A 70 cm long wire is to be cut into two pieces such that one piece will be $\frac{2}{5}$ as long as the other. How many centimetres will the shorter piece be?

### Concept & Strategy When one part is described as a fraction of another part, you can use the Ratio Method. If Piece A is $\frac{2}{5}$ of Piece B, the ratio of Piece A to Piece B is 2:5. The total length is divided into proportional parts. ### Step-by-Step Solution * **Establish the ratio:** Let the longer piece be $x$. The shorter piece is $\frac{2}{5}x$. The ratio of Shorter to Longer = $2 : 5$. * **Find the total parts:** Total parts = $2 + 5 = 7$ units. * **Determine the value of one unit:** We know the total wire length is 70 cm. $$7 \text{ units} = 70 \text{ cm}$$ $$1 \text{ unit} = 10 \text{ cm}$$ * **Calculate the shorter piece:** Shorter piece = 2 units $$2 \times 10 \text{ cm} = 20 \text{ cm}$$ ### Exam Strategy & Shortcut You can instantly solve this mentally by realizing the parts sum to $2+5=7$. The shorter piece is exactly $\frac{2}{7}$ of the *entire* wire. $$\frac{2}{7} \times 70 = 20 \text{ cm}$$ This bypasses setting up variables entirely. ### Common Pitfall A common trap is assuming the shorter piece is $\frac{2}{5}$ of the *total* wire length (which would yield $70 \times \frac{2}{5} = 28$). Carefully read whether it says "$\frac{2}{5}$ of the total" or "$\frac{2}{5}$ as long as the *other*". ### Final Answer **Therefore, the correct answer is 20.**