Subtraction

Subtraction is a way of taking away a number of things from a larger number. You take them away to find how many things are left. Only like things can be subtracted. That is, you cannot subtract apples from pencils.

Suppose you have a set of 8 oranges.

Suppose you want to take away a set of 5 oranges.

You find that you have 3 oranges left.

Learning to subtract

A question such as “3 from 6 is how many?” is a subtraction problem. To find out how many things are left in a subtraction problem, you can count or find the answer by thinking.

Subtraction by counting.

Here are two groups of chocolate cupcakes.

How many cupcakes are there in the first group? Count them. There are 6 cupcakes in the first group. Mary took 3 cupcakes from the second group. How many cupcakes are left in the second group? Count them. There are 3 cupcakes left. You counted to find how many cupcakes are left if you take 3 from 6. You discovered that 3 taken from 6 leaves 3.

Subtraction by thinking.

Tommy has 5 pennies.

He wants to spend 2 pennies for a pencil. How many pennies will Tommy have left? Cover 2 pennies in the picture. You should be able to tell how many pennies are left by just looking at the picture, without counting. You should learn to think “2 from 5 leaves 3.” This article will show you the facts you need to know to subtract by thinking. Thinking the answer is a quicker way of subtracting than counting.

You can learn to think the answer to a subtraction problem from what you know about addition. For example, you know that 3 and 2 are 5. This means that if you take 2 from 5, you have 3. You can practice this method of subtraction by writing the addition and subtraction facts in groups of four.

• 3 and 4 are 7
• 4 from 7 leaves 3
• 4 and 3 are 7
• 3 from 7 leaves 4

Subtraction questions.

Subtraction tells you how many things are left when you take away one set of things from another. It also lets you compare two sets of things. Suppose Mary has 5 balloons and Sue has 3 balloons.

To compare the two sets of balloons, you must find the difference between the two sets. You can find the difference by subtracting. When you subtract 3 from 5, you discover that the difference between the two sets is 2 balloons, or 2.

You can also use subtraction to find out how many more things are needed. Suppose John needs 12 pennies. He has 5 pennies. How many more pennies does he need?

When you subtract 5 from 12, you discover that John needs 7 more pennies to make 12.

Subtraction can tell you (1) how many things are left, (2) what the difference is, and (3) how many more things are needed.

Writing subtraction.

It is best to write your subtraction problems and their answers. This gives you a record of your thinking.

You can make a record with pictures.

The picture shows that 3 taken from 5 leaves 2.

You can write this in numbers and words: 3 from 5 leaves 2

But you must learn to write with numbers and signs: 5 – 3 = 2

The – sign means to subtract or take away. So 5 – 3 means “3 taken from 5.” We call the – sign the minus sign, and read 5 – 3 as “5 minus 3.” The = sign means that the sets on one side of the = sign are equal to the sets on the other side. Here is how it works:

There is another way to use numerals and signs.

Each part of a subtraction problem has a name. When we are subtracting to find out how many things are left, we call the answer the remainder. When we are subtracting to compare two groups or to find how many more things are needed, we call the answer the difference. We call the number being taken away or subtracted the subtrahend. The number from which the subtrahend is taken is called the minuend.

Subtraction facts.

By subtracting one group from another, you discover that 8 – 5 = 3, 6 – 3 = 3, and 12 – 5 = 7. We call these subtraction facts.

Each subtraction fact consists of a minuend, a subtrahend, and a remainder, or difference. You can discover each subtraction fact for yourself by counting and taking away one set of things from another. For example, you can practice by crossing off squares as you have seen crossing off done in an earlier example.

There are other subtraction facts. For example, any number minus itself is zero. Thus, 5 – 5 = 0 and 9 – 9 = 0. Also, any number minus zero is the number itself. Thus, 6 – 0 = 6 and 3 – 0 = 3. It is best to learn the subtraction facts so that you can recall them without stopping to work them out. You can use them to solve problems right away.

To learn the harder facts, it is sometimes useful to regroup. For example, many people find it easier to subtract numbers from 10. Suppose you wanted to solve the problem 14 – 7. You know that 14 is the same as one 10 and four 1’s. So you could regroup it like this.

First, you can take away 4. You know that 7 – 4 = 3, so you must still take away 3. Subtracting 3 from 10 is simple.

You can see that 10 – 3 = 7. So 14 – 7 = 7.

You can probably invent other ways to help you learn the subtraction facts.

Subtracting larger numbers

Subtracting larger numbers is not difficult, if you know the subtraction facts and understand the number system.

Subtracting 10’s and 100’s.

Suppose you have 5 dimes. This is the same as 50¢. Suppose you want to spend 3 dimes on a book. This is the same as 30¢. How much money will you have left? The problem is 5 dimes – 3 dimes or 50¢ – 30¢. You can find the answer by counting.

You can also find the answer by using the subtraction facts and thinking.

If you know that 5 – 3 = 2, you can see that 3 dimes taken from 5 dimes leaves 2 dimes. A dime is 10¢, so you can see that 50 – 30 = 20. The subtraction fact 5 – 3 = 2 helps you find the answer. You subtract 10’s the same way that you subtract 1’s. But you must write the remainder in the 10’s place. And you must remember to write in a zero to show that the remainder is 10’s, not 1’s.

Subtracting 100’s is done in the same way. Suppose you had to subtract 3 dollars from 5 dollars.

You subtract 100’s (and 1,000’s and so on) the same way that you subtract 1’s and 10’s. Once again, you can see how the subtraction fact helps you find the answer to the subtraction example.

Subtracting 10’s and 1’s.

Tom had 45 tickets to sell. He sold 23 of them. How many tickets should he have left? That is, what is 45 – 23? We call numbers such as 45 and 23 two-place numbers, because 45 has two places, four 10’s and five 1’s; and 23 has two places, two 10’s and three 1’s.

To subtract one two-place number from another, you begin by subtracting the 1’s: 5 – 3 = 2. Write the 2 in the 1’s place in the remainder.

Next, subtract the 10’s: 4 – 2 = 2. Remember that the 4 – 2 stands for 10’s, not 1’s. Write the 2 in the 10’s place in the remainder.

So Tom should have 22 tickets left.

Here is an example of subtracting three-place numbers.

First, subtract the 1’s: 7 – 3 = 4. Write the 4 in the 1’s place of the remainder. Next, subtract the 10’s: 4 – 2 = 2. Write the 2 in the 10’s place in the remainder. Next, subtract the 100’s: 6 – 1 = 5. Write the 5 in the 100’s place. Subtracting two- and three-place numbers is easy, but you must remember two things. You must subtract the 1’s, 10’s, 100’s, 1,000’s, and so on, in that order. Always begin at the right—in the 1’s place—and work to the left. Second, you must write your work carefully, so that the numbers of the remainders are in the proper places.

How to borrow.

When you subtract larger numbers, you often cannot solve a problem unless you know how to borrow. For instance, look at the example 62 – 27. How can you subtract seven 1’s from two 1’s? Borrowing helps solve this kind of example.

You cannot subtract seven 1’s from two 1’s. But you can take one of the 10’s in the minuend and change it into 1’s. Now you can solve the problem.

So 62 – 27 = 35. There were too many 1’s in the subtrahend to subtract. You “borrowed,” or changed a 10 from the 10’s part of the minuend into the 1’s. This is what borrowing means. You can also borrow 100’s, 1,000’s, and so on, in solving problems.

You do not have to write out a problem every time you borrow. You can think the steps and write in little numbers as a guide. Here is the same example:

First, you study the example. “I cannot take 7 from 2,” you think, “so I must change a 10 to 1’s.” You draw a line through the 6 in the minuend and write a 5 above it. This means that there are now five 10’s in the 10’s place instead of six. Next, you write a little 1 just above and to the left of the 2. This means that there are now twelve 1’s, instead of two.

Now you can do the subtraction. “Seven 1’s from twelve 1’s leave 5,” you think, and write a 5 in the 1’s place of the remainder. “Two 10’s from five 10’s leave 3,” you think, and write a 3 in the 10’s place of the remainder. This completes the example.

The same method of “borrowing” a 10 can be used for 100’s and 1,000’s.

First, you subtract one 1 from eight 1’s, and write a 7 in the 1’s place of the remainder. But you see that you cannot subtract six 10’s from two 10’s. You must borrow a 100, or ten 10’s, from the six 100’s in the minuend.

You draw a line through the 6 in the minuend and write a 5 above it. This means that there are now five 100’s in the 100’s place, instead of six. Next, you write in a little 1 just above and to the left of the 2. This means that there are now twelve 10’s, instead of two. Now you can finish the subtraction. Six 10’s from twelve 10’s leaves six. You write a 6 in the 10’s place of the remainder. Three 100’s from five 100’s leaves two. You write a 2 in the 100’s place of the remainder.

You use the same method for 1,000’s. You borrow a 1,000 just as you borrowed a 10 or a 100.

Checking subtraction

You should always check your work in subtraction to make sure that you have done it correctly.

Checking by subtraction.

One way to check a subtraction problem is to subtract the remainder from the minuend.

The new remainder should be the same as the old subtrahend. This checks your work.

Checking by addition.

A good way to check subtraction problems is by addition, because addition is the opposite of subtraction. You add the subtrahend and the remainder.

The sum of the addition should be the same as the old minuend in the subtraction problem.

Estimating

helps you know if your answer is reasonable. Try to estimate the answer before you work the problem. Here is an example:

This is almost the exact answer. You can estimate in larger numbers. For example, 476 is about 500, and 254 is about 250. Subtracting 500 – 250 gives you 250. This gives you a good idea of what the answer should be. Estimating the answer before you work a problem will save you time if you make a mistake, because you know about what the answer should be.

Subtraction rules to remember

Here are six rules that will help you solve subtraction problems.

1. Remember what subtraction means. You can find the answers to subtraction problems by counting. But it is quicker and easier to think the answers.

2. Learning the subtraction facts will help you think the answers to subtraction problems quickly.

3. Subtraction is the opposite of addition. Because of this, addition will help you learn the subtraction facts and check problems.

4. The subtraction facts help you subtract larger numbers to solve problems.

5. You can only subtract quantities of the same kind. That is, you must subtract 1’s from 1’s and 10’s from 10’s.

6. Subtraction answers three kinds of questions: how many are left, what is the difference, and how many more are needed.

Other ways to subtract

There are several ways of thinking out a subtraction problem. The method we have used is called the “take-away-borrow” method. Here is another example:

First, you see that you cannot take eight 1’s from two 1’s. You borrow a 10, making the minuend six 10’s and twelve 1’s. Then you subtract eight 1’s from twelve 1’s: 12 – 8 = 4. You write the 4 in the 1’s place in the answer. Next you subtract two 10’s from six 10’s: 6 – 2 = 4. You write the 4 in the 10’s place in the answer.

Another method is called the “addition-borrow” method.

The numbers are the same as in the “take-away-borrow” method, but the thinking is different. You see that you cannot take eight 1’s from two 1’s, and borrow a 10. Instead of subtracting eight 1’s from twelve 1’s, you think “what added to 8 makes 12?” You know that 8 + 4 = 12, so you write the 4 in the 1’s place in the answer. Instead of subtracting two 10’s from six 10’s, you think “what added to 2 makes 6?” You know that 2 + 4 = 6, so you write the 4 in the 10’s place in the answer.

A third method is called the “addition-carry” method or the “Austrian” method.

First, you see that you cannot take eight 1’s from two 1’s. Instead of borrowing, you add ten 1’s to the two 1’s: 2 + 10 = 12. Next, you think “what added to 8 makes 12?” You already know that 8 + 4 = 12, so you write the 4 in the 1’s place in the answer. Now you think “I added a 10 to the 1’s, so I must subtract a 10 from the 10’s.” To do this, you change the two 10’s in the subtrahend to three 10’s. You think “what added to 3 makes 7?” You know that 3 + 4 = 7, so you write the 4 in the 10’s place in the answer.

Fun with subtraction

Many games that can be played with the addition, multiplication, and division facts can be changed a little for the subtraction facts.

To play a game called More or Less, make a pack of 36 cards. Write the numbers from 1 to 18 separately on two sets of cards. There will be two cards for each number. Shuffle the cards and place the pile facedown. The leader of the game takes the first card and holds it up for the players to see. Suppose it is 14. The first player takes a card from the pile and shows it. Suppose it is 6. The player compares it with the 14 card and says “It is less.” Then he must tell how much less. In this case, the player would say “It is 8 less than 14.” He must find the answer by thinking the subtraction. Suppose the next player turns up 17. She compares it with the first card. She must say “It is more. It is 3 more than 14.” A player who gives the wrong answer is out of the game. When you have gone through the cards once, you can mix them up and play again with new numbers.