WHOLE NUMBER ARITHMETIC

• Addition
• Subtraction
• Multiplication
• Division
• Parentheses
• Average
• Go to the Whole number home page
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After each section there will be a series of exercises.
In each exercise you'll be asked to work out a problem and place your answer into a text box
After that, click on the button marked "Check my answer"
You'll then be told whether your answer is correct or not, and be given further options if warranted.

14,589 + 834

First line up the two numbers as shown:

                      14,589
                      +  834

Now look at the first column. You have 9 + 4. Now 9+4=13. Write the 3 directly underneath the 4 and "carry" the 1.

                          1
                      14,589
                      +  834
                           3

Moving on to the next column we have 8 + 3 + "our carried 1." 8+3+1=12. Write down the two and carry the 1 onto the third column

                         1
                      14,589
                      +  834
                          23

Now moving on to the third column we have 5 + 8 + "our borrowed 1". 5 + 8 + 1 = 14. Write down 4 and carry 1.

                       1
                      14,589
                      +  834
                         423

Now moving on we have 4 plus "our carried 1" giving us 5.

                      14,589
                      +  834
                       5,423

Lastly, we have our last column with 1 in it, and nothing else. Just write down the 1

                      14,589
                      +  834
                      15,423

14,589 + 834 = 15,423

Now it's your chance to prove you've been paying attention! Note:When placing your answers in the box, do not include commas

3,157 - 289

first line the numbers up as shown
Now we can immediately detect that in the first column we have 7 - 9.
Now 9 cannot be subtracted from 7, so we need to borrow from the 5.
Take one from the 5, and think of it as 4 (Note: the 5 has a strike through it, and a blue 4 appears above it).
Then take that borrowed one and think of the 7 as 17 (Note: the blue one immediately above the 7).
Now 17 - 9 = 8, which is written directly below the first column. Moving on we have the 4 (formerly 5) - 8.
Borrow from the 1, and for here on think of it as 0. Notice that the 1 now is striked though and 0 appears above it.
Think of the 4 as 14. Notice the 1 directly above the 4
14 - 8 =6 which is now written directly below the second column. Moving on we have 0 - 2
Borrow from the 3 and think of the 0 as 10.
The effects of this move are shown below In the last column we have a 2 with nothing to subtract from it
Write the 2 in that column

3,157 - 289 = 2,868

Here's a way to check your answer (Put that calculator down!): Add your solution (2,868) to the number you are subtracting (289), and you should get the original starting number (3,157)

                      2868
                      +289

8 + 9 = 17

                        1
                      2868
                      +289
                         7

6 + 8 + 1 = 15

                       1
                      2868
                      +289
                        57

8 + 2 + 1 = 11

                      1
                      2868
                      +289
                       157

2 + 1 = 3

                      2868
                      +289
                      3157

So 2868 + 289 is indeed 3157, and our answer to the subtraction problem has been verified!

Now let's go back to subtraction.
The above example showed how to borrow from one column to the next, but what if we can't borrow from the next column?
For the answer, study this next example

200 - 65

Now 5 cannot be taken from 0, so we would like to borrow from the second column.
However, the second column is another zero so we have to go on to the third column
Reduce the 2 in the third column to 1, and make the second column's 0 into 10 Now take the second column, which we currently think of as 10 and reduce it to 9
Using that borrowed one think of the first column's 0 as 10 Now subtract 5 from 10 leaving 5 Now subtract 6 from 9 leaving 3 Finally drop the 1 down, since there is nothing to subtract from it

200 - 65 = 135
Ready for your first assignment? (well even if you're not, here it come's anyway)
verify this by adding 65 and 135.

After completing a few exercises on your own, you'll be ready to graduate to multiplication
Note:When placing your answers in the box, do not include commas

• 3 * 4
• 3 x 4
• 3 · 4

Multiplication is a little more involved than addition or subtraction, but you can handle it!

We'll start out simple, with a 3 digit number being multiplied by a single digit

324 * 8

Okay line the problem as shown below

                      324
                        8

Now 8 * 4 = 32. Write down the 2 and carry the 3 {Yes, here we go with "carrying" again}

                       1
                      324
                        8
                        2
                                                                 

Now multiply 8 by 2 and add the carried 3
8 * 2 = 16.
16 + 3 = 19
Write down the 9 and carry the 1

                      1
                      324
                        8
                       92
                                                                   

Now 8 * 3 = 24 + 1 = 25

                     324
                       8
                    2592
                                                                  

324 * 8 = 2592

324 * 738

We already know 324 * 8 = 2592

In the same manner we can compute that 324 * 3 = 972

4 * 3 = 12. Write down 2 and carry 1. So far 324 * 3 = ??2
2 * 3 = 6 plus the carried 1 = 7. So far 324 * 3 = ?72
3 * 3 = 9 . So 324 * 3 = 972

324 * 7 = 2268
[Your assignment: verify this by doing the multiplication yourself]

Now write the problem as follows

                     324
                     738
                    2592
                    9720                                
                  226800
                              

Notice the second product has one 0 attached to it's end and the third product has two zeros attached to it's end.
If their were a fourth product it would have three zeros attached to it's end

Now We have three figure 2592, 9720, and 226800.
The final step is to add these numbers together.
You will find that their sum is 239112
[Your next assignment is to verify this by doing the addition yourself]

                     324
                     738
                    2592
                    9720                                
                  226800
                  239112

324 * 738 = 239,112

324 * 708

324 * 8 = 2592
324 * 0 = 0
324 * 7 = 2268

Line the problem up us shown

                  324
                  708
                 2592
               226800
                               

Now the "2268" still has two zeros attached to it, since it is still the third multiplication.
There is no mention of zero in the second position since it would make no difference in the final addition.

We solve the problem by adding 226800 and 2592 giving us a final product of 229392.
Verify this by doing the addition

                  324
                  708
                 2592
               226800
               229392

324 * 708 = 229,392

See multiplication isn't so bad after all!
Now you try it!

• 87535/287
  
• 87535 ¸ 287
  
• 

Don't panic yet, I'll get to this "long division" problem in a few minutes
I'll start with something a little easier
28,730/5

Okay we're going to divide 28,730 by 5
The first step is to write out our problem This time we're reading left to right
Now 5 is greater than 2, so obviously 5 cannot divide 2
Going on we have 28.
How many times does 5 go into 28?
The answer is five. 5 * 5 = 25
Okay write a 5 directly above the 8 AND write 25 directly below the 28. Now subtract 25 from 28. 28 - 25 = 3
Write the 3 directly below the 5 in 25. Notice the difference is smaller than what we are dividing by (5), if it were greater than or equal to what we are dividing by, then that indicates that our guess of how often the divisor (5) goes into the dividend (in this case 28) was too small.

Let's say you thought 5 went into 28 four times. 5 * 4 = 20 and 28 - 20 = 8
Since 8 is larger than the divisor (5), you know you made the wrong guess

Ready for the next step. Okay the next number in the dividend (2875) is 7. Drop the 7 down and put it next to the 3 making 37 Now how many times does 5 go into 37?
Seven. 5 * 7 = 35
Write a 7 directly above the 7 in 2875 AND write 35 directly below 37
Subtracting we find 37 - 35 = 2. So write a 2 directly below the 35
Notice that 2 is less than what we are dividing by (5) The next number in 2875 is 5. Drop the 5 down and put it next to the 2 to form 25 How many times does 5 go into 25?
Five. 5 * 5 = 25
Write a 5 directly above the 5 in 2875
Write 25 directly below 25.
Subtracting we find no remainder, so the division is now finished

2875/5 = 575

Here's a way to check your answer. Multiply your quotient (i.e the answer 575) by the divisor (5). The answer should be the original number (2875)

575 * 5

5 * 5 = 25
575 * 5 = ???5 carry a 2

7 * 5 = 35 + the carried 2 = 37
575 * 5 = ??75 carry a 3

5 * 5 = 25 + the carried 3 = 28
575 * 5 = 2875

Your solution has now been verified!

Okay now it's time to panic.
I'm going to tackle the problem listed above: 87535/287?

Division example 2

First write down the problem
Next starting with the 8 in 87535, and ask can 287 go into 8?
No
Go on to the next digit (7), and ask can 287 go into 87?
No
Go on to the next digit (5), and ask can 287 go into 875?
Yes
Now the question is how many times can 287 go into 875, without going over? (Just like playing "The Price is Right")
This may require a little guesswork
Let's try two. 2 * 287 = 574, and 875 - 574 = 301 which is actually larger than our divisor (287). Our guess of 2 was too small.
Okay let's try four. 287 * 4 = 1148 which is too big. (1148 is greater than 875). Our guess of 4 was too big.
Okay let's try three. 287 * 3 = 861, and 875-861 = 14
Write the 3 directly above the 5 in 875 and 861 directly below 875.
Write their difference, 14, directly below the 861

Bring down the next digit in 87535 (3), making the 14 into 143.

Now 287 cannot go into 143 so place a 0 above the 3 in 87535

Bring down the next digit in 87535 (5), making the 143 into 1435

Now how many times does 287 go into 1435?
Through trial and error, you will eventually come up with 5
[Note: Your is to verify this by trying 4 and 5 as possible answers]
287 * 5 = 1435
Write a 5 directly above the 5 in 87535 AND write 1435 directly below 1435
Subtracting we find that their is no remainder.


87535/287 = 305
Verify this by multiplying 305 by 287

Ready to try your hands on division?
You better be, cause here goes!
Again, do not put commas in your answer

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Division Exercise 1 1800/45

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Go to the next problem.
continue on to the parentheses section.

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Division Exercise 2 168402/381

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continue on to the parentheses section.

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Division Exercise 3 9872/8

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Go to the next problem.
continue on to the parentheses section.

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Division Exercise 4

2179926/2133

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Go to the next problem.
continue on to the parentheses section.

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Division Exercise 5 12350/25

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continue on to the parentheses section.

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PARENTHESES
We have already seen examples where two arithmetic operators were used

• when multiplying 575 by 5 at one point we wrote 7 * 5 = 35 + the carried 2 = 37
• When dividing 87535 by 287, we ended up subtracting the product of 3 and 287 (861) from 875

An easy way to write these problems is with the use of parentheses

The first example above could be written as (7 * 5 ) + 2
The parentheses says to first multiply 7 * 5
And then add 2

The second example above could be written as 871 - (287 * 3)
The parentheses says to first multiply 287 * 3
and then subtract this product from 871

Going forwards what if we had the expression (7 * 3 ) - 2
The parentheses would say first multiply 7 * 3 (yielding 21)
and then subtract 2 (yielding 19)

The important thing to remember is always do what is in the parentheses first

It's your turn again

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Parentheses Exercise 1 6 * (221 + 334)

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continue on to the average section.

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Parentheses Exercise 2 (334 - 122)/2

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continue on to the average section.

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Parentheses Exercise 3 (553 + 233) - 678

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continue on to the average section.

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Parentheses Exercise 4 (1947/11) + 234

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continue on to the average section.

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Parentheses Exercise 5 (1921 - 987) * 65

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continue on to the average section.

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AVERAGE

Sometimes you have to look for middle ground and that is the main purpose of averages

Suppose you have a list of numbers, and want to calculate the average
Your first step would be to add the numbers together
And your next step would be to divide this sum by the number of items added

Bear with me as I do some examples illustrating this concept, and then I'll let you compute averages on your own

Average example 1

Find the average of {33, 17, 21, 42, 32}

We have 5 numbers to add
33 + 17 + 21 + 42 + 32 = 145
The average = (The sum)/(The number of items added) = 145/5 = 29

Average example 2

Find the average of {212, 314, 138, 444}

We have 4 numbers to add
212 + 314 + 138 + 444 = 1108
The average = (The sum)/(The number of items added) = 1108/4 = 277

Ready to take some averages?

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Average Exercise 1 {22,47,48}

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Go on to the "factor & factoring" section.

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Average Exercise 2 {12,22,33,44,14}

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Average Exercise 3 {47,53}

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Average Exercise 4 {251,36,54,103}

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Average Exercise 5 {14,8,7,5,1}

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Go on to the "factor & factoring" section.

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