An artist has completed one-fourth of a rectangular oil painting. When he will paint another 100 square centimetres of the painting, he would complete three-quarters of the painting. If the height of the oil painting is 10 cm, determine the length (in cm) of the oil painting. (J.M.E.T., 2008)

### Concept & Fraction Logic This problem is solved by translating word fractions into a simple algebraic equation to find the total area, and then using the standard area formula for a rectangle to find the missing dimension. $$ \text{Area} = \text{Length} \times \text{Height (Breadth)} $$ ### Step-by-Step Solution 1. Let the total area of the rectangular oil painting be $A$ square centimeters. 2. The artist has already completed $\frac{1}{4}$ of the painting, which is $\frac{A}{4}$. 3. Painting another $100 \text{ cm}^2$ brings the total completed portion to $\frac{3}{4}$ of the painting. We can set up the equation: $$ \frac{A}{4} + 100 = \frac{3A}{4} $$ 4. Isolate the $A$ terms on one side to solve for the total area: $$ 100 = \frac{3A}{4} - \frac{A}{4} $$ $$ 100 = \frac{2A}{4} $$ $$ 100 = \frac{A}{2} $$ 5. Multiply by 2: $$ A = 200 \text{ cm}^2 $$ 6. We are given that the height (which acts as the breadth of the rectangle) is $10 \text{ cm}$. Let the length be $L$. $$ L \times 10 = 200 $$ $$ L = \frac{200}{10} = 20 \text{ cm} $$ ### Exam Strategy & Shortcut You can solve this mentally using parts. The difference between completing $\frac{3}{4}$ and $\frac{1}{4}$ of the painting is $\frac{2}{4}$, which simplifies to $\frac{1}{2}$ (half of the painting). If half the painting is $100 \text{ cm}^2$, the full painting is $200 \text{ cm}^2$. Divide by the given height of $10 \text{ cm}$ to immediately get a length of $20 \text{ cm}$. ### Common Pitfall A potential pitfall is solving for the total area ($200 \text{ cm}^2$) and mistakenly selecting an option matching that number if it were available, rather than taking the final step to find the specific dimension (length) requested. ### Final Answer Therefore, the correct answer is **20**.