Test: Ratio And Proportion, Indices, Logarithms - 4 - CA Foundation MCQ

The fourth proportional to three numbers a, b, c is given by:
Fourth Proportional = (b × c) / a
Given:
a = 4
b = 6
c = 8

Substitute the values:
Fourth Proportional = (6 × 8) / 4
= 48 / 4
= 12

To find the ratio A:B:C given A:B = 3:2 and B:C = 3:5, follow these steps:

• Write the ratios: A:B = 3:2 and B:C = 3:5.
• Make the B terms equal in both ratios. Here, B is already 3 in both.
• Combine the ratios to get A:B:C:
  • A is 3
  • B is 2 (from A:B) and 3 (from B:C)
  • C is 5
• Adjust to make a single ratio: Multiply the parts of A:B by the B value in B:C and vice-versa:
  • A = 3 × 3 = 9
  • B = 2 × 3 = 6
  • C = 5 × 2 = 10
• Thus, the combined ratio is 9:6:10.

a/ 3 = b/ 4 = c/7

a/c = 3/ 7 ; b/ c = 4/7

(a+b+c) / c ⇒ a/ c+ b/ c + 1

⇒ 3/7 + 4/7+ 1 

⇒ 1 + 1

⇒ 2 

Step 1: Set up the equation

The sum of the parts in the ratio is:
4x + 5x + 6x = 750
Combine the terms:
15x = 750

Step 2: Solve for x

Divide both sides by 15:
x = 750 / 15 = 50

Step 3: Calculate each part

• First part: 4x = 4 × 50 = 200
• Second part: 5x = 5 × 50 = 250
• Third part: 6x = 6 × 50 = 300

So, the three parts are Rs. 200, Rs. 250, and Rs. 300.

Step 4: Verify the total

200 + 250 + 300 = 750, which matches the given total.

Step 5: Check the ratio

The ratio of the parts 200:250:300 simplifies to:
200 / 50 = 4, 250 / 50 = 5, 300 / 50 = 6, which is indeed 4:5:6.

Step 6: Match with the options

• a) (200, 250, 300): Matches our result exactly.
• b) (250, 250, 250): This is a 1:1:1 ratio, not 4:5:6, and incorrect.
• c) (350, 250, 150): This does not match the ratio 4:5:6 (350:250:150 simplifies to 7:5:3), and the total is 350 + 250 + 150 = 750, but the ratio is wrong.
• d) none of these: Since option (a) matches, this is incorrect.

Final Answer:

The division of Rs. 750 into 3 parts in the ratio 4:5:6 is (200, 250, 300).
Correct Option: a)

Let the third proportion be x

12 : 18 = 18 : x

x = 18*(18/12)

x = 18*(3/2)

x = 54/2

x = 27

Let the fourth proportion be x

⇒ 2 , 5 , 8 , x are in proportion

∴ 2 : 5 = 8 : x

∴ 2×x = 5×8 (∵Product of extremes = Product of means)

∴ 2x = 40

x = 20

Hence the fourth proportion is 20.

GIVEN

The sum of the ages of 3 persons is 150 years. 10 years ago their ages were in the ratio 7:8:9.

TO FIND

Their Present ages

SOLUTION

Let the ages of the person 10 yrs ago be 7x, 8x, 9x according to the question.

Present ages will be 7x + 10, 8x+10 and 9x + 10

Now according to the Question,

(7x + 10) + (8x + 10) + (9x + 10) = 150

=> 24x + 30 = 150

=> 24x = 150 - 30

=> 24x = 120

=> x = 5

Therefore Present ages are

7x + 10 = 45yrs

8x+10 = 50yrs

9x + 10 = 55yrs

Hence their Present Ages are 45,50 and 55yrs. (ANS)

From a/b = c/d, by componendo we add the terms: (a + b)/b = (c + d)/d.

From a/b = c/d, by dividendo we subtract the terms: (a - b)/b = (c - d)/d.

Now divide the componendo result by the dividendo result:

((a + b)/b) ÷ ((a - b)/b) = ((c + d)/d) ÷ ((c - d)/d).

Simplifying both sides gives (a + b)/(a - b) = (c + d)/(c - d). Hence the operation is called Componendo and Dividendo.

• To find the missing number (*), we use the property of proportions: a/b = c/d.
• Here, we have 12/16 = */20.
• Cross-multiply to solve: 12 * 20 = 16 * *.
• This simplifies to 240 = 16 *.
• Now, divide both sides by 16: * = 240 / 16.
• Calculating gives * = 15.
• Thus, the correct answer is 15.

Recognize that 256 equals 2⁸. Therefore, (256)^-1/8 becomes (2⁸)^-1/8, which simplifies to 2^-1 or 1/2. Multiplying by 2 yields 2 × 1/2 = 1. Thus, option a is the correct answer.