Chapter: Multiplication (Three Digits)
Hello, Super Mathematicians! Welcome to the exciting world of multiplication with bigger numbers. Have you ever wondered how a shop knows how many sweets are in 100 jars? Or how many bricks are needed to build a big wall? They use multiplication!
In this chapter, we're going to learn how to multiply three-digit numbers. It might sound like a big challenge, but don't worry! We'll break it down into simple, easy steps. By the end, you'll be able to solve these big problems like a pro. Let's get started!
Quick Recap: What is Multiplication?
Remember, multiplication is just a fast way of doing repeated addition. It's like having a superpower to add numbers really quickly!
For example, $$4 \times 3$$ is the same as saying $$4 + 4 + 4$$, which equals 12.
Quick Review Box
Knowing your multiplication tables (or times tables) is super important. It makes everything much easier. Keep practicing them every day!
Multiplying a 3-Digit Number by a 1-Digit Number
Let's warm up with something you might have seen before. We will multiply a big number by a small number, using the column method.
Step-by-Step Guide: No Carrying
Let's try to solve $$123 \times 3$$.
We always start from the right side, with the Ones place!
Step 1: Multiply the Ones
First, multiply the Ones digit of the top number (3) by the bottom number (3).
$$3 \times 3 = 9$$
Write the 9 in the Ones place in your answer.
Step 2: Multiply the Tens
Next, multiply the Tens digit of the top number (2) by the bottom number (3).
$$2 \times 3 = 6$$
Write the 6 in the Tens place.
Step 3: Multiply the Hundreds
Finally, multiply the Hundreds digit of the top number (1) by the bottom number (3).
$$1 \times 3 = 3$$
Write the 3 in the Hundreds place.
So, $$123 \times 3 = 369$$. See? You did it!
Step-by-Step Guide: With Carrying Over
Sometimes, the answer in a column is 10 or more. When this happens, we have to do something called carrying over. It's like passing a number to its neighbour for safekeeping!
Let's try $$145 \times 4$$.
Step 1: Multiply the Ones
$$5 \times 4 = 20$$
This is a two-digit number! We write the 0 in the Ones place and carry the small 2 over to the top of the Tens column.
Step 2: Multiply the Tens (and add the carried number!)
$$4 \times 4 = 16$$
Now, add the little 2 that we carried over: $$16 + 2 = 18$$.
Write the 8 in the Tens place and carry the small 1 over to the top of the Hundreds column.
Step 3: Multiply the Hundreds (and add the carried number!)
$$1 \times 4 = 4$$
Now, add the little 1 that we carried over: $$4 + 1 = 5$$.
Write the 5 in the Hundreds place.
Awesome! The final answer, or product, is 580.
Common Mistake to Avoid!
A very common mistake is forgetting to add the number you carried over. Always look for a little number at the top of the column before you write your final digit down!
Key Takeaway
When you multiply, work from right to left (Ones -> Tens -> Hundreds). Always remember to add any numbers you have carried over!
Level Up! Multiplying a 3-Digit Number by a 2-Digit Number
You're doing great! Now, let's try multiplying by a two-digit number. Don't worry if this seems tricky at first. It's just two small problems combined into one!
Think of it like this: to solve $$234 \times 21$$, we first solve $$234 \times 1$$ and then we solve $$234 \times 20$$. Finally, we add the two answers together!
Step-by-Step Guide
Let's solve $$234 \times 21$$ together.
Part 1: Multiply by the Ones digit (1)
First, we ignore the '2' in '21' and just multiply 234 by 1. This is easy!
$$234 \times 1 = 234$$
Write this as the first line of your answer.
Part 2: Multiply by the Tens digit (2) and use the Magic Zero!
Now we're going to multiply by the '2' in '21'. But wait! That '2' is in the Tens place, so it really means 20.
To show this, we put a placeholder zero (our Magic Zero!) in the Ones place on the second line. This is a VERY important step!
Now, we can multiply 234 by 2, just like we did before.
$$4 \times 2 = 8$$
$$3 \times 2 = 6$$
$$2 \times 2 = 4$$
Write 468 next to the zero.
Part 3: Add them up!
The final step is to add your two answer lines together. Draw a line and add them up, column by column.
The final answer is 4,914. Great job!
Key Takeaway
The placeholder zero is your best friend when multiplying by the Tens digit. It makes sure all your numbers line up correctly!
Checking Your Answer with Estimation
How do you know if your big answer is correct? You can use estimation! This is like making a smart guess to see if your answer is in the right ballpark.
Let's say we want to estimate the answer for $$156 \times 34$$.
1. Round the numbers to make them easy to work with.
- 156 is close to 200.
- 34 is close to 30.
2. Multiply the rounded numbers.
- $$200 \times 30 = 6000$$
The real answer to $$156 \times 34$$ is 5304. Since 5304 is close to our estimate of 6000, we can be pretty sure our answer is correct! If we had gotten 530, we would know we made a mistake.
Solving Word Problems
Multiplication is everywhere! Let's see how we can use it to solve a real-world problem.
Problem: A farmer has 115 apple trees. If each tree grows 25 apples, how many apples does the farmer have in total?
Steps to Solve:
1. Find the key numbers: 115 trees and 25 apples.
2. Decide the operation: We need to find the total of many equal groups, so it's multiplication.
3. Solve the calculation: $$115 \times 25$$
$$ \begin{array}{@{}c@{\,}c@{}c@{}c} & & \overset{2}{1} & \overset{2}{1} & 5 \\ \times & & & 2 & 5 \\ \hline & & 5 & 7 & 5 \\ + & 2 & 3 & 0 & 0 \\ \hline & 2 & 8 & 7 & 5 \\ \end{array} $$4. Write the final answer: The farmer has 2,875 apples in total.
Key Takeaway
You are now a master of three-digit multiplication! Remember to take it one step at a time, watch out for carrying, use the magic zero, and check your work with estimation. Keep practicing and you'll be a maths superhero!