In modular arithmetic, if the divisor is 12, it is called as 12 hour clock arithmetic.

Because the divisor is 12, the remainders must be less than or equal to 11.

So, the remainder are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

"0" corresponds to the hour "12"

"1" corresponds to the hour "1"

"2" corresponds to the hour "2"

"3" corresponds to the hour "3"

"4" corresponds to the hour "4"

"5" corresponds to the hour "5"

"6" corresponds to the hour "6"

"7" corresponds to the hour "7"

"8" corresponds to the hour "8"

"9" corresponds to the hour "9"

"10" corresponds to the hour "10"

"11" corresponds to the hour "11"

**Case 1 :**

Let us say the time now is 3 hours in 12 hours clock. Find the time after 15 hours from now.

If you want to find the time after 15 hours from now, you have to add 15 to 3 and divide the result by 12.

3 + 15 = 18

When 18 is divided by 12, the remainder is 6.

So, time after 15 hours from now (3 hours) is 6 hours.

**Case 2 :**

Let us say the time now is 2 hours in 12 hours clock. Find the time after 8 hours from now.

If you want to find the time after 8 hours from now, you have to add 8 to 2 and divide the result by 12.

2 + 8 = 10

Here 10 is less than 12. So, we can not divide 10 by 12.

So, time after 8 hours from now (2 hours) is 10 hours.

**Case 3 :**

Let us say the time now is 5 hours in 12 hours clock. Find the time 32 hours back.

Subtract 32 from 5.

5 - 32 = -27

Here, we get a negative value.

In the case of getting a negative value, we have to find the next integer which is exactly divisible by 12.

That is, we have to get the next integer after 27 which is exactly divisible by 12.

It is 36.

Now, we have to write -27 in terms of 36.

So, -27 can be written as

-27 = -36 + 9

Therefore, the time 32 hours back was 9 hours.

**Problem 1 : **

It is 9 hours now in a 12 hour clock. What will be the time after 18 hours.

**Solution :**

Now, the time is 9 hours. We want to know the time after 18 hours.

To get answer for our question, we have to do the following steps.

**Step 1 : **

Add 18 to 9.

9 + 18 = 27

**Step 2 : **

Divide 27 by 12.

27 / 12

**Step 3 : **

Take the remainder, when 27 is divided by 12.

The remainder is 3.

So, the time after 18 hours will be 3 hours.

**Problem 2 : **

If the time now in a 12 hour clock is is 9 hours, what was the time 71 hours back?

**Solution :**

Now, the time is 9 hours. We want to know the time 71 hours back.

To get answer for our question, we have to do the following steps.

**Step 1 : **

Subtract 71 from 9.

That is,

9 - 71 = -62

**Step 2 : **

We get negative value in step 1 and also 62 is not divisible by 12.

So, find the next integer after 62 which is exactly divisible by 12.

That is 72.

**Step 3 :**

Write -62 in terms of 72.

So, -62 can be written as

-62 = -72 + 10

Therefore, the time 71 hours back was 10 hours.

**Problem 3 : **

Lily was assigned a work when the time was 7 hours in a 12 hours clock. She completed the work after 72 hours. At what time did she complete the work?

**Solution :**

From the question, we want to know the time 72 hours after 7 hours.

To get answer for our question, we have to do the following steps.

**Step 1 :**

Add 72 to 7.

72 + 7 = 79

**Step 2 :**

Divide 79 by 12.

79 / 12

**Step 3 :**

Take the remainder, when 79 is divided by 12.

The remainder is 7.

So, the time after 72 hours will be 7 hours.

**Problem 4 : **

Mr. Johnson started a work at 4 hours in a 12 hours clock. He was given 38 hours time to complete the work. But he took 3 hours more to complete the work. At what time did he complete the work.

**Solution :**

From the question, we want to know the time 41 hours (38 + 3) after 4 hours.

To get answer for our question, we have to do the following steps.

**Step 1 :**

Add 41 to 4.

41 + 4 = 45

**Step 2 : **

Divide 45 by 12.

**Step 3 : **

Take the remainder, when 45 is divided by 12.

The remainder is 9.

So, he completed the work at 9 hours.

**Problem 5 : **

Now the time is 6 hours and Mr. Lenin is at home. Lenin was in Washington 81 hours before. At what time was he in Washington ?

**Solution :**

Now, the time is 6 hours. We want to know the time 81 hours before.

To get answer for our question, we have to do the following steps.

**Step 1 :**

Subtract 81 from 6.

That is,

6 - 81 = -75

**Step 2 :**

We get negative value in step 1 and also 75 is not divisible by 12.

So, find the next integer after 75 which is exactly divisible by 12.

That is 84.

**Step 3 :**

Write -75 in terms of 84.

So, -75 can be written as

-75 = -84 + 9

Therefore, Mr. Lenin was in Washington at 9 hours.

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