# Decimals

This section covers:

You actually know a lot about decimals already since you probably do a lot of shopping. The “cents” part of the money is like the “hundredths” part of decimals – it’s one hundredth of a dollar. That’s because there are a hundred cents in a dollar. When you buy a new pair of ballet slippers for $49.99, this is a decimal, since there are some numbers after a “period”. Think of a decimal as typically being “in between” regular numbers. You can divide the 49.99 into 40 dollars + 9 dollars + 99 cents. The 4 is in the “tens” field (since $$4\times 10=40$$), the 9 is in the “ones” field (since $$9\times 1=9$$), and the decimal point represents the start of the fractional part of the number (less than 1). The first 9 after the decimal point represents the “tenths” (since $$\displaystyle \frac{1}{{10}}\times \,9\,=\,.1\,\times \,9=.9$$), and the second 9 represents the “hundredths” (since $$\displaystyle \frac{1}{{100}}\times 9\,\,=.01\,\,\times 9\,\,=.09$$). (With money, the first 9 after the decimal would be the dimes, and the second 9 would be the pennies). Notice that to the right of the decimal point, you add “th” to the places; think of “th” as meaning “part of”. Remember that $$\displaystyle \frac{1}{{10}}$$ and $$\displaystyle \frac{1}{{100}}$$ are fractions, since they are the quotient of two numbers. Here’s a diagram of the different “places” in a number. Again, the decimal divides the number into the part that is a “whole” and the part that is a “part” or fraction. # Adding and Subtracting Decimals Now let’s talk about adding, subtracting, multiplying and dividing decimals. When we add decimals, we have to line up the decimal point and decimal places. This is because we have to add the tens to the tens, the ones to the ones, the tenths to the tenths, the hundredths to the hundredths, and so on. For this reason, we usually add down (vertically) when we add decimals. We may still have to carry when we add, like we did earlier, and borrow when we subtract, but again, we must line up the decimals perfectly. < Let’s say we have to pay$49.99 for the ballet shoes and we also paid $132.99 for other stuff we bought (shoes and a dress). The total amount that we’d have to pay (not including tax) is: Make sure you totally understand the carrying here. We start from the right (remember, this is the opposite from reading), and when the two numbers add up to over 10 (9 + 9 = 18), we put the 8 down below and “carry” the 1 to the next place to the left. Then we add the 199 and come up with 19. We put the 9 down, and carry the 1 to the next place to the left. We repeat this until we’re done. So the total amount we had to pay was$182.98.

Subtracting decimals is similar, but we may have to “borrow” (like we did without decimals) instead of “carry”. Again, line up the decimals and subtract like you did before. For example, if your grandma gave you $200 to spend and you buy a new purse that was$30.42 (including tax), the amount you had left could be determined this way: This one is a little tricky since we have so many 0‘s on the top.  We start with the rightmost 0, and have to borrow to make it a 10. But since the next number (the tens place) is a 0, we can’t borrow from it until we turn it into a 10 and borrow again from the next 0 (the hundreds place). When we do borrow, we are left with a 9. We do this all the way until we get to the 2 and have to make it a 1. We have $169.58 left! Make sure you go through this example a couple of times to get it! # Multiplying and Dividing Decimals Multiplying and dividing decimals are a little different. We do not need to line up the decimal places, but we need to do some counting with the decimal places for multiplying. For dividing, we will do some lining up; I know that’s a little weird! Let’s start with multiplication. Let’s say we have four pieces of jewelry that we’d like to buy, and each one costs$9.99. We’ll use our math (math is great for shopping!!) to multiply $$9.99\times 4$$. It’s always better to do these types of multiplications vertically, and, as we mentioned in the multiplication section, put the “longer” number (number with more digits) on top. We do the multiplication just like we learned earlier, and then we count the number of places to the right of the decimal in all the numbers we’re multiplying to get the number of places to count over (from the right) in the final answer.

So, with our example, we count over two places for the 9.99, and then we have to go back two places from the right when we get the final answer: It costs $39.96 to buy our four beautiful jewelry pieces without tax (we’ll talk about tax later). Let’s look at another example. Note that, when we got the final answer, we had to put the decimal point over four places from the right, since we had two decimal places in the first number and two decimal places in the second number: Dividing with decimals is a little trickier. We do the division part the same way as we learned before, but we need to be careful with the decimal points. If we have a decimal point on the outside (the divisor), we need to move it over to the rightmost place. Then, to make up for this, we need to move the decimal point on the inside (the dividend) the same number of places. This way we are always dividing by whole numbers, which is what we want. We then always put the decimal place straight up from where it is in the inside (the dividend), and leave it there to get the answer. Note that with decimal division, if the number doesn’t go in perfectly (we have remainders), we might possibly have to keep going forever (these are called irrational numbers, since, like people, we can never get an exact answer)! So usually we just round to 2 decimal places (like with money), or how many decimal places you are asked to round to. Let’s work to “undo” what we did with multiplication in the last two examples. We are first going to divide$39.99 by 4:

 Long Division Notes Bring the decimal straight up first.   Since the 4 doesn’t go into the 3, we have to see if it goes in the 39. It does 9 times, so we put the 9 above the 39.   Then we multiply 9 by 4 to get 36, subtract 36 from 39 to get 3, and bring the next 9 down.   We continue like this until either we don’t have a remainder or we get the number of decimal places we want in the answer.   With money, we just have to continue until we have 2 decimal places. In this problem, we would actually round to \$10, since we have a remainder of 3; we’ll talk about rounding later.

Now, in a more complicated case, let’s divide .1827 by .63. The first thing we need to do is move the decimal points over on both the outside (the divisor) and the inside (the dividend). Since we moved the decimal point two places to the right to get to the end on the outside, we need to move it over two places to the right on the inside. Now it’s in between the 8 and 2 on the inside, and we move it straight up. Then we can do the long division; in this case, we have no remainders. This means that .63  x .29  = .29  x .63  = .1827. One other note about decimals: if you have any zeros (0‘s) to the right of a decimal point without another number, you can get rid of those zeros – they do nothing:

$$\begin{array}{l}.2000000=.2\\1000.4300=1000.43\end{array}$$

# Rounding Decimals Many times with decimals or even with regular numbers, you have to do some rounding up or down, since we only want the answer in a certain number of decimal places. One good example of this is money, where we only need two decimal places for the cents.

Rounding isn’t difficult; you just start at the position that you’re asked to round to (like the hundredths place) and look at the next number to the right. If that number is 5 or higher (5, 6, 7, 8or 9), you add one more number to number you’re rounding, and drop all the numbers after it. If the number is 4 or lower (4, 3, 2, 1, or 0), you just get rid of all the numbers to the right. Remember if you’re rounding a number to the left of the decimal point, you may have to add 0’s to make up for the digits you’ve dropped.

Here are some examples:

• Round 1.555 to the nearest hundredths place: 1.56.   Since the next number to the right (thousandths place) is 5, we round the hundredths place to 6 and get rid of the 5.
• Round .4049444 to the nearest hundredths place: .40. Since the next digit to the right (the thousandths place) is a 4, we leave the hundreds place as is and get rid of the 49444 digits.
• Round 100,245 to the nearest hundreds place: 100,200. Since the next digit to the right (tens place) is a 4, we leave the hundreds place as is, get rid of the 45 digits and add 2 0’s to make up for them (since we are at the left of the decimal point).

# Decimals That Repeat Note that in many cases, you’ll find that a decimal repeats when you do the division (in one place, or two places, or even more!). These numbers are called repeating decimals, and are again a part of what we call irrational numbers (since we never get an exact answer). We indicate a decimal that repeats by putting a line over the number or numbers that actually repeats. Try these by hand or on your calculator to see what happens.

Here are some common (and not so common) repeating decimals. Note that the first number is what we call a fraction, which is talked about in the Fractions section.

 \displaystyle \begin{align}\frac{1}{3}=3\overset{{.33333..}}{\overline{\left){{1.00000..}}\right.}}=.33333….=.\overline{3}\\3\text{ Repeats}\\\text{Round 2 places}:\text{ }.33\end{align} \begin{align}\frac{2}{3}=3\overset{{.66666..}}{\overline{\left){{2.00000..}}\right.}}=.66666….=.\overline{6}\\\text{66 Repeats}\\\text{Round 2 places: }\text{.67}\end{align} \begin{align}\frac{2}{{11}}=11\overset{{.1818..}}{\overline{\left){{2.0000..}}\right.}}=..1818….=.\overline{{18}}\\\text{18 Repeats}\\\text{Round 2 places: }\text{.18}\end{align} \begin{align}\frac{3}{7}=3\overset{{.4285714..}}{\overline{\left){{7.0000000..}}\right.}}=.428571….=.\overline{{428571}}\\\text{428571 Repeats}\\\text{Round 2 places: }\text{.43}\end{align}

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