Welcome to the World of Decimal Division!
Hello, Math Explorer! Are you ready for an exciting new adventure? In this chapter, we're going to learn how to divide decimals. It might sound tricky, but don't worry! It's just like sharing, but with numbers that have a decimal point.
Why is this important? Well, we use decimal division all the time! When we split a bill with friends, figure out the price per item, or cut a piece of ribbon into equal parts, we're using decimal division. By the end of these notes, you'll be a pro!
Quick Review: What is Division?
Division is all about sharing equally or finding out how many times one number fits into another. The main parts are:
- Dividend: The number being divided (the total amount you have).
- Divisor: The number you are dividing by (how many groups you are sharing into).
- Quotient: The answer to the division problem!
Example: In $$10 \div 2 = 5$$, 10 is the dividend, 2 is the divisor, and 5 is the quotient.
Part 1: The Magic of Moving Decimals
Dividing by 10, 100, and 1000
Dividing by 10, 100, or 1000 is like a magic trick! You don't need to do long division. All you have to do is move the decimal point to the left.
Think of it this way: dividing makes a number smaller, and moving the decimal point to the left also makes it smaller. It's a perfect match!
Here's the simple rule:
Count the number of zeros in the divisor (10, 100, or 1000). That's how many places you move the decimal point to the LEFT.
- To divide by 10 (one zero), move the decimal 1 place to the left.
- To divide by 100 (two zeros), move the decimal 2 places to the left.
- To divide by 1000 (three zeros), move the decimal 3 places to the left.
Let's see it in action!
Example 1: Divide a decimal by 10
$$145.8 \div 10 = ?$$
The number 10 has one zero. So, we move the decimal point one place to the left.
$$14.5.8 \rightarrow 14.58$$
So, $$145.8 \div 10 = 14.58$$
Example 2: Divide a whole number by 100
$$234 \div 100 = ?$$
Wait, where's the decimal point in 234? For any whole number, the decimal point is hiding at the end! So, 234 is the same as 234.0.
The number 100 has two zeros, so we move the point two places to the left.
$$234. \rightarrow 2.34$$
So, $$234 \div 100 = 2.34$$
Example 3: Adding placeholder zeros
$$5.6 \div 1000 = ?$$
We need to move the point three places to the left, but there's only one digit! What do we do? We add placeholder zeros in the empty spots.
$$5.6 \rightarrow .56 \rightarrow .056 \rightarrow .0056$$
So, $$5.6 \div 1000 = 0.0056$$ (We add a zero at the front to make it look nice!)
Key Takeaway
To divide by 10, 100, or 1000, just slide the decimal point to the left. The number of places you slide is the same as the number of zeros you're dividing by!
Part 2: The Main Event: Decimal Division
Okay, now let's look at division problems that need a bit more work. We'll use the long division method you already know, with one or two new, easy steps.
Case 1: Decimal ÷ Whole Number
This is when you're sharing a decimal amount into whole groups. Imagine sharing a 7.5 metre rope among 3 friends.
Step-by-Step Guide:
- Set it up: Write the problem in the long division format.
- Float the Point: The very first thing you do is "float" the decimal point from the dividend straight up into the answer line (the quotient).
- Divide: Now, just ignore the decimal point and divide like you normally would with whole numbers!
Example: $$8.4 \div 4 = ?$$
Step 1 & 2: Set it up and float the point.
The decimal in 8.4 goes straight up.
Step 3: Divide.
How many 4s in 8? Two. (Write 2 above the 8).
How many 4s in 4? One. (Write 1 above the 4).
Your answer is 2.1!
What if you have a remainder? Just add a zero to the end of the dividend and keep going! A decimal like 8.4 is the same as 8.40 or 8.400.
Key Takeaway
When dividing a decimal by a whole number, the most important rule is: Float the decimal point straight up! Then, just divide normally.
Case 2: Whole Number ÷ Whole Number (with a Decimal Answer)
Sometimes, when you divide two whole numbers, you get a remainder. Instead of just writing "R 1", we can get a more exact decimal answer!
Imagine sharing 5 chocolate bars between 2 people. Each gets 2 bars, and there is 1 left over. We can split that last bar in half, so they each get 2.5 bars!
Step-by-Step Guide:
- Start dividing: Begin the long division as you normally would.
- Hit a wall? When you get a remainder and have no more numbers to bring down, don't stop!
- Add a Decimal and a Zero: Add a decimal point and a zero to the end of your dividend (e.g., 5 becomes 5.0).
- Float the Point: Float that new decimal point straight up into your answer.
- Keep Dividing: Bring down the zero and continue dividing until you have no remainder.
Example: $$15 \div 2 = ?$$
$$ \begin{array}{r} 7.5 \\ 2 \enclose{longdiv}{15.0} \\ -14\phantom{.} \\ \hline 10 \\ -10 \\ \hline 0 \end{array} $$2 goes into 15 seven times, with a remainder of 1.
We add a decimal point and a zero to 15, making it 15.0. We float the point up.
We bring down the 0 to make 10.
2 goes into 10 five times.
The answer is 7.5!
Watch Out! A Common Mistake
Remember to add the decimal point to the dividend before you add any zeros. If you just add a zero, you change 15 into 150, which is a different problem!
Case 3: Dividing By a Decimal (The BIG Trick!)
This is the most important part of decimal division. There is one BIG rule we must follow:
You can't divide by a decimal!
It's too confusing. But don't worry, we have a super-clever trick to change the problem into one we already know how to do. The goal is to make the divisor a whole number.
The Analogy: Think of it like a balancing scale. To keep it balanced, whatever you do to one side, you MUST do to the other. Here, we want to change the divisor, so we must also change the dividend in the exact same way.
The Step-by-Step Trick:
- Look at the divisor. Move its decimal point all the way to the right to make it a whole number.
- Count the hops! Count how many places you moved the decimal point.
- Now look at the dividend. Move its decimal point the exact same number of places to the right. You might need to add zeros.
- Rewrite the problem. You now have a brand new, easier problem!
- Float the NEW decimal point up and solve it just like in Case 1 or 2.
Example 1: Whole Number ÷ Decimal
$$18 \div 0.9 = ?$$
The divisor is 0.9. To make it a whole number (9), we move the decimal one place to the right.
$$0.9 \rightarrow 9.$$ (1 hop)
Now we must do the same to the dividend, 18. Remember, 18 is 18.0.
$$18.0 \rightarrow 180.$$ (1 hop)
Our new, easy problem is $$180 \div 9$$. The answer is 20!
Example 2: Decimal ÷ Decimal
$$2.4 \div 0.8 = ?$$
Make the divisor (0.8) a whole number. Move the decimal one place right.
$$0.8 \rightarrow 8.$$ (1 hop)
Do the same to the dividend (2.4). Move the decimal one place right.
$$2.4 \rightarrow 24.$$ (1 hop)
The new problem is $$24 \div 8$$. The answer is 3!
Example 3: A trickier one!
$$5.25 \div 0.5 = ?$$
Make the divisor (0.5) a whole number. Move the decimal one place right.
$$0.5 \rightarrow 5.$$ (1 hop)
Do the same to the dividend (5.25). Move the decimal one place right.
$$5.25 \rightarrow 52.5$$ (1 hop)
The new problem is $$52.5 \div 5$$. Now we just float the new point up and solve! The answer is 10.5.
Key Takeaway
To divide by a decimal, remember this phrase: "Slide in the divisor, slide in the dividend!" Once the divisor is a whole number, the problem becomes easy.
Part 3: Rounding and Estimating
Rounding Your Answer
Sometimes, a division problem can go on forever! For example, $$10 \div 3 = 3.33333...$$ We can't write '3's all day, so we often need to round our answer.
When we round, we use a special symbol: ≈, which means "approximately equal to".
Quick Rounding Review:
- Find the place you need to round to (e.g., the tenth or hundredth).
- Look at the digit to its right.
- If that digit is 5 or more, you round up (add one to your rounding digit).
- If that digit is 4 or less, you let it rest (the rounding digit stays the same).
Example: Solve $$2 \div 7$$ and round to the nearest hundredth.
$$2 \div 7 = 0.2857...$$
The hundredths place is the 8. The digit next to it is 5.
Since it's 5 or more, we round the 8 up to 9.
So, $$2 \div 7 \approx 0.29$$.
Estimating Your Answer
Before you even start dividing, it's a great idea to estimate the answer. This helps you know if your final answer is sensible.
How? Just round the numbers in the problem to "friendly" numbers that are easy to work with.
Example: You need to solve $$24.3 \div 4.9$$
Let's estimate! 24.3 is close to 25. And 4.9 is very close to 5.
The problem is about the same as $$25 \div 5$$.
Our estimated answer is 5.
The real answer to $$24.3 \div 4.9$$ is 4.959..., which is very close to 5! Our estimate tells us we are on the right track!
Key Takeaway
Estimating is your secret weapon to catch mistakes. Rounding helps you give a neat answer when the division gets messy.
Part 4: Let's Solve Some Problems!
Now let's use our new skills to solve some real-world problems.
Problem 1:
A packet of 5 identical pens costs $12.50. How much does one pen cost?
Thinking: We need to share the total cost ($12.50) among the 5 pens.
Equation: $$12.50 \div 5$$
Solving: This is a decimal divided by a whole number. We just float the point up and divide!
The answer is $2.50 per pen.
Problem 2:
Mr. Chan has a wooden plank that is 4 metres long. He needs to cut it into smaller pieces that are each 0.8 metres long. How many small pieces can he get?
Thinking: We need to see how many times 0.8 fits into 4.
Equation: $$4 \div 0.8$$
Solving: We are dividing by a decimal! We must use our trick.
Move the decimal in the divisor (0.8) one place right to get 8.
Move the decimal in the dividend (4.0) one place right to get 40.
Our new problem is $$40 \div 8$$.
The answer is 5. He can get 5 small pieces.
Chapter Summary: You're a Decimal Division Expert!
The Most Important Rules to Remember:
- Dividing by 10, 100, 1000: Move the decimal point to the LEFT.
- Decimal ÷ Whole Number: The golden rule is FLOAT THE POINT UP! Then divide as usual.
- Dividing by a Decimal: This is the BIG ONE! You MUST make the divisor a whole number first. Remember: "Slide in the divisor, slide in the dividend!"
- Estimating: Always try to estimate first to see if your final answer makes sense. It's the best way to be a careful math detective!
Great job working through this chapter. Keep practising, and soon decimal division will be as easy as 1, 2, 3!