Does the prospect of multiplying or dividing decimal numbers fill you with dread? Would you rather wrestle great white sharks with both ands AND both feet tied behind your back? Don’t worry. Multiplying and dividing decimals isn’t that hard at all — and we’ll show you how to do it.
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Decimals: A Quick Run-Through
In case you’re a little hazy on the subject, here’s what you need to know about decimal numbers: Everything to the left of the decimal place is just an ordinary number; everything to the left of the decimal is a fraction, and the farther right you go, the smaller the fraction gets. The first place to the right of the decimal is tenths (0.1), the second place is hundredths (0.01), the third place is thousandths (0.001), and so on. Just divide by ten one more time for each step you go to the right. (Don’t worry about that zero to the left of the decimal; it’s there to make it easier to see the decimal point, but it doesn’t change the value of the number.)
OK, Let’s Multiply
You know the basics of multiplication with whole (non-decimal) numbers: start from the right, and if the product is greater than 10, you carry the tens’ part one step to the left, like this:
1234 x 5 ----- 6170
That’s basically what you do when you multiply decimals. The only difference is that you have to keep track of the decimal point.
Stick To the Point
How does this work? Let’s start by multiplying a decimal number by a whole number; we’ll use the above example, but with a decimal point in the top number:
12.34 x 5 ------ 61.70
The easy part is that you only need to keep track of the decimal when you get to the number just to its right, and all you really need to remember is that if you carry anything over, when you multiply that number, the part that you carry gets carried to the left side of the decimal:
0.3 x 5 ---- 1.5
Everything else stays on the same side of the decimal as it originally was located. The only difference is that if you carry from the number just to the right of the decimal, you carry it to the number just to the left of the decimal.
Decimals x Decimals
“That’s easy enough, but what if you’re multiplying a decimal number by a decimal number? How do you keep track of two decimal points?”
All it takes is a tiny trick, one that isn’t even a trick at all: when you multiply decimals, take at the number of digits (not the values of the digits – just how many digits there are) to the right of the decimal point in each number that you multiply, and add them. The product should have that many digits to the right of its decimal point.
How Many Digits?
You already did it here, but it wasn’t obvious:
12.34 x 5 ------ 61.70
12.34 has two digits to the right of the decimal, and 5 has none; 61.70 has two digits to the right of its decimal, because 2 + 0 = 2. Now look what happens if we use 0.5 instead of 5:
12.34 x 0.5 ------ 6.170
12.34 still has two digits to the right, but 0.5 now has one digit to the right. 2 + 1 = 3, so 6.170 has three digits to the right, and not just two. (Note that we count the right-hand zero in the product as a digit, because it’s the result of multiplying numbers that weren’t zero, but we wouldn’t count right-hand zeroes in the numbers that we’re multiplying.)
Dividing decimals is actually easier than multiplying them, if anything. Whole-number division is just the opposite of multiplication, of course: start from the left, move toward the right, and carry anything that’s left over (the remainder) to the right. But if you think in terms of decimal numbers, you can keep going, right past the decimal:
1234 / 5 = 246.2
We could have just called it 246 with a remainder of 4, but if we put a decimal after the 4 in 1234, we can finish the job and divide it out all the way. And just as with multiplication (only going the opposite way), when you get to the number just to the left of the decimal point, you carry the remainder over to the first digit to the right of the decimal point, and if the number too small to divide, you carry the division over one place to the right:
1.5 / 5 = 0.3
Decimals / Decimals
“But you’re dividing by whole numbers. Let’s see some real decimal-on-decimal division!”
And now we get to the really good trick: the secret to dividing decimal numbers by decimal numbers is that you don’t do it. Instead, when you have two decimal numbers and you want to divide them, you shift the decimal points for both of them to the right an equal number of times, until the number that you’re dividing by turns into a while number.
Shift, Divide, and Conquer
How does it work? Let’s divide 37.248 by 2.4.
First, we shift the decimals for both numbers one place to the right, to turn them into 372.48 and 24.
Note that we shifted both decimal points the same distance to the right.
Also note that we only need to shift them so the number that we’re dividing by is a whole number. The number that we’re dividing into can be a decimal, because we know how to divide decimals by whole numbers:
372.48 / 24 = 15.52
Check Your Work
But is dividing 372.48 by 24 really the same as dividing 37.248 by 2.4? If it is, then multiplying 2.4 by 15.52 should give us 37.248. You can step through it and check, multiplying the two decimal numbers just as we showed you earlier:
15.52 x 2.4 ------ 37.248
It does work, and it works because we shifted both decimal points the same number of places. It’s the same as doing this: 200 / 40 = 5, and 20 / 4 = 5, only we’re shifting the numbers on the right side of the decimal place, instead of the left side.
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