Let's explore the division problem 821 divided by 5. We want to find how many groups of 5 can be made from 821, or how many times 5 goes into 821. We'll represent this problem in both horizontal format and long division format to familiarize ourselves with different representations.
Now let's work through the long division process step by step. First, we divide 8 by 5, which gives us 1 with a remainder of 3. We write the 1 above the division bracket. Next, we bring down the 2 to make 32. Then we divide 32 by 5, which gives us 6 with a remainder of 2. We write the 6 above the division bracket. Then we bring down the 1 to make 21. Finally, we divide 21 by 5, which gives us 4 with a remainder of 1. Our final answer is 164 with a remainder of 1.
Our division gives us 164 with a remainder of 1. This means that 821 equals 5 times 164 plus 1. There are different ways to express this answer. We can write it as a whole number with remainder: 164 R1. We can express it as a mixed number: 164 and 1 fifth. Or we can convert it to a decimal: 164.2. All these forms represent the same mathematical relationship.
Let's verify our answer using multiplication. If we multiply 164 by 5, we should get 820. Then adding the remainder 1 gives us 821, confirming our division was correct. We can break down 164 times 5 as 100 plus 60 plus 4, all multiplied by 5, which equals 500 plus 300 plus 20, giving us 820. Adding the remainder 1 confirms our original number 821. We can also use an estimation method: 800 divided by 5 is 160, and 21 divided by 5 is 4 with remainder 1, so 821 divided by 5 is approximately 164 with remainder 1, confirming our exact calculation.
Let's consider a real-world example. If you have 821 stickers to distribute equally among 5 children, each child gets 164 stickers, and there will be 1 sticker left over. This visual shows 5 children, each receiving a pile of 164 stickers represented by these orange dots. The single orange dot at the bottom represents the one leftover sticker. This type of division is useful in many practical scenarios, such as dividing money, organizing items into groups, or calculating rates. Understanding division helps us solve everyday problems involving fair distribution of resources.