1.6+Division

Section 1.6 - Division
You might find division written several different ways:



Plenty of vocabulary, but the important one is **Quotient - the answer to a division problem**.


 * Division is the Opposite of Multiplication**

Division will be much easier if you know your times tables, because you will probably do simple multiplication to do division problems, even big, hairy long division problems.

Even though division uses different terms, like dividend, divisor, and quotient. Division can also be understood using multiplication terms.



The only time, I think, that division is not the opposite of multiplication is when zero is involved. Zero, divided by anything, is zero. However, you cannot divide anything by zero - any division by zero problem is considered "undefined". It will produce an error on a calculator if you try. Think of it this way, if you have 6 candy bars and divide them evenly between 3 friends, each would get 2 - 6 divided by 3 is 2. If you have zero candy bars and distribute them between 3 friends, each unhappy friend would get zero candy bars - 0 divided by 3 is 0. However, if you have 6 candy bars to distribute, but no friends, you cannot divide them - you will never be able to divide them evenly - it simply can't be done, just like division by zero is always "undefined."
 * Division by Zero**


 * Factor Pairs**

This is where the multiplication tables come into play. Factor pairs are the two numbers that multiply together, exactly, to get a product. 6 x 5 = 30. Nothing else multiplied by 5 = 30, so 6 and 5 must go together (multiplied) to get 30. 6 and 5 are factor pairs of 30. However, there are other factor pairs that make 30. 3 and 10 are also factor pairs of 30. So are 2 and 15 and 1 and 30.

So which number, multiplied by 4, equals 30? No whole number does. That means that 4 is not a factor of 30 because it doesn't evenly divide into it. Factors must evenly divide into a number.


 * Divisibility Tests**

There are some shortcuts to help you find factors, or numbers that will evenly divide into other numbers.

2 - If a number is even, 2 will be a factor. The number is even if it ends in 0, 2, 4, 6, or 8.

3 - If you add up the digits of a number and the sum is divisible by 3, the whole number is divisible by 3. For example, if you add up the digits in 2,316, you get 12 (2+3+1+6 = 12), which is divisible by 3 (3 x 4 = 12). That means that 2,316 is also divisible by 3. Three time something = 2,316. It might require long division to figure out that 3 x 772 = 2,316.

4 - If the last two digits in the number are divisible by 4, the whole number is. You don't add those last two digits up, you treat them like a two digit number. 54,621 is not divisible by 4, because 21 (last two digits) is not.

5 - All numbers ending in either 5 or 0 are divisible by 5.

10 - If it ends in a 0, it is divisible by 10.

There are other rules for other digits as well, but I don't think they are overly important. 7 is very complicated, and 6 and 9 are multiples of other, easier digits anyway.


 * Long Division**

Although basic division can usually be done in your head, long division is really not that hard, or that long for that matter. It is very useful, and will be used in many chapters in this course.



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 * Remainders**

When you get to the end of a long division problem and have a remainder left over, it may not mean what you think it does. For example, If you are dividing up a large group of 200 peace-loving hippies and trying to get them on buses to get to the rally, but the buses only hold 60 people each, you will fill up 3 buses, but have 20 people left over. The remainder of 20 in this case does mean 20 people, but if you look at the answer to the division problem, the quotient, it will say 3 r 20, or 3 remainder 20. The three does not refer to the number of people, but the number of full buses. Obviously it isn't 3 remainder 20 buses.

In reality, the remainder is a part of a bus, or a fraction of the bus. The fraction, (which we address in Chapter 4), is made up of the remainder over the divisor. In this case 20 over 60, which reduces to one third. 3 and 20/60 or three and a third describes how many buses would be needed.


 * Application Problems**

Hopefully you are saying, "But wait, how can I schedule a third of a bus, when they only come in whole buses?!" Sometimes with real-world applications, you will need to round up. You will have to order 4 buses in order to accommodate the peace rally, even though the last bus may not even be half full.

Division also enables you to work with perimeter and area again, but in a different way. Say you had a square shape and knew that the perimeter was 400 inches. How long would each side be?

What if a rectangle had an area of 80 square meters and you knew that one of the sides was 8 meters long? How could you use the formula for area of a rectangle, and division, to find the length of the missing side? A = L x W 80 is equal to 8 times something. If you divide 80 by 8 (or use your vast knowledge of times tables), you will find that the other side must be 10 meters long.

**Where to from here?**
1.1 Place Value 1.2 Rounding and Estimating Numbers 1.3 Addition 1.4 Subtraction 1.5 Multiplication 1.6 Division 1.7 Exponents 1.8 Order of Operations 1.9 Prime Numbers
 * Chapter One Review**
 * Chapter One Homework**