Question: Yuri’s old wheelchair wheels each weigh $840$g and the new wheels are $27\%$ lighter. Calculate the weight of a new wheel. Give your answer correct to the nearest gram.
1. **State the problem:**
We need to find the weight of the new wheelchair wheel which is $27\%$ lighter than the old wheel weighing $840$ grams.
2. **Formula and explanation:**
If the new wheel is $27\%$ lighter, it means it weighs $100\% - 27\% = 73\%$ of the old wheel's weight.
3. **Calculate the weight of the new wheel:**
$$\text{New weight} = 840 \times \frac{73}{100}$$
4. **Perform the multiplication:**
$$840 \times 0.73 = 613.2$$
5. **Round to the nearest gram:**
The weight of the new wheel is approximately $613$ grams.
**Final answer:** The new wheelchair wheel weighs $\boxed{613}$ grams.