Subtracting Decimals

These exercises combine learning to subtract with multiple digits and subtracting decimals. Note the top number, the number you are taking another number away from is called the minuend. The bottom number, the one you are taking away from the top number is called the subtrahend. The answer to the subtraction problem is called the difference. As long as the minuend is larger than the subtrahend you can just take the bottom number from the top number. If the number you are taking away is larger than the number you are taking it away from, take the top number away from the bottom number and then put a minus (-) sign in front of it to show that it is negative. You can check and see that this works on the number line.

Example:

Look at the number line below. Start at the minuend 3, count 7 graduations to the left because we are taking away 7. you will land on -4.

By reversing the order of subtraction and putting a minus sign in front of the answer when the subtrahend is larger than the minuend we can solve these problems with ease. A good way to check a subtraction problem is to add the difference back to the subtrahend to make sure that it equals the value of the minuend. (-4)+(7)=3 Or since we have reversed the order of subtraction in this problem we could 3+4=7. On the number line you would start at -4 and count 7 graduations to the right landing on 3. Just as in Adding and Decimals (a previous Homemade Math lesson), we make sure and line up the decimal points and place value before beginning to solve the problem. We had carries in addition, but will have to barrow in subtraction. You will find it easier at first to rewrite the problems using the built in scratchpad or use a pencil and paper and put the larger number on top. When i say to put the larger number on top, i am talking about the absolute value meaning to disregard the sign of the number and look only at its value and not if it is positive or negative or which way it would be moving on the number line. To determine which number has the larger absolute value, start at the far left or most significant digits and compare. If there is no number shown for a place value, use 0 for that value. Examples are:

Graphics are screensots from Homemade Math's built in scratchpad.

If the minuend in the most significant place value is larger than the subtrahend digit in the most significant place value then the minuend is larger and stays on top. If the subtrahend's most significant place value is larger than the minuend's most significant place value then the subtrahend is larger and it will be easier to solve by placing the subtrahend on top. An example is:

In these exercises the most significant place value is the one's place. If the numbers in the one's place happen to be the same or equal for the minuend and subtrahend, move one place value to the right or to the tenths place value and compare as in the example below.

If the one's and tenth place value have equal values, you'll have to compare the hundredth place values like below. 3 Is greater than 2 so the minuend stays on top.

There are only 3 place values to deal with in these Homemade Math exercises but the method is the same no matter how many place value you have. Start at the far left digits and compare. until you find one greater than the other. Use a 0 if no digit is shown for a place value. Now that you have your subtraction problem arranged for ease of solving, start at the far right or least significant digits and take the bottom number from the top and place the answer directly under the least significant digits in the problem. Hundredth place in this case.

Next move one place value to the left or to the tenth place values and take the bottom number from the top and place the result directly under the tenth place values in the problem.

Bring the decimal point down and place it directly under the decimal points in the problem. Move another place value to the left which is the one's place value in this case and take the bottom number from the top placing the result directly under the one's place values in the problem.

The above problem is completely solved. Each digit in the minuend is larger than each digit in the subtrahend, so we didn't need to do any barrowing. Now look at an example where we need to barrow.

6.34 is larger than 5.27 but when we try and take 7 away from 4 we run into a snag. We need to barrow 1 from the next higher significant digit, the 3 that is in the tenth's place making the 3 a 2 and the 4 a 14.

Since there are 10 one hundredths in each unit in the tenth place, we end up with 10+4 hundredths or 14 for the least significant digit and 2 for the tenth place digit. Now we can proceed solving the subtraction problem.

14 Minus 7 is 7 so we place a 7 directly under the hundredth place. Move left to the tenth place and 2-2=0 so we place a 0 directly under the tenth place in the problem. Bring the decimal point down. Move to the one's place and 6-5=1 so we place a one directly under the one's place value in the problem. You can check and see that 6.34-5.27=1.07 by adding 1.07 to 5.27. It will equal 6.34. Now we will look at another problem that requires more barrowing. 5.13-3.42

Start at the far right. 3-2=1 so place a 1 directly under the hundredth place in the problem. Move to the tenth place. 1 is smaller than 4 so we have to barrow from the one's place. Each unit in the one's place has 10 tenths in it so we end up with 10+1 tenths or 11 tenths in the tenth's place and 4 in the one's place. 11-4=7 So place a 7 directly under the tenth place in the problem. 4-3=1 So place a 1 under the one's place in the problem for a final difference of 1.71. You can check and see that it is correct by adding 1.71 to 3.42 and it will equal 5.13. The next example is of a number containing 0 in the minuend.

First go to the far right least significant digits. 1 Is smaller than 6 we need to barrow from the next higher place value but there is a 0 in the tenths place. There are a couple of ways to handle this. Most people cross out the 0 and then go to the next higher place value and cross that number out and reduce it by 1 so that the 2 in the minuend will become a 1. Then make the 0 in the minuend a 10. Then cross out the 10 and barrow 1 from it and making it a 9 and the 1 in the least significant digits place an 11.

Then carry out the subtraction working from left to right. Make sure it is correct by adding the subtrahend back to the minuend. Another way to handle barrowing and 0 in the problem 2.01-1.76 is to cross out the 0 and make it a -1, make the 1 in the hundredth place an 11, take 6 from 11 and put your 5 under the hundredth place column. Move left to the tenth place column and you have -1-7. Barrow from the one's place making the 2 in the one's place a 1. Add the barrow to your tenth place {10+(-1)} giving you 9. Each one's place unit contains 10 tenth place units so you end up with 10+(-1) or 9. 9-7=2, So place a 2 under the tenth place column. I don't go out to the thousandth place in these exercises but am showing an example of barrowing with multiple zeros in the minuend just for your future reference.

0-7 in the thousandth place makes us barrow 1 from the hundredth place, making the 0 in the thousandth place a 10. The 0 in the hundredth place becomes a -1. Normally we just cross the 0 in the hundredth place out and knowing we will have to barrow go to the tenth place and barrow a 1, making our hundredth place 10+(-1) or 9. Since the tenth place is also a 0 we know we are going to have to barrow and go to the one's place and take 1 from it, making it a 7 and the tenth place a 9. Now we can perform the subtraction. Start at the far right thousandth place. 10-7=3, So place a 3 under the thousandth place. 9-5=4, So place a 4 under the hundredth place. 9-3=6, So place a 6 under the tenth's place. Bring down the decimal point. 7-7=0, You can place a 0 under the one's place or just leave it blank. .643 And 0.643 are both correct. Now for some examples where the subtrahend is larger than the minuend. I show how i solve them in Homemade Math with the built in scratchpad tiles.

Since 9.12 is larger than 8.56 we take 8.56 away from 9.12 and then put a minus sign in front of the difference to show that it is negative. I just do the problems from the bottom to the top instead of from the top to the bottom when the subtrahend is larger than the minuend. 2 Is smaller than 6, so barrow 1 from the 1 in the tenth's place making it a 0 and the 2 in the hundredth place a 12. Now 12-6=6 so put a 6 above the hundredth place in the problem.

0 Is less than 5, so barrow 1 from the 9 in the one's place making it an 8 and the 0 in the tenth's place a 10. Take 5 from 10 and place a 5 above the tenth's place in the problem. Bring up the decimal point. 8-8=0, So place a 0 above the one's place in the problem or you can leave it blank. Don't forget the minus sign. -0.56 And -.56 are both correct.