If three equal cubes are placed adjacently in a row, then the ratio of the total surface area of the new cuboid to the sum of the surface areas of the three cubes will be

### Concept & Obscured Surface Area When solid shapes are joined, the faces that touch each other become "hidden" and are no longer part of the total exterior surface area. For $n$ cubes placed in a row, $n-1$ pairs of faces are hidden. $$TSA_{\text{cuboid}} = 2(lb + bh + hl)$$ ### Step-by-Step Solution 1. **Let the side of each cube be $a$.** - Surface area of one cube = $6a^2$. - Sum of surface areas of three distinct cubes = $3 \times 6a^2 = 18a^2$. 2. **Determine the dimensions of the new cuboid:** - Three cubes placed in a row make a cuboid of length $l = 3a$. - Breadth $b = a$. - Height $h = a$. 3. **Calculate the Total Surface Area (TSA) of the new cuboid:** - $TSA = 2(lb + bh + hl)$ - $TSA = 2((3a)(a) + (a)(a) + (a)(3a))$ - $TSA = 2(3a^2 + a^2 + 3a^2) = 2(7a^2) = 14a^2$. 4. **Determine the ratio:** - Ratio = $TSA_{\text{cuboid}} : \text{Sum of } TSA_{\text{cubes}}$ - Ratio = $14a^2 : 18a^2 = 14 : 18$. - Simplifying by dividing by 2 yields $7 : 9$. ### Exam Strategy & Shortcut Instead of formulas, simply count exposed square faces. - 3 independent cubes have $3 \times 6 = 18$ faces. - When 3 cubes are joined in a line, 2 internal joints hide $2 \times 2 = 4$ faces. - Exposed faces = $18 - 4 = 14$. - The ratio is instantly $14 : 18 = 7 : 9$. ### Common Pitfall Students often assume the surface area remains exactly the same when objects are joined, forgetting to subtract the internal, hidden faces where the cubes touch. ### Final Answer Therefore, the correct answer is **7 : 9**.