Long division and division with decimals

Long division

Definitions

Long division

A multi-step process for dividing numbers, especially when the quotient isn’t a whole number.

Partial products

Intermediate results you get while multiplying in a multi-step multiplication problem.

Long division is a step-by-step process for breaking larger division problems into smaller, manageable parts. The same repeating cycle works whether you’re dealing with whole numbers, remainders, or decimals — once you learn the pattern, you can apply it to any division problem.

Divide, Multiply, Subtract, Bring Down!

Here’s what each step does:

• Divide to choose the next digit of the quotient.
• Multiply to find the amount represented by that quotient digit.
• Subtract to see what remains.
• Bring down the next digit (or a $0$ if you’re continuing into decimals) so you can repeat the cycle.

Example:

What is $13÷4$?

This is asking: “How many times does $4$ go into $13$, and what is left over?”

First, set up long division with $13$ inside the division bracket and $4$ outside.

Start by dividing:

$$\begin{array}{rl}4\times 1& =4\\ 4\times 2& =8\\ 4\times 3& =12\\ 4\times 4& =16\end{array}$$

Since $16$ is too large, use $12$. This means $4$ goes into $13$ three times. Write $3$ on top.

Multiply the quotient digit by the divisor to find how much of $13$ has been used:

$$3\times 4=12$$

Subtract to find what is left over:

$$13-12=1$$

There are no more digits to bring down. Since $4$ cannot go into $1$, the division stops with a remainder.

Our final answer:

$$13÷4=3r1$$

Steps taken:

• Divide to find how many times the divisor fits
• Multiply the quotient digit by the divisor
• Subtract to find the remainder
• Write the remainder

Knowledge check: What is $15÷4$?

(spoiler)

$$3r3$$

Long division with larger numbers

The steps stay the same when working with larger numbers, but you may need to look at more than one digit at a time. If the divisor cannot go into the first digit, use the first two digits to begin the division process. From there, continue the same cycle: divide, multiply, subtract, and bring down.

Example:

Now let’s try long division with a larger number.

What is $234÷7$?

Set up the problem with $7$ outside and $234$ inside.

• $7$ can’t go into $2$, so use $23$.
• $7$ goes into $23$ three times because $7\times 3=21$. Write $3$ on top.
• Write $21$ under $23$ and subtract:

$$23-21=2$$

Now bring down the next digit.

• Bring down the $4$ to make $24$.
• $7$ goes into $24$ three times because $7\times 3=21$. Write another $3$ on top.
• Write $21$ under $24$ and subtract:

$$24-21=3$$

There are no more digits to bring down, and $7$ can’t go into $3$, so $3$ is the remainder.

Our final answer:

$$234÷7=33r3$$

Steps taken:

• Divide to find how many times the divisor goes into the number
• Multiply the quotient digit by the divisor
• Subtract to find what remains
• Bring down the next digit
• Repeat the process until no digits remain
• Write the remainder

Knowledge check 1: What is $156÷8$?

(spoiler)

$$19r4$$

Knowledge check 2: What is $412÷6$?

(spoiler)

$$68r4$$

Long division with decimals

Sometimes you want the quotient written as a decimal instead of using a remainder. Writing the answer as a decimal lets you keep dividing until the remainder becomes $0$.

To do that, add a decimal point and zeros to the number inside the division bracket. When you add a decimal to the dividend, also place a decimal point in the quotient above. Each time you “bring down,” you can bring down another $0$. The division is complete when the remainder is $0$.

Example:

What is $5÷4?$

Start the same way you would if you were stopping with a remainder.

• $4$ goes into $5$ one time, so write $1$ on top.
• Multiply: We multiply $1\times 4=4$
• Subtract: We subtract $5-4=1$

At this point you have a remainder of $1$, but we’ll keep going by switching to decimals.

• Place a decimal in the quotient and after the $5$ (so you’re working with $5.0$).
• Bring down the $0$ to make $10$.

Now repeat the cycle:

• Divide: $4$ goes into $10$ two times.
• Multiply: $2\times 4=8$.
• Subtract: $10-8=2$.

Bring down another $0$:

• Bring down $0$ to make $20$.
• Divide: $4$ goes into $20$ five times.
• Multiply: $5\times 4=20$.
• Subtract: $20-20=0$.

A remainder of $0$ means the division is finished.

Our final answer:

$$5÷4=1.25$$

Steps taken:

• Divide to find how many times the divisor goes into the number
• Multiply the quotient digit by the divisor
• Subtract to find the remainder
• Add a decimal and bring down a zero
• Repeat the process until the remainder is $0$

Knowledge check: What is $7÷4$? Write your answer as a decimal.

(spoiler)

$$1.75$$