Express the total number of days contained in k weeks and k additional days together, in terms of k.

Introduction / Context:
This is a straightforward algebra and calendar question about converting weeks and extra days into a total number of days. The expression k weeks k days means that we have k full weeks plus k extra single days. The goal is to express the total number of days purely in terms of k, using a simple algebraic expression.

Given Data / Assumptions:
- There are exactly 7 days in one week. - We are given k weeks and k additional days. - k is a positive integer standing for some number of weeks and days. - We must find a single algebraic expression for the total number of days.

Concept / Approach:
The key idea is to use the fact that each week equals 7 days. If we have k weeks, then the number of days from the weeks alone is 7 multiplied by k. We then add the extra k days to this quantity. This gives a linear expression in k. Finally, we can factor or simplify the expression if needed to get the most compact form. The result must be dimensionally consistent, representing a number of days.

Step-by-Step Solution:
Step 1: Convert weeks to days. Since 1 week has 7 days, k weeks have 7 * k days. Step 2: The extra k days are given in addition to the days from the weeks. Step 3: Total days T can be written as: T = 7k (from the weeks) + k (from the extra days). Step 4: Combine like terms in T. Both terms contain k, so factor k. Step 5: T = 7k + k = (7 + 1) * k. Step 6: Simplify inside the brackets to T = 8k. Step 7: Therefore, the total number of days in k weeks and k days is 8k days.

Verification / Alternative check:
To verify, plug in specific values for k. For example, if k equals 1, then 1 week and 1 day equals 7 + 1 = 8 days, which matches 8k = 8 * 1 = 8 days. If k equals 2, then 2 weeks and 2 days equals 2 * 7 + 2 = 14 + 2 = 16 days, and 8k equals 8 * 2 = 16 days again. Trying several values of k confirms that the formula 8k days accurately represents the total days for any positive integer k.

Why Other Options Are Wrong:
- 8 + k days suggests only 8 fixed days plus k, which does not scale correctly when k changes. - 7 + 1k days is essentially 7 + k days, again failing to reflect the 7k days contributed by the weeks. - 7/k days is dimensionally incorrect, since weeks and days should add, not divide, and this expression becomes smaller as k grows, which is clearly wrong. - 7k + 1 days gives only one extra day beyond the weeks, regardless of k, and so cannot match k extra days.

Common Pitfalls:
Students sometimes mix up multiplication and addition of units, writing 7 + k instead of 7k + k. Another mistake is to misread k weeks k days as something like k plus k weeks, which inverts the quantities. Always remember that number of days equals weeks multiplied by 7 plus standalone days, and that algebraic simplification can be done only after this unit conversion step.

Final Answer:
The total number of days in k weeks and k days together is 8k days.