W is directly proportional to U. If W = 5 when U = 3, find U when W = $\frac{2}{7}$

Analysis

The question involves Direct Variation. When a variable $W$ is directly proportional to $U$, it means that as $U$ increases, $W$ increases at a constant rate. Mathematically, this is expressed as $W = kU$, where $k$ is the constant of proportionality.

Concept

1. Find the constant ($k$): Use the initial values given ($W=5, U=3$) to determine the value of $k$.
2. Form the equation: Substitute $k$ back into the general formula $W = kU$.
3. Solve for the unknown: Use the new value of $W$ to find the corresponding value of $U$.

Step-by-Step Solution

Step 1: Find $k$ Given $W = kU$: $5 = k \times 3$ $k = \frac{5}{3}$

Step 2: Form the equation Now that we know $k = \frac{5}{3}$, our specific equation is: $W = \frac{5}{3}U$

Step 3: Solve for $U$ when $W = \frac{2}{7}$ Substitute $W = \frac{2}{7}$ into the equation: $\frac{2}{7} = \frac{5}{3}U$

To isolate $U$, multiply both sides by the reciprocal of $\frac{5}{3}$, which is $\frac{3}{5}$: $U = \frac{2}{7} \times \frac{3}{5}$ $U = \frac{2 \times 3}{7 \times 5}$ $U = \frac{6}{35}$

Conclusion

Comparing our result to the options provided, we can see that our calculated value matches option A.

Exam Tip: In variation problems, always ensure you find the constant ($k$) first. A common mistake students make is trying to use ratios incorrectly; writing out $W = kU$ clearly at the start of your working prevents this error.