Column Division and Rules of Calculation

Welcome to Long Division and the Rules of Calculation!

Hello everyone! In 4th-grade math, we will learn how to divide numbers with more digits, along with handy rules that make calculations super easy. You might feel that "long division looks long and difficult...", but don't worry! Once you learn the rhythm and steps, you will be solving them smoothly, just like solving a puzzle. Let's master them together while having fun!

1. Let's Learn Math Vocabulary (Sum, Difference, Product, and Quotient)

In 4th grade, four important words appear that describe the answers to calculations. You will use these words throughout your math journey, so let's make sure to remember them!

・Sum: The answer to an addition problem (e.g., \(8\) in \(5 + 3 = 8\))
・Difference: The answer to a subtraction problem (e.g., \(2\) in \(5 - 3 = 2\))
・Product: The answer to a multiplication problem (e.g., \(15\) in \(5 \times 3 = 15\))
・Quotient: The answer to a division problem (e.g., \(2\) in \(6 \div 3 = 2\))

[Key Point]
The "answer to a division problem" is called the quotient. When a question asks you to "find the quotient," just remember: "Aha, that means I just need to find the answer to the division problem!"

2. The Basic Rhythm of Long Division

Whether you divide by a 1-digit number or a 2-digit number, the steps of long division are always the same four repeated steps. Say it out loud and remember the rhythm!

"Divide (Estimate)" \(\rightarrow\) "Multiply" \(\rightarrow\) "Subtract" \(\rightarrow\) "Bring down"

Dividing by a 1-digit number (Example: \(74 \div 3\))

1. Divide: There are two \(3\)s in the tens digit \(7\), so write a quotient of \(2\) above the tens place (Divide).
2. Multiply: Calculate \(3 \times 2 = 6\), and write \(6\) below the \(7\) (Multiply).
3. Subtract: Calculate \(7 - 6 = 1\) (Subtract).
4. Bring down: Bring down the ones digit \(4\) to make it \(14\) (Bring down).
Next, repeat the same steps for the ones place.
5. Divide: There are four \(3\)s in \(14\), so write \(4\) above the ones place (Divide).
6. Multiply: Write \(3 \times 4 = 12\).
7. Subtract: Calculate \(14 - 12 = 2\). Since there are no more numbers to bring down, the remainder is \(2\).
The answer is a quotient of \(24\) with a remainder of \(2\) (\(24\) R \(2\)).

Mastering the Check Equation

You can easily check if your division answer is correct by using the following equation:

Divisor \(\times\) Quotient \(+\) Remainder \(=\) Dividend

Let's check our previous problem \(74 \div 3 = 24\) R \(2\):
\(3 \times 24 + 2 = 72 + 2 = 74\)
It gives us back the dividend \(74\)! This tells us our calculation is completely correct.

[Common Mistake]
There is an absolute rule for the size of the remainder.
The remainder must always be smaller than the divisor.
If you calculate \(74 \div 3\) and get a "remainder of \(5\)", it means you can still subtract another \(3\), so your estimated quotient was too small! Whenever you subtract during long division, always check that the remainder is smaller than the divisor.

3. Dividing by a 2-Digit Number (The Trick of Estimating)

When the divisor has 2 digits, it might be hard to see right away what number to put in the quotient. That's when we use the trick of "estimating".

Example: \(86 \div 23\)

Think of the divisor \(23\) as a nice, round number: "about \(20\)".
Think of the dividend \(86\) as "about \(80\)".
Then you can ask yourself, "How many \(20\)s are in \(80\)?" Since \(8 \div 2 = 4\), we can estimate our quotient to be \(4\).

Let's try it out:
\(23 \times 4 = 92\)
Uh-oh! It's larger than \(86\)! We cannot subtract it.
When this happens, decrease your estimate by \(1\) to make it \(3\).
\(23 \times 3 = 69\)
Now we can subtract!
\(86 - 69 = 17\)
The remainder \(17\) is smaller than the divisor \(23\), which is perfect.
The answer is \(3\) with a remainder of \(17\) (\(3\) R \(17\)).

[Helpful Tip]
You don't need to feel discouraged at all if your estimate isn't right on the first try! Even math experts adjust their estimates by making them \(1\) bigger or smaller. Remember that "adjusting your estimate is totally normal," and tackle your long division with confidence!

Dividing a 3-Digit Dividend (Example: \(324 \div 18\))

1. First, look at the hundreds digit. There are no \(18\)s in \(3\), so nothing goes in the hundreds place of the quotient.
2. Next, look at the first two digits, \(32\). There is one \(18\) in \(32\). Write \(1\) in the tens place of the quotient.
3. Write \(18 \times 1 = 18\), and calculate \(32 - 18 = 14\) (Subtract).
4. Bring down the ones digit \(4\) to make it \(144\) (Bring down).
5. Now consider \(144 \div 18\). Thinking of \(18\) as about \(20\), you might estimate \(14 \div 2 = 7\), but let's test higher: \(18 \times 8 = 144\) fits exactly! Write \(8\) in the ones place.
6. Since \(144 - 144 = 0\), there is no remainder.
The answer is \(18\).

4. Properties of Division (Magic Tricks to Simplify Calculations)

Division has very interesting and useful properties.

[Important Rule of Division]
If you multiply or divide both the dividend and the divisor by the same number, the quotient (the answer) does not change.

For example, take \(12 \div 4 = 3\).
・Multiply both by \(2\): \((12 \times 2) \div (4 \times 2) = 24 \div 8 = 3\) (The quotient stays \(3\)!)
・Divide both by \(2\): \((12 \div 2) \div (4 \div 2) = 6 \div 2 = 3\) (The quotient stays \(3\)!)

Using This Rule to Easily Divide Large Numbers

For example, look at a calculation like \(1500 \div 500\).
If you divide both the dividend and divisor by \(100\), you can cross off two zeros from the end of each number.
\(1500 \div 500 = 15 \div 5 = 3\)
You can do it mentally in a flash!

[Common Mistake: Watch out when there's a remainder!]
When you simplify by crossing off zeros, you must be careful with the place value of the remainder.
Example: \(700 \div 200\)
Suppose you divide both numbers by \(100\) and calculate \(7 \div 2 = 3\) R \(1\).
However, the actual remainder is not \(1\)!
Recall the check equation: \(200 \times 3 + 1 = 601\), which does not bring you back to \(700\).
Because you divided both numbers by \(100\) to compute, you must multiply the remainder by \(100\) to restore it to its original size!
The correct answer is a quotient of \(3\) with a remainder of \(100\).
(Check: \(200 \times 3 + 100 = 700\), which is perfect!)

5. Order of Operations and Parentheses (　)

When an expression contains a mix of \(+\), \(-\), \(\times\), \(\div\), and parentheses (　), there is a set order to calculate them.

[Rules for Order of Operations]
1. Calculate inside the parentheses (　) first!
2. Next, calculate multiplication \(\times\) and division \(\div\)!
3. Finally, calculate addition \(+\) and subtraction \(-\)!
* When operations of the same level are side by side, calculate them from left to right.

Let's Practice the Order with Examples

Example 1: \(10 + 5 \times 4\)
Multiplication comes before addition!
Calculate \(5 \times 4 = 20\) first, then \(10 + 20 = 30\).

Example 2: \((15 - 7) \times 3\)
Inside the parentheses gets top priority!
Calculate \(15 - 7 = 8\) first, then \(8 \times 3 = 24\).

Example 3: \(40 - 12 \div (2 + 4)\)
Step 1 (Parentheses): \(2 + 4 = 6\)
Step 2 (Division): \(12 \div 6 = 2\)
Step 3 (Subtraction): \(40 - 2 = 38\)
As long as you solve step by step following the rules, you can solve even the longest expressions without mistakes!

6. Properties of Operations (Smart Tricks for Easier Calculations)

There are three very convenient properties for changing the order and grouping of calculations.

1. Commutative Property (Swapping numbers gives the same answer)

In addition and multiplication, swapping the order of the numbers does not change the answer.
・Addition: \(a + b = b + a\) (e.g., \(27 + 58 = 58 + 27\))
・Multiplication: \(a \times b = b \times a\) (e.g., \(4 \times 9 = 9 \times 4\))
* Be careful: this cannot be done with subtraction or division (for example, \(8 - 3\) is not the same as \(3 - 8\)).

2. Associative Property (Regrouping gives the same answer)

When adding or multiplying three numbers, you get the same answer whether you calculate the first pair or the last pair first.
・Addition: \((a + b) + c = a + (b + c)\)
・Multiplication: \((a \times b) \times c = a \times (b \times c)\)

Clever Trick Example: \(25 \times 36 \times 4\)
Calculating from left to right requires difficult long multiplication for \(25 \times 36\), but if you change the order of multiplication...
\(25 \times 4 \times 36 = (25 \times 4) \times 36 = 100 \times 36 = 3600\)
Look at that! You can solve it mentally without any long multiplication!
The pairs \(25 \times 4 = 100\) and \(125 \times 8 = 1000\) are best friends in math, so be sure to memorize them!

3. Distributive Property (Distributing and Factoring)

You can multiply a number outside parentheses by each number inside the parentheses.
・\((a + b) \times c = a \times c + b \times c\)
・\((a - b) \times c = a \times c - b \times c\)
You can also do the reverse: "combine terms when the same multiplication is shared."

Clever Trick Example: \(102 \times 6\)
Split \(102\) into \(100 + 2\).
\((100 + 2) \times 6 = 100 \times 6 + 2 \times 6 = 600 + 12 = 612\)

Clever Trick Example: \(74 \times 8 + 26 \times 8\)
Both parts share "\(\times 8\)", so group them together inside parentheses.
\((74 + 26) \times 8 = 100 \times 8 = 800\)
This is much simpler than doing two separate long multiplications and helps prevent calculation mistakes!

7. Writing Equations Using a Box (□)

Using □ or △ to represent an unknown quantity and writing an equation directly from a word problem is another key skill in 4th grade.

Problem: You bought \(4\) bottles of juice that each cost the same price, and a snack that cost \(100\) yen. The total cost was \(660\) yen. How much does one bottle of juice cost?

Let the price of one bottle of juice be □ yen, and write the story directly as an equation:
\(□ \times 4 + 100 = 660\)

How to solve for □:
1. The cost of 4 bottles of juice is the total minus the cost of the snack:
\(□ \times 4 = 660 - 100 = 560\)
2. The cost of 1 bottle of juice is found by dividing that amount by \(4\):
\(□ = 560 \div 4 = 140\)
Answer: One bottle of juice is \(140\) yen.

By using □ for the unknown value and writing out the equation just as the story unfolds, you can organize your thoughts clearly and find the answer by working backwards.

Summary (Key Review Points!)

・The answer to a division problem is the quotient. Compute with the rhythm of "Divide \(\rightarrow\) Multiply \(\rightarrow\) Subtract \(\rightarrow\) Bring down"!
・The check equation is: "Divisor \(\times\) Quotient \(+\) Remainder \(=\) Dividend".
・The remainder is always smaller than the divisor!
・Dividing both the dividend and the divisor by the same number keeps the quotient the same (but be careful with the place value of the remainder).
・Follow the order of operations: Inside parentheses (　) \(\rightarrow\) Multiplication & Division \(\rightarrow\) Addition & Subtraction.
・Use handy pairs like \(25 \times 4 = 100\) and the distributive property to make your calculations a breeze!