GED Mathematical Reasoning: Operations With Decimals

Multiplying with decimals

To multiply with decimals, multiply as you would with whole numbers – by first writing the numbers vertically. Then, multiply each digit in the top number by each digit in the bottom number.

After you’ve finished multiplication process by adding down, the last step is to determine the placement of the decimal point in the final answer. To do this, count the number of decimal places in each of the numbers being multiplied and add those numbers together. That tells you the number of decimal places in the product (answer).

 

Dividing with decimals

Step 1. Place each number in its proper place using the division bracket. The dividend (the number being divided up) sits underneath the bracket. The divisor (the number you’re dividing by), sits outside the bracket and to the left.

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Step 2. Move the decimal point in the divisor (the number sitting outside the bracket) to the right until the decimal appears at the end of the number.

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Step 3. Move the decimal point in the dividend (the number under the bracket) to the right that same number of places. The decimal point does not have to appear at the end of the dividend, but you may need to add place-holding zeros if it goes beyond the last digit.

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Step 4. Bring the decimal point straight up to sit on top of the bracket – that is where the decimal point should appear in the final answer.

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Step 5. Divide as usual.

Example 1

To order materials for a fencing project, Patricia needs to determine the distance around the outside edge of her yard (which is known as the perimeter). She measured each of the four sides and the lengths are shown below (in yards). What is the total distance?

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Notice the key word “total”. That implies we need to add the four given lengths together.

The process for adding decimals is similar to the process for adding whole numbers.  We’ll first stack the numbers vertically -taking care to line up the decimal points.

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Add zeros before the left-most digit or after the right-most digit to ensure that each number has the same number of digits. We’ll call these place-holding zeros. So we’ll add place-holding zeros to the values 45.25, 29.75 and 37.5 so that all numbers contain 5 digits – two digits before the decimal point and three digits after.

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Bring the decimal point straight down into the answer area. Then, add down starting with the right-most column.

Starting with the thousandths column, zero plus 5 plus zero plus zero is 5. We will place a five in the thousandths column.

Next, we move to the hundredths column. 5 plus 2 plus 5 plus zero is 12. We’ll place a 2 in the hundredths column and carry a 1 to the tenths column. Now let’s add the tenths column. 1 plus 2 plus zero plus 7 plus 5 is 15 so we’ll place a 5 in the tenths column and carry the 1. The decimal point should be sitting just to the left of that 5.

Now let’s add the ones column. 1 plus 5 plus 1 plus 9 plus 7 is 23 so we’ll place a 3 in the ones column and carry a 2 to the tens column. Finally, we’ll add the tens column. 2 plus 4 plus 4 plus 2 plus 3 is 15. We’ll write the digits 1 and 5 left of the 3 to arrive at our final result of 153.525 yards.

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Example 2

Yesterday’s high temperature was 89.7 degrees. A heat wave was forecast and today’s high temperature reached 102.3 degrees. By how many degrees did the temperate rise from yesterday to today?

 

Since we are comparing one amount to another, we will subtract. For now we will always subtract the smaller number from the larger number so 102.3 will appear above 89.7.

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Add place-holding zeros to ensure that each number has the same number of digits. So we’ll add a zero to the left of the 8 making 89.7 look like 089.7.

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Bring the decimal point straight down into the answer area. Then, subtract down starting with the right-most column.

We begin with the tenths place. Since we cannot subtract 7 from 3, we’ll borrow from the 2 in the ones place, which makes that 2 a 1 and the 3 a 13. 13 minus 7 is 6 so we’ll place a 6 in the tenths column. The decimal point should be sitting just to the left of the 6.

Now let’s subtract the ones column. Since we cannot subtract 9 from 1, we look to the tens column to borrow, but there’s a zero there with nothing to offer. Therefore, we must borrow from the hundreds column, making that 1 a zero to make the zero in the tens place a 10. Now we can borrow 1 from the ten, making it a 9. And our 1 becomes 11. 11 minus 9 is 2, which we’ll place in the ones column – just to the left of the decimal. Going to the tens place, 9 minus 8 is 1, which we’ll put in the tens place. Finally for the hundreds place, zero minus zero is zero.

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So our final answer appears to be 012.6 degrees. We can, however, drop that preceding zero since it’s unnecessary, making our final answer 12.6 degrees.

 

Example 3

Walker’s car uses about 1.2 gallons of gas to drive the distance from home to work. He drives this distance twice a day, five days each week. How many total gallons of gas does Walker use to travel to and from work each week?

 

When reading this problem, you should note the key word “twice” which implies we will multiply by two in order to find the amount of gas used each day. We’ll need to multiply again to determine the about of gas used for the week.

First, we’ll multiply 1.2 times 2 to determine how many gallons of gas are used each day. Stack 1.2 above 2 and multiply.

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2 times 2 is 4, which we’ll write in the ones column. 2 times 1 is 2, which we’ll write in the tens column. Now we need to determine the placement of the decimal point. Let’s evaluate the number of decimal places in each number we  multiplied. In the number 1.2, there is one decimal place. In the number 2 there are no decimal places. 1 plus zero is 1. Therefore we need to place the decimal point in our answer so the answer has one decimal place. Place your pencil to the far right of the answer 24 and move one place value to the left. That’s where the decimal point goes, making the result of 1.2 times 2 equal to 2.4.

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This means that Walker uses 2.4 gallons of gas to drive to and from work each day. Now let’s calculate how many gallons of gas he uses each week by multiplying this amount by 5. Stack 2.4 above 5 and multiply.

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5 times 4 is 20. So we’ll write a zero in the ones column and carry the 2. 5 times 2 is 10 plus 2 is 12, which we’ll write to the left of the zero.

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Now we need to determine the placement of the decimal point. In the number 2.4, there is one decimal place. In the number 5 there are no decimal places. 1 plus zero is 1. Therefore we need to place the decimal point in our answer so the answer has one decimal place. Place your pencil to the far right of the answer 120 and move one place value to the left. That’s where the decimal point goes, making the result of 2.4 times 5 equal to 12.0 or 12 gallons.

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Example 4

Penny enjoys making fabric dolls to benefit a local charity. Penny has 4.25 meters of fabric. If each fabric doll requires 0.25 meters of fabric, how many dolls can Penny make?

 

Note that Penny will be splitting the fabric up to make each doll. This implies that we will use division to solve this problem. We’ll divide 4.25 by 0.25.

First, let’s place 4.25 under the division bracket with the 0.25 sitting outside the bracket to the left.

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Next, we’ll move the decimal place in the divisor 0.25 two places to the right so that the decimal sits at the end of the number, making it 25.

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Then, we’ll move the decimal place in the dividend two places to the right, making it 425.

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We’ll move the decimal point straight up to sit on top of the bracket to ensure we don’t forget the decimal in our final answer.

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Now, let’s divide.

Does 25 go into the first digit of the dividend? No – 25 does not go into 4. Does 25 go into the first two digits of the dividend? Yes – 25 will go into 42 one time. So we’ll write a 1 on top of the division bracket, above the 2.

Now, multiply 1 times 25. The result is 25. Place that under the 42 and subtract. 25 subtracted from 42 is 17. We’ll place that result under the subtraction bar and then bring down the 5. The 175 showing is what we’ll use for our new dividend.

Now we repeat the process by asking ourselves: How many times does 25 go into 175? The answer is 7 times and we’ll place that 7 on top of the division bracket above the 5 and just left of the decimal point. 7 times 25 equals 175. When we write the result of 175 under the existing 175 and subtract, we get a remainder of zero and there are no other digits to bring down. Therefore, the division process is complete. The final result is showing on top of the division bracket as being 17.0 or 17 dolls.

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As a final note, using the calculator to perform operations with decimals is very similar to doing so with whole numbers. Just be sure to use the decimal point key to type the decimal point in the appropriate place as you enter the values being calculated.

So for example, to divide 4.25 by 0.25, we’ll press the 4 key, then the decimal point key, then 2 then 5. After that, press the key containing the division sign. Next, press the decimal point key followed by the key for 2 then 5. And finally, press the ENTER or equal-sign key to display the final answer of: 17

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