hour is 90/12
So the ratio of their earning is Anand : Promod = 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]

Given:

• Ratio of average temperature between P and Q: P : Q = 11 : 12
• Ratio of average temperature between P and R: P : R = 9 : 8

Let the average temperature of P, Q, and R be 11x, 12x, and 8y respectively. Since P is common in both ratios, we equate the values of P:

11x = 9y

From this, we find the relationship between x and y:

x = 9y / 11

Now, substitute x = 9y / 11 into Q = 12x:

Q = 12 × (9y / 11) = 108y / 11

Next, calculate the ratio of Q : R:

Q : R = (108y / 11) : 8y

Simplify the ratio:

Q : R = 108 : 88 = 27 : 22

The ratio of the average temperature between Q and R is: 27 : 22

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

We are given two ratios:
p : q = 2 : 3
x : y = 4 : 5

We need to find the value of:

(5px + 3qy) : (10px + 4qy)

Step 1: Express p and q in terms of a common variable
Let p = 2k and q = 3k, where k is a constant.

Step 2: Express x and y in terms of a common variable
Let x = 4m and y = 5m, where m is another constant.

Step 3: Substitute values into the expression
(5px + 3qy) : (10px + 4qy)

Substituting values:

(5(2k)(4m) + 3(3k)(5m)) : (10(2k)(4m) + 4(3k)(5m))

Step 4: Simplify numerator and denominator
Numerator:

5(2k)(4m) = 40km, 3(3k)(5m) = 45km

40km + 45km = 85km

Denominator:

10(2k)(4m) = 80km, 4(3k)(5m) = 60km

80km + 60km = 140km

Step 5: Simplify the ratio
85km : 140km = 85 : 140 = 17 : 28

Thus, the value of the expression is:

17 : 28

19-x / 31-x = 1 / 4
76 - 4x = 31 - x
-3x = -45
x = 45 / 3
x = 15
So, the 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 formula to find the mean proportional between two numbers \(a\) and \(b\) is:

Mean Proportional = √(a × b)

Substituting the given numbers (a = 25) and (b = 81):

Mean Proportional = √(25 × 81) = √2025 = 45

Therefore, the mean proportional between 25 and 81 is 45.

Let the required number be x . The problem states:

X/26 = 6/13

Solving for  x:
x = (6 / 13) × 26
x = 156 / 13
x = 12

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.