1. **State the problem:** We have three lengths of wood in the ratio $2 : 5 : 7$. The difference between two of them is 30 cm. We want to verify if Kofi's statement that the sum of the three lengths must be at least 120 cm is correct. 2. **Express the lengths:** Let the three lengths be $2x$, $5x$, and $7x$ respectively. 3. **Use the difference condition:** The difference between two lengths is 30 cm. We check all possible differences: - $5x - 2x = 3x$ - $7x - 5x = 2x$ - $7x - 2x = 5x$ These differences could be 30 cm. 4. **Find $x$ for each case:** - If $3x = 30$, then $x = 10$ - If $2x = 30$, then $x = 15$ - If $5x = 30$, then $x = 6$ 5. **Calculate sums for each $x$:** - For $x=10$: sum = $2(10) + 5(10) + 7(10) = 20 + 50 + 70 = 140$ - For $x=15$: sum = $2(15) + 5(15) + 7(15) = 30 + 75 + 105 = 210$ - For $x=6$: sum = $2(6) + 5(6) + 7(6) = 12 + 30 + 42 = 84$ 6. **Check the sums against 120 cm:** - For $x=10$ and $x=15$ sum is more than 120 cm. - For $x=6$ sum is less than 120 cm. 7. **Conclusion:** Since one possible sum is $84$ cm, which is less than 120 cm, Kofi's statement that the sum must be at least 120 cm is **not always correct**. **Final answer:** Kofi is incorrect because the sum can be less than 120 cm depending on which lengths differ by 30 cm.