1. **State the problem:** We need to fence a rectangular garden with dimensions 3 m by 6 m without cutting or bending any fence pieces. The fence pieces come in lengths 0.6 m, 1.8 m, and 2.1 m. We want to find the smallest number of fence pieces needed to cover the entire perimeter. 2. **Calculate the perimeter:** The perimeter $P$ of a rectangle is given by $$P = 2 \times (\text{length} + \text{width})$$ Substitute the values: $$P = 2 \times (6 + 3) = 2 \times 9 = 18 \text{ meters}$$ 3. **Determine how to cover 18 meters using the fence pieces without cutting:** We want to find the minimum number of pieces from lengths 0.6 m, 1.8 m, and 2.1 m that sum exactly to 18 m. 4. **Check multiples of each fence size:** - Using only 0.6 m pieces: $$\frac{18}{0.6} = 30 \text{ pieces}$$ - Using only 1.8 m pieces: $$\frac{18}{1.8} = 10 \text{ pieces}$$ - Using only 2.1 m pieces: $$\frac{18}{2.1} \approx 8.571... \text{ (not an integer, so not possible)}$$ 5. **Try combinations to reduce the number of pieces:** - Using 1.8 m pieces only requires 10 pieces. - Try mixing 2.1 m and 0.6 m pieces to get 18 m exactly with fewer pieces. 6. **Set up equation for combination:** Let $x$ be the number of 2.1 m pieces and $y$ be the number of 0.6 m pieces. $$2.1x + 0.6y = 18$$ We want to minimize $x + y$ with $x,y \geq 0$ integers. 7. **Solve for $y$:** $$0.6y = 18 - 2.1x$$ $$y = \frac{18 - 2.1x}{0.6} = 30 - 3.5x$$ Since $y$ must be an integer, $3.5x$ must be an integer, so $x$ must be even. 8. **Try even values for $x$:** - For $x=0$: $y=30$, total pieces $=30$ (too many) - For $x=2$: $y=30 - 3.5 \times 2 = 30 - 7 = 23$, total pieces $= 2 + 23 = 25$ (more than 10) - For $x=4$: $y=30 - 14 = 16$, total pieces $= 4 + 16 = 20$ (more than 10) - For $x=6$: $y=30 - 21 = 9$, total pieces $= 6 + 9 = 15$ (more than 10) - For $x=8$: $y=30 - 28 = 2$, total pieces $= 8 + 2 = 10$ (equal to 10) 9. **Try mixing 1.8 m and 2.1 m pieces:** Let $a$ be number of 1.8 m pieces and $b$ be number of 2.1 m pieces. $$1.8a + 2.1b = 18$$ Solve for $a$: $$a = \frac{18 - 2.1b}{1.8} = 10 - 1.166\overline{6}b$$ $a$ must be integer, so $1.166\overline{6}b$ must be integer. Try values of $b$: - $b=0$: $a=10$, total $=10$ - $b=3$: $a=10 - 3.5 = 6.5$ (not integer) - $b=6$: $a=10 - 7 = 3$, total $= 6 + 3 = 9$ - $b=9$: $a=10 - 10.5 = -0.5$ (not valid) 10. **Check $b=6$, $a=3$ solution:** Total pieces $= 6 + 3 = 9$ pieces. Check length: $$1.8 \times 3 + 2.1 \times 6 = 5.4 + 12.6 = 18$$ 11. **Conclusion:** The smallest number of pieces needed is 9, using 3 pieces of 1.8 m and 6 pieces of 2.1 m. **Final answer:** $$\boxed{9}$$ pieces of fence are needed.