1. **Stating the problem:** We have a frequency distribution showing the distances (in multiples of 1000 km) that wires lasted before becoming unusable for 60 wires. 2. **Organizing the data:** The given data is: | Distance (1000 km) | Frequency (f) | |-------------------|---------------| | 18 - 24 | 7 | | 24 - 30 | 1 | | 30 - 36 | 6 | | 36 - 42 | 5 | | 42 - 48 | 3 | | 48 - 54 | 1 | 3. **(i) Find the mode(s) of diffusion:** Mode is the value(s) that appear most frequently. From frequencies, the highest frequency is 7 in class 18 - 24. So, the mode class is 18-24. 4. **(ii) Find the mean distance:** We use the midpoint of each class as the class mark ($x$): $\text{Class midpoints} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$ | Distance | Midpoint ($x$) | Frequency ($f$) | |----------|----------------|-----------------| | 18-24 | $\frac{18+24}{2} = 21$ | 7 | | 24-30 | $\frac{24+30}{2} = 27$ | 1 | | 30-36 | $\frac{30+36}{2} = 33$ | 6 | | 36-42 | $\frac{36+42}{2} = 39$ | 5 | | 42-48 | $\frac{42+48}{2} = 45$ | 3 | | 48-54 | $\frac{48+54}{2} = 51$ | 1 | Calculate $f \times x$ for each class: $7 \times 21 = 147$ $1 \times 27 = 27$ $6 \times 33 = 198$ $5 \times 39 = 195$ $3 \times 45 = 135$ $1 \times 51 = 51$ Sum of frequencies: $7+1+6+5+3+1 = 23$ (Note: The problem states a 60-year sample, but frequencies sum to 23; we'll proceed with actual sum 23.) Sum of $f \times x = 147 + 27 + 198 + 195 + 135 + 51 = 753$ Mean distance $= \frac{\sum f x}{\sum f} = \frac{753}{23} \approx 32.74$ (thousands of km) So, the mean distance is approximately $32,740$ km. 5. **(iii) Estimate the annual tire cost for a businessman:** Given: - Number of cars = 10 - Distance driven per car per year = 25,000 km - Each tire costs Rs. 10,000 Find the average distance a wire lasts: approx. 32,740 km (from above) Number of wire replacements per car per year = $\frac{25,000}{32,740} \approx 0.764$ (i.e. about 0.764 times per year) Assuming each car has 4 tires, total tires replaced in 10 cars: Total tire replacements = $10 \times 4 \times 0.764 = 30.56$ Total cost = $30.56 \times 10,000 = 305,600$ **Final answers:** (i) Mode class: 18-24 (thousands of km) (ii) Mean running distance: approx. $32,740$ km (iii) Estimated annual tire cost: Rs. 305,600