Decimals

Multiplying Decimals

To multiply decimals, you basically ignore the decimal, multiply as usual, then place decimal in the appropriate spot in your final answer. The following gives instructions on doing this without error.

First, is is always a good idea to place the number that contains the most places on top. Whether it is larger or smaller in value is really insignificant. All that matters is being able to multiply the numbers in a way to keep track of them. Once this is done, ignore the decimal (for now) and multiply the way you normally would do. Once the final answer is obtained, count how many numbers (including zeroes) are included to the right of the decimal in each problem. Since this is what seems to mix most people up, a few examples are now included. We are concentrating on counting decimal spaces here, not actually multiplying the out yet.

Example Multiply 5.6 x 7.8
There is one number, 6, to the right of the decimal in 5.6.
There is one number, 8, to the right of the decimal in 7.8.
This gives a total of 2 numbers.
In the final answer, the decimal will be placed 2 places to the left of the last number on the right.

Example Multiply 5.45 x 12.2
There are two numbers, 4 and 5, to the right of the decimal in 5.45.
There is one number, 2, to the right of the decimal in 12.2.
This gives a total of 3 numbers.
In the final answer, the decimal will be placed 3 places to the left of the last number on the right.

Beware if you add decimal or zeroes. Any zeroes you add as place holders are counted!
Know that a whole number without a decimal has zero numbers to the right.
You do not need to add zeroes after a whole number when multiplying, if you stick with the rule of placing the number with the most places on top

Example  Multiply 6 x 7.89
You could change 6 to 6.00 so it has the same places as 7.89. This is counterproductive. Leave the 6 alone and place it on the bottom.
In this example,
there are two numbers,8 and 9, to the right of the decimal in 7.89.
There are NO  numbers to the right of the decimal in 6.
This gives a total of 2 numbers.
In the final answer, the decimal will be placed 2 places to the left of the last number on the right.

Example Multiply 2.5 x 7.3
Here both 2.5 and 7.3 have the same spaces. It does not matter which one goes on top. Always line up the bottom number so the last number on the right is directly below the last number on the right of the number on top. Since in this example they both have the same number of places, this is not as critical.
Remember to shift one place over when multiplying by 7.
You could place a zero as a place holder under the 5 in 75.
Keep you numbers in neat columns!
There is one number, 5, to the left of the decimal in 2.5.
There is one number, 3, to the left of the decimal in 7.3.
That makes a total of 2. The decimal in the final answer goes 2 places to the left of the final number on the right.
Here 5 is the first, 2 is the second, so it goes just before the 2.

Example  Multiply 6.2 x 4.85

To Make Things Easy:
Don't add zeroes to 6.2.
Don't worry that 6.2 is smaller than 4.85.
Always put the number with the least spaces on the bottom.
Line up the numbers so that numbers on the far right are aligned.
Ignore the decimal. It is the same as doing 485 x 62.

Where does the decimal go?
There are two numbers,8 and 5, to the right of the decimal in 4.85.
There is one number, 2, to the right of the decimal in 6.2.
The decimal will go 3 places to the left.
Remember 30.070 is the same as 30.07.

Yes you can drop zeroes in certain cases!

Example Multiply 4.50 x 7.2
4.50 is the same as 4.5. You may drop the zero if you wish. But, you can keep it too. Either way, you will get the same answer.

And remember, 32.400 is the same as 32.40, which is the same as 32.4.
A zero to the right of a decimal point counts as a number!
In the first case you would move the decimal 3 places to the left.
In the second case you would move the decimal 2 places to the left.

If you have followed these lessons all the way through form the beginning, you should remember the part about becoming a "math person." Here's where that applies to decimals. If you are on the last step(or only step) of a problem, chances are the decimal is irrelevant. In other words, just pay attention to the important numbers. 12.5 x 6.5 will have the same digits as 125 x 65, just no decimals. Consider the following problem:

34.56 x 2.3 = ?

Answers:
a. 79.488
b. 67.249
c. 56.488
d. 23.396
e. 75.276

We just need to do 3456 x 23 = 79488. The only answer that has those digits is answer number one. The only time this does not work is when you have more than one answer with the same digits. Then you need to add the decimal in the correct place. However, if you were thinking like a math person, you would have chosen answer a right off. How? Because there is a 3 and 6 on the end. The answer must end with an 8 because 6 x 3 = 18. Doing 2 x 34 = 68 gives us a lower estimate. That is, the answer must be greater than 68. There is only one answer greater than 68 and ends with an 8. Now are you catching on? If you approach each CBEST math problem by thinking in logical terms, your fear of failing will go away. Also notice how there is little math involved in the alternate way. Remember, the more arithmetic you do, the more you are going to make an error.

CBEST Math Tip: Sometimes you can ignore the decimals and just look at the digits.

How about dividing decimals? This always gets certain people in a panic. Forget about the decimals and let the answers help you. Consider the following problem:

445.28 ÷ 25.3 = ?

a. 12.53
b. 20.39
c. 31.28
d. 17.6
e. 12.6

You know you need to take 25.3 into 445.28. However, since all answers have different digits, we can ignore the decimals. All that matters are the digits you are left with. So, think of it as 253 going into 44528. First you figure out how many times 253 will go into 445. Look at the answers. They begin with a 1,2, and 3. That means, 253 goes into 445 1,2, or 3 times. This eliminates guessing with a bunch of numbers out of the blue. 1 x 253 = 253, and 2 x 253 = 506, which is too high. So, we know our answer begins with a one. This eliminates answers b and c. (Remember to check the answers after each step). But now let's think like a math person. We are looking for a number that when we multiply it by 25.3, we get 445.28. This number must have a 6 on the end because that is the only way to multiply anything by 3 and get an 8 on the end. (3 x 6 = 18). Now we have only 2 choices d and e. You now can skip the division and multiply 17.6 x 25.3 and see what you get. If you get 445.28, that's the answer! If you don't, then you know e MUST be the answer. But remember, do arithmetic twice!

     Because the only decimals you will be comparing are less than 1, we will concentrate only on those. The trick is to change each decimal to the same units. You do that by adding zeroes, if necessary, to the end of the numbers. In other words, each one will have the same number of decimal places. Then, eliminate zeroes on the front, and the decimal, and you are left with regular numbers that are easily compared.
Example: Which is larger, 0.0123 or 0.002 ?
Note: The zero in front of the decimal is only a place holder.
First, notice 0.0123 has four places to the right and 0.002 has three. We will add a zero to the end of 0.002 to make it 0.0020 and now has four. Drop the zeroes in front, the two numbers are now:
123 and 20. Which is larger? 123 is larger than 20, so 0.0123 is larger than 0.002.
Example: How do the following decimals compare?
0.05, 0.006 , and 0.018
Again, there are at most three places past the decimal in all three numbers. We will add a zero to the first to give it three like the rest. The new numbers are:
0.050, 0.006 and 0.018.
We did not change the other two because they had three already.
Dropping the zeroes gives us:
50, 6, and 18.
Easy to see which is larger and smaller.
Note:Do not drop zeroes in between 2 nonuser numbers.
0.0506 would become 506, not 56.