Ratio And Proportion, Indices, Logarithms - 3 - Free MCQ Practice Test

We are given the equation:

2^x × 3^y × 5^z = 360

Let's first factor 360 into its prime factors:

360 = 2³ × 3² × 5

Now we can compare the powers of each prime factor on both sides of the equation.

• The prime factor 2 appears as 2³ on the right-hand side.
• The prime factor 3 appears as 3² on the right-hand side.
• The prime factor 5 appears as 5¹ on the right-hand side.

So, we can equate the exponents of the corresponding primes:

• x = 3 (for 2^x = 2³)
• y = 2 (for 3^y = 3²)
• z = 1 (for 5^z = 5¹)

Thus, the values of x, y, and z are:

x = 3, y = 2, z = 1

The correct answer is a) 3, 2, 1.

= [log₁₀ 25 - log₁₀(2³) + log₁₀(4)²]
= (1.397940009 - 0.903089987 +  1.204119983)

= 1.698970005

2^x-2^x-1 =4 , then taking common 2^x-1
2^x-1(2-1) = 4 , 2^x-1 =4 ,
2^x-1=2², simplify
x-1 = 2, x = 3 then
x^x = 3³ = 27
hence 27 is the required answer

• 49+a=3
  68+a  4
• 4 x (49+a) = 3 x (68+a)
• 196 + 4a = 204 + 3a
• a=8

Let's denote:

• A's investment = Rs. 1,26,000
• B's investment = Rs. 84,000
• C's investment = Rs. 2,10,000

The total investment is the sum of their individual investments: Total investment = Rs. 1,26,000 + Rs. 84,000 + Rs. 2,10,000 = Rs. 4,20,000

Now, we find the ratio of each person's investment to the total investment:

• A's ratio = Rs. 1,26,000 / Rs. 4,20,000 = 3/10
• B's ratio = Rs. 84,000 / Rs. 4,20,000 = 1/5
• C's ratio = Rs. 2,10,000 / Rs. 4,20,000 = 1/2

Now, we multiply each person's ratio by the total profit to find their share of the profit:

• A's share = (3/10) * Rs. 2,42,000 = Rs. 72,600
• B's share = (1/5) * Rs. 2,42,000 = Rs. 48,400
• C's share = (1/2) * Rs. 2,42,000 = Rs. 1,21,000

Calculation of the Ratios a : b : c : d

Given the relationships:

• (a : b = 2 : 3)
• (b : c = 4 : 5)
• (c : d = 6 : 7)

To find (a : b : c : d), we need to align the ratios so each segment transitions smoothly into the next:

1. Adjust (a : b) to match (b : c): Multiply (a : b = 2 : 3) by 4/3 to align 'b' with (b : c = 4 : 5), giving (a : b = 8 : 12).
2. Adjust (b : c) for matching with (c : d): Multiply (b : c = 4 : 5) by 6/5 to match 'c' in (c : d = 6 : 7), resulting in (b : c = 24 : 30).
3. Adjust \(c : d\): Directly use (c : d = 6 : 7) and adjust so 'c' is 30, giving (c : d = 30 : 35).
4. Final alignment: Multiply (a : b = 8 : 12) by 2 to ensure consistency, making (a : b : c : d = 16 : 24 : 30 : 35).

The finalized ratios are thus:

a : b : c : d = 16 : 24 : 30 : 35

log₄(x² + x) - log₄(x + 1) = 2
⇒ log₄((x² + x) / (x + 1)) = 2
⇒ (x(x + 1)) / (x + 1) = 4²
⇒ x = 16

- Let the fourth proportional be y.
- According to the concept of proportionality, x:2x :: (x+1):y
- This can be written as x/2x = (x+1)/y
- Cross multiplying, we get y = 2(x+1)
- Simplifying, we get y = 2x + 2
Therefore, the fourth proportional to x, 2x, and (x+1) is 2x+2, which is option C.
so just multiply x+1 by 2=2(x+1)=2x+2

• x = 3^1/3 + 1/3^1/3
• x = (3^2/3 + 1)/ 3^1/3
• 3^1/3x =  (3^2/3 + 1)
• Cubing both sides
• 3x³ = (3^2/3 + 1)³
• 3x³ = 3² + 1 + 3.3^2/3(1+ 3^2/3)
• 3x³ = 10 + x.3.3
• 3x³ -  9x = 10

Given the equation:

log_a b + log_a c = 0

Step 1: Apply the logarithmic sum property
The sum of two logarithms with the same base can be combined:
log_a b + log_a c = log_a (b ⋅ c)
So the equation becomes:
log_a (b ⋅ c) = 0

Step 2: Understand the meaning of log_a x = 0
From logarithmic properties, we know that:
log_a x = 0 implies x = a⁰
And since a⁰ = 1, we get:
x = 1

Step 3: Apply this to b ⋅ c
From log_a (b ⋅ c) = 0, we conclude:
b ⋅ c = 1
Final Result:
If log_a b + log_a c = 0, then:
b ⋅ c = 1

Let log₅ 625 = x.

Then, 5^x = 625 = 5⁴ or x = 4.

Let log₂(log₅ 625) = y.

Then, log₂ 4 = y or 2^y= 4 = 2² or y = 2.

∴ log₂(log₅ 625) = 2.

A/B = 2/5

so in (10A+3B)/(5A+2B)

divide both numerator and denominator by B

(10A/B +3)/(5A/B +2)

put A/B value

this becomes (10 * 2/5 + 3)/(5 * 2/5 + 2)

=7/4 = 7:4

The sub-duplicate ratio is obtained by taking the square roots of both terms in the given ratio. Thus, the calculations are as follows:

• Square root of 81: √81 = 9
• Square root of 64: √64 = 8

This leads to the ratio:

• Resulting ratio: 9:8

Subduplicate Ratio: The subduplicate ratio m:n is the ratio √m : √n.
25:36
√25 : √36
5 : 6

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

Compounded ratio = (2/3 × 9/4 × 5/6 × 8/10)

First, multiply the numerators together:
2 × 9 × 5 × 8 = 720

Then, multiply the denominators together:
3 × 4 × 6 × 10 = 720

So the compounded ratio is:
720 / 720 = 1 : 1

Thus, the correct option is:
1 : 1

Substitute the given ratio into the formula for the sub-triplicate ratio.
a∶b
√a∶√b
√8 : √27
2:3

The triplicate ratio of 2:3 is 
(2/3)³   =    8/27

Let the number of students in the two classes be 5x and 7x respectively.

According to the problem, if 10 students leave from each class, the remaining students in the two classes would be (5x - 10) and (7x - 10) respectively.

It is given that the ratio of the remaining students is 4:6, which simplifies to 2:3. Therefore, we can set up the equation:

(5x - 10) / (7x - 10) = 2 / 3

Cross-multiplying to solve for x:

3(5x - 10) = 2(7x - 10)

Expanding both sides:

15x - 30 = 14x - 20

Bringing like terms together:

15x - 14x = -20 + 30

x = 10

Substituting x = 10 back into the expressions for the number of students:

For the first class:
5x = 5(10) = 50

For the second class:
7x = 7(10) = 70

Thus, the number of students in each class is 50 and 70, respectively.

The correct answer is: Option 4: 50, 70

The antecedent is the numerator of the fraction, and the consequent is the denominator of the fraction.
Let the consequent of the fraction be x, when antecedent is 15.

Given:
15/x = 3/4

Rearranging:
x = (15 × 4) / 3

Simplifying:
x = 20

Final Answer: x = 20

The duplicate ratio of 3:4 is calculated by squaring both terms, resulting in:

• 3²:4² = 9:16

( a) 1:4  is right. 

Explanation:- 
A duplicate ratio of 3:4, will be the square of the ratio, so this will be  9:16
And the compounded ratio will be pa roduct of two ratios:-
4:9 * 9:16
=> 1:4 
( 9 will be cancelled and 16 by 4 )

The given ratio is 5:7, where 5 is the antecedent (the first term) and 7 is the consequent (the second term).

The inverse ratio would flip these, becoming 7:5.

If the consequent of the inverse ratio is 5, this confirms that the inverse ratio is indeed 7:5, as the consequent (second term) is 5.

You're asked for the antecedent of the inverse ratio. Since we've established that the inverse ratio is 7:5, the antecedent (first term) is 7.

Therefore, the correct answer is: 7

2x+7x+11x=180 [ as sum of all angles of triangle is 180 ]
20x=180
x=9
2(9) 7(9) 11(9)
18 ,63, 99

Given as :

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

Two numbers are in the ratio 2:3.

Let the two numbers are 2x and 3x.

If 4 be subtracted from each, they are in the ratio 3:5.

The value of x is 8.

The two numbers are

2x = 2 * 8 = 16

3x = 3 * 8 = 24

Therefore the two numbers are 16 and 24.

(x²y+xy²):(x³ +y³) divide boht numerator and denominator by y³

(x²/y² + x/y) / ((x/y)³ + 1)

put value of x/y = 3/4

(9/16 + 3/4) / (27/64 + 1)

= 12/13

We don't know the numbers so let it be xX

7x,10x and they said that the difference of two numbers is 105

10x-7x=105

3x=105

X=105/3

X=35

So the numbers are 245,350