Sometimes in your work you will need to subtract angles in degrees and minutes. Subtraction works the same way as it does with whole numbers. There is only one difference: Carrying and borrowing is done in multiples of 60 between the degrees, minutes or seconds columns, but it is done in multiples of 10 within a column. Watch and the video and see how it is done.

### A Short Review

Subtraction can be with and without borrowing. Borrowing within the minutes and seconds column is 10-based, borrowing between columns is 60-based.

To subtract 20 minutes from 15, borrowing is necessary from the degrees.

5° becomes 4° and 60`, this 60` gets added to the 15` already in the minutes column. 20` can now be subtracted from 75`, which = 55`. Further borrowing for seconds reduces this 55` to 54`.

5° 15` 20”

– 2° 20` 25”

2° 54` 55"

### Worksheet: Level 1

### Worksheet: Level 1 Sample Questions

- Decide whether borrowing is needed in the following subtractions in DMS and;
- Specify where exactly borrowing is needed, if any:

Remember: Compare every digit in the second line with the digits directly above them.

If a digit in the second line is bigger than the digit above it, borrowing is needed from the left of that column. This borrowing could be from the tens column in the same unit or from the next

unit column.

**Examples:**

15° 25` 35”

– 05° 04` 13”

No borrowing is needed.

15° 25` 35”

– 05° 06` 13”

Borrowing is needed in the minutes column.

15° 25` 35”

– 05° 04` 36”

Borrowing is needed from minutes to seconds.

**Question:**

89° 54` 23”

1. – 52° 31` 32”

### Worksheet: Level 1 Answer Key

Answer:

**Borrowing is needed from the minutes to the seconds column.**

### Download Worksheet: Level 1

### Worksheet: Level 2

### Worksheet: Level 2 Sample Questions

**Examples:**

**Question:**

*Subtract in DMS*:

1. 21° 51` 01” from 31° 55` 22”

**Further example:**

i. Subtract 35° 51` 32” from 53° 15` 23”

### Worksheet: Level 2 Sample Answer Key

**Answer:**

31° 55` 22”

- 21° 51` 01”

** 10° 04` 21”**

No borrowing.

### Download Worksheet: Level 2

### Worksheet: Level 3

### Worksheet: Level 3 Sample Questions

**Question:**

Subtract in DMS. Remember: Subtraction can involve borrowing or no borrowing:

- An apprentice is cutting a piece of metal by hand tool, on which an angle of 82°19`34.12” is laid out in pencil. The cut veers away from the pencil line by 2°18`28.71” and makes the original angle smaller. What is the ﬁnished angle on the metal?

**Further example:**

- What is the new heading if an aircraft turns 18° 41` 54” from heading 332° 40` 31”?

### Worksheet: Level 3 Sample Answer Key

**Answer:**

1. ** 80° 01` 05.41”**

Borrowing within the seconds column is shown but not for the degrees column. Borrowing between degrees and mins shown with carry over.

### Download Worksheet: Level 3

### Infosheet: Advanced

### Additional Information: Another way to get the answer (no borrowing)

**Another Way to Get the Answer**

**Task:**

Subtract 15°09`21” from 48°15`38”* (no borrowing)*

Layout your calculations neatly, so you can review, track changes, correct or learn from them. One way a layout can look is like this:

### Additional Information: Another way to get the answer (with borrowing left-to-right)

**Another way to get the answer**

Subtract 04°11`22” from 33° 06` 17” *(with borrowing left-to-right)*

Layout your calculations neatly, so you can review, track changes, correct or learn from them.

One way a layout can look is like this:

### Additional Information: Another way to get the answer

Subtract 04° 11` 22” from 33° 06` 17” (with borrowing left-to-right)

**Another way to get the answer:**

1. Identify that subtraction needs to be done in DMS.

2. Recognize that ° means degrees, ` means minutes and “ means seconds.

3. Set up the problem:

Write up the 2 numbers in DMS, carefully lining up the digits in 3 columns like this:

degrees minutes seconds

- degrees minutes seconds

With the given numbers, it should look like this:

33° 06` 17”

- 04° 11` 22”

Borrowing for **minutes**:

4. Recognize that borrowing is necessary since the minutes amount to be subtracted (at the bottom) is bigger than the minutes to be subtracted from (number on top).

5. Set up the problem:

degrees – 1 = new degrees

6. Calculate: 33 – 1 = 32

7. Take a pencil and write down 32 and cross out the 33

8. Recognize that the borrowing/carrying amount between degrees and minutes is 60, since there are 60 minutes in a degree.

9. Set up the problem:

minutes + 60 = new minutes

10. Calculate: 6 + 60 = 66

11. Take a pencil and write down 66 and cross out the 6 and the 60

Borrowing for **seconds**:

12. Recognize that borrowing is necessary since the seconds amount to be subtracted (at the bottom) is bigger than the seconds to be subtracted from (number on top).

13. Set up the problem:

minutes – 1 = new minutes

14. Calculate: 66 – 1 = 65

15. Take a pencil and write down 65 and cross out the 66

16. Recognize that the borrowing/carrying amount between minutes and seconds is 60, since there are 60 seconds in a minute.

17. Set up the problem:

seconds + 60 = new seconds

18. Calculate: 17 + 60 = 77

19. Take a pencil and write down 77 and cross out the 17and the 60

Now subtraction can begin:

20. Calculate

77 – 22 = 55

21. Take a pencil and write down 55

22. Write the “ sign after the 55

23. Check back: determine that the correct way to write the answer is 55”

24. Calculate

65 – 11 = 54

25. Take a pencil and write down 54

26. Write the ` sign after the 54

27. Check back: determine that the correct way to write the answer is 54`

28. Calculate

32 – 4 = 28

29. Take a pencil and write down 28

30. Write the ° sign after the 28 Now you`re done with the math.

31. Check back: determine that the correct way to write the answer is 28°

32. The last step is to check your work: make sure everything was copied and written correctly then determine that the correct way to write the answer is 28° 54` 55”

### Additional Information: Another way to get the answer using a scientific calculator

**Another way to get the answer, using a scientific calculator:**

1. Enter 33 **DMS **6 **DMS **17 **DMS**

2. Press – (minus)

3. Enter 4 **DMS **11 **DMS **22 **DMS**

4. Press **=**

The number on the display changes to 28° 54` 55”

Get a $20-or-so dual-display scientific calculator, even if you can`t take it with you into the exam hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you can`t do without.

### Download Infosheet: Advanced