Great-Grand-Pa is a lovely person, but, sometimes, can be quite difficult (he is set in his own ways). He must have his eggs boiled exactly for fifteen minutes (from the moment the egg is placed in room-temperature water on the stove), not a second less nor a second more.

Imagine that one day Great-Grand-Pa asks you to boil an egg for him. Imagine that you, and Great-Grand-Pa, do not have any way to measure time other than two hourglasses.

The larger hourglass takes exactly eleven minutes for all the sand to descend.

The smaller hourglass takes exactly seven minutes for all the sand to descend.

(a) How do you measure exactly fifteen minutes with the two hourglasses?

(b) Suppose that Great-Grand-Pa is also very impatient and wishes to have his egg served in fifteen minutes, what would you do? What is the shortest time you need to cook Great-Grand-Pa's eggs to his standard of perfection?

(Visualize some manipulations of the two hourglasses and a timeline to boil the egg.)

(Hint: Draw a sketch of the two hourglasses, and a timeline, or a storyboard, of when to turn the hourglasses and when to boil the egg.)

Let's solve the problem step-by-step, visualizing the manipulations of the hourglasses and the sequence of events needed to measure **exactly 15 minutes**.

### Part (a): Measuring Exactly 15 Minutes
You have two hourglasses:
- The larger one takes 11 minutes to empty.
- The smaller one takes 7 minutes to empty.

You need to time exactly 15 minutes using these hourglasses.

#### Steps to Measure 15 Minutes:
1. **Start both hourglasses at the same time**.
   - The 11-minute and 7-minute hourglasses are both flipped and sand begins to flow.

2. **After 7 minutes**, the smaller hourglass runs out of sand.
   - At this point, **4 minutes** of sand are still left in the 11-minute hourglass.

3. **Flip the 7-minute hourglass immediately** (while the larger one continues running).

4. **When the 11-minute hourglass runs out (at 11 minutes)**:
   - At this moment, **11 minutes** have passed in total.
   - The 7-minute hourglass still has **3 minutes** of sand remaining (since it was flipped after 7 minutes).

5. **Flip the 11-minute hourglass immediately** after it runs out (at 11 minutes).
   - The 7-minute hourglass will run out after 3 more minutes, bringing the total time to **14 minutes**.

6. **When the 7-minute hourglass runs out (at 14 minutes)**, flip it again immediately.
   - At this point, you have **1 more minute** to measure.

7. **Let the 7-minute hourglass run for 1 minute**, and when 1 minute passes, you will have measured **exactly 15 minutes**!

### Part (b): Shortest Time to Boil the Egg
Now, imagine Great-Grand-Pa is very impatient and wants his egg to be cooked as soon as possible, which means starting the egg boiling immediately. You need to minimize the time wasted.

#### Shortest Method:
To minimize waiting time, you can follow the same hourglass manipulation process, but **start boiling the egg immediately** when you start the hourglasses. Then follow the same sequence:

- As soon as both hourglasses are flipped, **start boiling the egg**.
- After 15 minutes of the above method, the egg will be perfectly cooked to Great-Grand-Pa's standards.

So, **the shortest time is 15 minutes**, as you can't start the egg earlier without having a way to accurately measure the cooking time from the start.

### Visual Timeline:
Here’s a rough timeline of events based on the steps:
1. **0 min**: Start both hourglasses and place the egg in the water.
2. **7 min**: 7-minute hourglass finishes. Flip the 7-minute hourglass.
3. **11 min**: 11-minute hourglass finishes. Flip the 11-minute hourglass.
4. **14 min**: 7-minute hourglass finishes. Flip the 7-minute hourglass again.
5. **15 min**: 1 minute passes (after 14 minutes), and the egg is perfectly cooked.

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Would you like more details or a visual timeline? Here are five related questions to extend this scenario:

1. How could you use only one of the hourglasses to measure specific time intervals like 1 minute or 4 minutes?
2. What if the hourglasses had different times, say 13 minutes and 5 minutes, could you still measure 15 minutes?
3. Can you think of other creative ways to measure time with just two hourglasses?
4. How would the process change if Great-Grand-Pa wanted his egg cooked for 10 minutes instead?
5. Could this strategy be adapted if you had hourglasses that weren't perfect (e.g., one ran slightly faster)?

**Tip**: When solving time measurement puzzles, always focus on the points where the hourglasses run out, as these are moments when new time intervals can be marked.

Great-Grand-Pa is a lovely person, but, sometimes, can be quite difficult (he is set in his own ways). He must have his eggs boiled exactly for fifteen minutes (from the moment the egg is placed in room-temperature water on the stove), not a second less nor a second more. Imagine that one day Great-Grand-Pa asks you to boil an egg for him. Imagine that you, and Great-Grand-Pa, do not have any way to measure time other than two hourglasses. The larger hourglass takes exactly eleven minutes for all the sand to descend. The smaller hourglass takes exactly seven minutes for all the sand to descend. (a) How do you measure exactly fifteen minutes with the two hourglasses? (b) Suppose that Great-Grand-Pa is also very impatient and wishes to have his egg served in fifteen minutes, what would you do? What is the shortest time you need to cook Great-Grand-Pa's eggs to his standard of perfection?

Learn how to measure exactly 15 minutes using two hourglasses of 11 minutes and 7 minutes each. This problem involves time measurement strategies and covers key concepts of modular arithmetic. Suitable for students in Grades 5-7.

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