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The problem provided can be broken down and solved systematically. Here's the problem stated:

1. A husband leaves his home for the office at 7 a.m., traveling at 30 km/hr.
2. After 10 minutes (at 7:10 a.m.), his wife realizes that he has left his ID card at home. She decides to ride a bike towards his office to deliver the ID card.
3. The task is to determine the minimum speed at which the wife should ride the bike so that the husband receives the ID card just when he reaches his office at 8 a.m.

### Step-by-Step Solution:

1. **Calculate the distance traveled by the husband in 10 minutes:**

   - Speed of the husband = 30 km/h
   - Time traveled = 10 minutes = \( \frac{10}{60} = \frac{1}{6} \) hours
   - Distance traveled by the husband in 10 minutes = Speed × Time
     \[
     \text{Distance} = 30 \times \frac{1}{6} = 5 \text{ km}
     \]

2. **Determine the total distance between the home and the office:**

   - The husband needs to travel from home to office in 1 hour.
   - Speed = 30 km/h, so the total distance to the office:
     \[
     \text{Total Distance} = 30 \times 1 = 30 \text{ km}
     \]

3. **Calculate the remaining distance the husband needs to cover:**

   - At 7:10 a.m., the husband has 55 minutes left (until 8 a.m.) to reach the office.
   - In 55 minutes, he will cover:
     \[
     \text{Remaining Distance} = 30 \times \frac{55}{60} = 30 \times \frac{11}{12} = 27.5 \text{ km}
     \]

4. **Determine the time available for the wife to catch up with her husband:**

   - The husband will take 55 minutes to cover 27.5 km.
   - The wife will cover the same distance (5 km to where the husband was at 7:10 a.m. plus the distance traveled by the husband) in 50 minutes (the time remaining until the husband reaches his destination).

5. **Set up the equation for the wife’s speed:**

   - Let the wife’s speed be \( v \) km/h.
   - The wife needs to cover 27.5 km in 50 minutes (or \( \frac{5}{6} \) hours).
   - We set up the equation:
     \[
     v \times \frac{5}{6} = 27.5
     \]
     \[
     v = \frac{27.5 \times 6}{5} = \frac{165}{5} = 33 \text{ km/h}
     \]

### Final Answer:
The wife should ride the bike at a minimum speed of **33 km/h** to ensure that the husband receives the ID card just as he reaches his office at 8 a.m.

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Do you have any questions or need further details?

### Relative Questions:
1. How would the solution change if the husband had traveled at a different speed?
2. What if the wife left 15 minutes after the husband? How would that affect her required speed?
3. If the wife's bike had a maximum speed of 30 km/h, what would be the earliest time she could catch up with the husband?
4. How does the problem change if the husband and wife started at different times but with different speeds?
5. What if there was a delay in the wife realizing that the ID card was forgotten? How would that affect her speed?

### Tip:
Always make sure to convert all units to the same system (e.g., minutes to hours) to avoid mistakes when calculating speed or distance.

A husband leaves his home for office at 7 a.m. by travelling at 30 km/hr. 10min later his wife realizes that he has left his ID card at home. So she quickly collects the same and rides a bike towards the husband's office. What should be the minimum speed at which the wife should ride the bike so that her husband receives the ID card just when he reaches his office at 8 a.m?

This problem involves calculating the minimum speed at which the wife should ride a bike to deliver an ID card to her husband, who is traveling at 30 km/hr. The problem covers concepts of distance, speed, and time, with step-by-step solutions suitable for Grades 9-10.

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