Explanation:

Let’s break it down:

- Here’s our table:

| Class Interval | Frequency |

|----------------|-----------|

| 0 – 30 | 4 |

| 30 – 60 | 5 |

| 60 – 90 | 7 |

| 90 – 120 | 4 |

- The total frequency = 4 + 5 + 7 + 4 = 20.

- Median is at the 10th and 11th value. Both land in the 60–90 group (cumulative freq. before it is 9 from previous classes; so 10th and 11th are in the 60–90 group).

- Median formula: \( \text{Median} = l + \frac{\left(\frac{N}{2} - F\right)}{f} \times c \), which gives median class lower limit = 60, F = 9, f = 7, c = 30.

- Median: \(60 + \frac{1}{7} \times 30 = 60 + 4.29 \approx 64.29\).

- Mode is in highest frequency group (60–90, f1 = 7, f0 = 5, f2 = 4).

- Mode formula: \( \text{Mode} = l + \frac{f_1-f_0}{2f_1-f_0-f_2} \times c \)

- Plug in: \( l = 60, f_1 = 7, f_0 = 5, f_2 = 4, c = 30 \)

- \( \text{Mode} = 60 + \frac{7-5}{2 \times 7 - 5 - 4} \times 30 = 60 + \frac{2}{5} \times 30 = 60 + 12 = 72 \)

- So P = 64.29, Q = 72.

Now, 7(Q-P) = 9R:

- \( 7(Q-P) = 7(72 - 64.29) = 7 \times 7.71 = 53.97 \)

- \( 9R = 53.97 \implies R = \frac{53.97}{9} \approx 6 \)

- So, Option 1 (6) is what we’re looking for.

Here’s what’s happening with each option:

- Option 1: 6 — Matches calculation. Correct

- Option 2: 5 — Off by a bit.

- Option 3: 3 — Not it.

- Option 4: 1 — Way too small.