Ms. B and Ms. D are writing problems for an algebra book. Ms. B started with ten
problems already written, and she can write six problems an hour. Ms. D had no
problems written, but she writes ten problems an hour. When will Ms. D have the same
number of problems written as Ms. B? How many problems will they each have written
at that time?

Let's define variables to represent the number of problems written by each person after \( t \) hours.

- **Ms. B** starts with 10 problems and writes 6 problems per hour. Therefore, her total number of problems after \( t \) hours is:

\[
\text{Problems by Ms. B} = 10 + 6t
\]

- **Ms. D** starts with 0 problems but writes 10 problems per hour. Therefore, her total number of problems after \( t \) hours is:

\[
\text{Problems by Ms. D} = 10t
\]

We are asked to find when Ms. D will have the same number of problems as Ms. B. This occurs when:

\[
10 + 6t = 10t
\]

### Solving the equation:

\[
10 + 6t = 10t
\]

Subtract \( 6t \) from both sides:

\[
10 = 4t
\]

Now, divide both sides by 4:

\[
t = \frac{10}{4} = 2.5
\]

So, after 2.5 hours, Ms. D will have written the same number of problems as Ms. B.

### Finding the number of problems they each have written at that time:

Substitute \( t = 2.5 \) into either equation. Using Ms. D’s equation:

\[
\text{Problems by Ms. D} = 10 \times 2.5 = 25
\]

Therefore, after 2.5 hours, both Ms. B and Ms. D will have written 25 problems.

### Conclusion:
After 2.5 hours, both Ms. B and Ms. D will have written 25 problems.

Would you like more details, or do you have any questions?

Here are 5 related questions you might consider:
1. What if Ms. B wrote problems at a different rate, say 8 problems per hour?
2. How long would it take Ms. D to have double the number of problems as Ms. B?
3. If Ms. D started with 5 problems instead of 0, how would that change the result?
4. What if both Ms. B and Ms. D started writing at the same time but with different starting numbers of problems?
5. How many problems would Ms. B have after 4 hours, and how does that compare to Ms. D?

**Tip:** Setting up equations to represent real-world situations can make it easier to solve word problems!

Ms. B and Ms. D are writing problems for an algebra book. Ms. B started with ten problems already written, and she can write six problems an hour. Ms. D had no problems written, but she writes ten problems an hour. When will Ms. D have the same number of problems written as Ms. B? How many problems will they each have written at that time?

Learn how to solve a linear system problem where Ms. B and Ms. D are writing problems at different rates. This problem involves linear equations and solving for the time when both will have written the same number of problems. Suitable for students in Grades 6-8.

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