CA Foundation P3 · Chapter 1

Ratio and Proportion, Indices and Logarithms MCQs with Answers

9 multiple-choice questions on Ratio and Proportion, Indices and Logarithms for CA Foundation P3 Quantitative Aptitude. Try each one before revealing the answer and explanation.

1. Question 1

   Two numbers are in the ratio 5 : 8. If 9 is added to each number, the ratio becomes 2 : 3. The two numbers are:

   • A) 40 and 64
   • B) 45 and 72
   • C) 54 and 81
   • D) 50 and 80
   Show answer & explanation

   Answer: B) 45 and 72

   Let the numbers be 5x and 8x. Then (5x + 9)/(8x + 9) = 2/3, so 15x + 27 = 16x + 18, giving x = 9. The numbers are 5 x 9 = 45 and 8 x 9 = 72. Check: 54/81 = 2/3. The option 54 and 81 gives the numbers after adding 9, not the original numbers.

2. Question 2

   The duplicate ratio of 3 : 7 is:

   • A) √3 : √7
   • B) 27 : 343
   • C) 7 : 3
   • D) 9 : 49
   Show answer & explanation

   Answer: D) 9 : 49

   The duplicate ratio of a : b is a² : b². Hence the duplicate ratio of 3 : 7 is 3² : 7² = 9 : 49. √3 : √7 is the sub-duplicate ratio, 27 : 343 is the triplicate ratio and 7 : 3 is the inverse ratio.

3. Question 3

   The fourth proportional to 4, 10 and 14 is:

   • A) 5.6
   • B) 28
   • C) 35
   • D) 20
   Show answer & explanation

   Answer: C) 35

   If x is the fourth proportional, then 4 : 10 = 14 : x, so 4x = 10 x 14 = 140 and x = 35. The value 5.6 comes from wrongly computing 4 x 14 / 10.

4. Question 4

   The mean proportional between 12 and 75 is:

   • A) 30
   • B) 43.5
   • C) 45
   • D) 15
   Show answer & explanation

   Answer: A) 30

   The mean proportional between a and b is √(ab). Here √(12 x 75) = √900 = 30. The value 43.5 is the arithmetic mean (12 + 75)/2, which is a common confusion.

5. Question 5

   The value of (xᵃ/xᵇ)^(a+b) · (xᵇ/xᶜ)^(b+c) · (xᶜ/xᵃ)^(c+a) is:

   • A) x^(a+b+c)
   • B) 0
   • C) x^(2(a+b+c))
   • D) 1
   Show answer & explanation

   Answer: D) 1

   Each factor simplifies using (xᵃ/xᵇ)^(a+b) = x^((a−b)(a+b)) = x^(a² − b²). The product is x^((a² − b²) + (b² − c²) + (c² − a²)) = x⁰ = 1. Note x⁰ equals 1, not 0.

6. Question 6

   If 3ˣ = 5ʸ = 15ᶻ, then z equals:

   • A) x + y
   • B) xy/(x + y)
   • C) (x + y)/xy
   • D) √(xy)
   Show answer & explanation

   Answer: B) xy/(x + y)

   Let 3ˣ = 5ʸ = 15ᶻ = k. Then 3 = k^(1/x), 5 = k^(1/y) and 15 = k^(1/z). Since 15 = 3 x 5, k^(1/z) = k^(1/x + 1/y), so 1/z = 1/x + 1/y = (x + y)/xy. Therefore z = xy/(x + y).

7. Question 7

   Given log 2 = 0.3010 and log 3 = 0.4771, the value of log 72 is:

   • A) 1.8572
   • B) 1.3801
   • C) 2.0333
   • D) 1.8062
   Show answer & explanation

   Answer: A) 1.8572

   72 = 8 x 9 = 2³ x 3². So log 72 = 3 log 2 + 2 log 3 = 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542 = 1.8572. Taking 3² as 3 gives 1.3801, and swapping the powers (2² x 3³ = 108) gives 2.0333.

8. Question 8

   If log₄ x = 2.5, then x is equal to:

   • A) 10
   • B) 16
   • C) 32
   • D) 64
   Show answer & explanation

   Answer: C) 32

   log₄ x = 2.5 means x = 4^2.5 = (2²)^2.5 = 2⁵ = 32. The value 10 comes from multiplying 4 by 2.5, and 16 and 64 are 4² and 4³ respectively.

9. Question 9

   If a² + b² = 7ab (a, b > 0), then log((a + b)/3) is equal to:

   • A) log a + log b
   • B) ½(log a + log b)
   • C) ½ log(a + b)
   • D) log a · log b
   Show answer & explanation

   Answer: B) ½(log a + log b)

   Adding 2ab to both sides gives a² + 2ab + b² = 9ab, i.e. (a + b)² = 9ab. Hence (a + b)/3 = √(ab). Taking logs, log((a + b)/3) = ½ log(ab) = ½(log a + log b).