GED Mathematical Reasoning: Multiplying and Dividing Whole Numbers
A local animal shelter rescues an average of 36 animals per month. What is the total number of animals rescued on average for one year?
To solve this problem, we could employ repeated addition and add 36, 12 times (since there are 12 months in one year). Or, to be more efficient, we can use multiplication.
Let’s talk briefly about multiplying whole numbers before we work through this example.
- Symbols that denote multiplication include: x (which looks like the letter “ex”), * (the asterisks symbol), or a dot ( · )
- Numbers that are being multiplied together are call “factors.” The result of a multiplication problem is called the “product.”
- To multiply whole numbers, it is important to be very familiar with the multiplication table shown here.
To use the multiplication table, note that each row and column header intersect at the product of those two factors. For example, the product of 7 and 6 is 42.
Notice that any number multiplied by zero is equal to zero.
And the product of any number and 1 is itself.
As a side note: Memorizing the multiplication table will be an asset to you in all levels of math. An effective way to become familiar with these multiplication facts is by using flashcards.
Now, back to our example.
To set up the problem, step 1 is to write the numbers 36 and 12 vertically (one on top of the other).
For step 2, we’ll multiply each digit in the top number by each digit in the bottom number, carrying as needed. When multiplying by the ones digit, start the answer row in the ones column. When multiplying by the tens digit, move left one space and start the answer row in the tens column.
We will start the multiplication process by multiplying each digit in the number 36 by 2. Then, we’ll multiply each digit in the number 36 by 1.
When we multiply 2 times 6, the result is 12 so we implement the “carry” technique and write the 2 in the ones column and carry the 1 over to the top of the tens column.
Now, multiply 2 times 3. The result is 6. You may be wondering: How do we account for the 1 that we carried?
The answer is: We add it to the product of 2 times 3 and place that result in the tens column. Therefore, the number we place in the tens column, to the left of the 2, is a 7.
Next, we’ll multiply each digit of 36 by the 1. When we do this, pay special attention to where we place the result.
When we multiply 1 times 6, the result is 6. We place the 6 in the tens column, underneath the 7.
Now multiply 1 and 3 and place the result of 3 to the left of the 6.
Step 3 is to add down. To determine our final answer, we add down starting with the ones column – treating any blank space as a zero. It may help to actually write zeros in the blank spaces.
2 plus 0 equals 2. We place the result in the ones column. 7 plus 6 equals 13. We place the 3 in the tens column and carry the 1 to the hundreds column. 1 (plus zero) plus 3 is equal to 4, which we will place in the hundreds column.
Our final result is: 432 animals.
As you can see, multiplying whole numbers can be a very tedious process. However, when one of the factors is a power of ten, such as 10, 100, or 1000 the process is much simpler.
- To multiply a number by 10, add one zero to the end of the number. For example, 452 times 10 is equal to 4520 or 4,520.
- To multiply a number by 100, add two zeros to the end of the number. For example, 452 times 100 is equal to 45200 or 45,200.
- To multiply a number by 1000, add three zeros to the end of the number. For example, 452 times 1000 is equal to 452000 or 452,000.
Now let’s move on to division.
Drew drove 495 miles on a full tank of gas. If his car’s gas tank holds 15 gallons of gasoline, what was Drew’s miles per gallon?
To find the number of miles driven per gallon of gas, we divide 495 by 15.
Here are some important facts about division:
- Division is the opposite operation of multiplication.
- To divide is to determine how many times a number will go into another number. For a simple example, there are five 2’s in the number 10, so we say 10 divided by 2 is 5.
- There are a few symbols that denote division. They are:
The division sign: ÷
The fraction bar: —
The backslash: /
And the division bracket:
And now for some definitions.
- The number we divide into is called the dividend. The number we divide by is called the divisor. The result of a division problem is called the quotient.
- When we divide a dividend by a divisor and there is a quantity left over, we call this leftover quantity the remainder. For example, 7 divided by 3 equals 2 (since 3 goes in to 7 two times). And there is a remainder of 1.
As for the process itself, division with larger numbers typically requires us to use a process called long division.
Long division is the process of:
Step 1. Asking: How many times does the divisor go into the dividend (or part of the dividend)?
Step 2. Multiplying
Step 3. Subtracting
Step 4. Bringing down
And then repeating this process all over again until the remainder is smaller than the divisor and there are not other digits to bring down.
Going back to our example, to solve the problem using the division bracket, 495 goes underneath the bracket and the 15 goes on the outside of the bracket on the left side.
We first ask ourselves: Does 15 go into the first digit of the dividend 495? The answer is no – the number 15 is too large to go into the first digit, which is 4.
Then, we ask: Does 15 go into the first two digits of the dividend? The answer is yes – the number 15 does go into 49.
Now, let’s begin the long division process – using our four-step loop.
Step 1. 15 goes into 49 three times (since 15 times 3 equals 45).
We write the 3 on top of the division bracket, directly above the 9.
Step 2. Now we multiply 3 times 15, which – as previously stated – is 45.
We write 45 directly underneath the 49.
Step 3. Next, we subtract 45 from 49.
The result is 4.
Step 4. Carry down the next digit in the dividend. It is a 5.
Now repeat the 4 step process using the 45 as the new dividend.
Step 1. 15 goes into 45 three times (since 15 times 3 equals 45).
We write the 3 on top of the division bracket, above the 5.
Step 2. Now we multiply 3 times 15, which – as we stated – is 45.
We write 45 underneath the 45 to prepare for the subtraction step.
Step 3. Next, we subtract 45 from 45.
The result is 0.
Since the remainder is zero and there are no other digits to carry down, the process is complete. Our quotient is 33. Therefore, the miles per gallon is equal to 33.