Ratio and Proportion

Q1: Ratio of number of men and women in an office is 12 : 13.

(i) We can definitely say that there are 13 women in the office.

Step 1: The ratio 12 : 13 means men : women = 12 : 13, but the actual numbers can be any multiples of 12 and 13.
So, number of women can be 13, or 26, or 39, etc.
Hence, statement (i) is not definitely true.

(ii) We can say that there can be 26 women and 24 men in the office.

Step 2: Multiply ratio by 2: \[ 12 \times 2 = 24 \quad \text{men}, \quad 13 \times 2 = 26 \quad \text{women} \] Hence, statement (ii) is true.
Answer: b. only (ii)

Q2: Three numbers are in the ratio 2 : 3 : 5 and sum of their squares is 608. The numbers are:

Step 1: Let the numbers be \(2x\), \(3x\), and \(5x\).
Step 2: Sum of squares = \[ (2x)^2 + (3x)^2 + (5x)^2 = 608 \\ 4x^2 + 9x^2 + 25x^2 = 608 \\ 38x^2 = 608 \\ x^2 = \frac{608}{38} = 16 \\ x = 4 \]Step 3: Numbers are: \[ 2 \times 4 = 8, \quad 3 \times 4 = 12, \quad 5 \times 4 = 20 \]Answer: a. 8, 12, 20

Q3: 0.6 of a number equals 0.09 of another number. The ratio of the numbers is:

Step 1: Let the two numbers be \(x\) and \(y\). Given: \[ 0.6 \times x = 0.09 \times y \]Step 2: Rearranging: \[ \frac{x}{y} = \frac{0.09}{0.6} = \frac{9}{100} \div \frac{6}{10} = \frac{9}{100} \times \frac{10}{6} = \frac{9 \times 10}{100 \times 6} = \frac{90}{600} = \frac{3}{20} \]Step 3: Ratio \(x : y = \frac{3}{20}\) or \(3 : 20\).
Answer: c. 3 : 20

Q4: If A : B = 2 : 3 and 2 : A = 1 : 2, then the value of B is:

Step 1: Given: \[ A : B = 2 : 3 \\ \frac{A}{B} = \frac{2}{3} \] and \[ 2 : A = 1 : 2 \\ \frac{2}{A} = \frac{1}{2} \]Step 2: From second ratio: \[ 2 \times 2 = 1 \times A \\ A = 4 \]Step 3: Using \(A = 4\) in first ratio: \[ \frac{4}{B} = \frac{2}{3} \\ 2B = 12 \\ B = 6 \]Answer: a. 6

Q5: A fraction bears the same ratio to \(\frac{1}{27}\) as \(\frac{3}{7}\) does to \(\frac{5}{9}\). The fraction is:

Step 1: Let the required fraction be \(x\). Given: \[ \frac{x}{\frac{1}{27}} = \frac{\frac{3}{7}}{\frac{5}{9}} \]Step 2: Simplify the right side: \[ \frac{\frac{3}{7}}{\frac{5}{9}} = \frac{3}{7} \times \frac{9}{5} = \frac{27}{35} \]Step 3: Now, \[ \frac{x}{\frac{1}{27}} = \frac{27}{35} \\ x = \frac{1}{27} \times \frac{27}{35} = \frac{1}{35} \]Answer: d. \(\frac{1}{35}\)

Q6: In a class, the number of boys is more than the number of girls by 12% of the total strength. The ratio of boys to girls in the class is:

Step 1: Let the total strength be \(T\).
Number of boys = \(B\), number of girls = \(G\).
Given: \[ B = G + 0.12 T \] and \[ B + G = T \]Step 2: From total strength, \[ T = B + G = (G + 0.12 T) + G = 2G + 0.12 T \\ T – 0.12 T = 2G \\ 0.88 T = 2G \\ G = \frac{0.88 T}{2} = 0.44 T \]Step 3: Now, \[ B = T – G = T – 0.44 T = 0.56 T \]Step 4: Ratio of boys to girls: \[ B : G = 0.56 T : 0.44 T = 0.56 : 0.44 = \frac{56}{44} = \frac{14}{11} \]Answer: c. 14 : 11

Q7: The speeds of three cars are in the ratio 3 : 4 : 5. The ratio between the times taken by them to travel the same distance is:

Step 1:Speed ratio = 3 : 4 : 5
Time taken is inversely proportional to speed when distance is same.
Therefore, time ratio = inverse of speed ratio = \[ \frac{1}{3} : \frac{1}{4} : \frac{1}{5} = \frac{20}{60} : \frac{15}{60} : \frac{12}{60} = 20 : 15 : 12 \]Answer: b. 20 : 15 : 12

Q8: The ratio between two numbers is 3 : 4 and their LCM is 180. The first number is:

Step 1: Let the two numbers be \(3x\) and \(4x\).
The LCM of \(3x\) and \(4x\) is: \[ \text{LCM} = \text{LCM}(3,4) \times x = 12x \]Step 2: Given LCM = 180, so: \[ 12x = 180 \Rightarrow x = \frac{180}{12} = 15 \]Step 3: First number = \(3x = 3 \times 15 = 45\)
Answer: c. 45

Q9: If \(x : y = y : z\), then \(x^2 : y^2\) is:

Step 1: Given: \[ \frac{x}{y} = \frac{y}{z} \]Step 2: From the proportion, we have: \[ y^2 = x \times z \]Step 3: Now, calculate \(x^2 : y^2\): \[ x^2 : y^2 = x^2 : (x z) = \frac{x^2}{x z} = \frac{x}{z} \\ x : z \]Answer: c. \(x : z\)

Q10: The sum of three numbers is 174. The ratio of second number to the third number is 9 : 16 and the ratio of first number to the third one is 1 : 4. The second number is:

Step 1: Let the three numbers be \(x\) (first), \(y\) (second), and \(z\) (third).
Given: \[ y : z = 9 : 16 \\ y = 9k, \quad z = 16k \] and \[ x : z = 1 : 4 \\ x = \frac{1}{4} z = \frac{1}{4} \times 16k = 4k \]Step 2: Sum of three numbers is: \[ x + y + z = 174 \\ 4k + 9k + 16k = 174 \\ 29k = 174 \\ k = \frac{174}{29} = 6 \]Step 3: Second number: \[ y = 9k = 9 \times 6 = 54 \]Answer: a. 54