Long Division with Decimals
Solve Division Problems with Step-by-Step Work, Decimals, and Remainders
Long Division with Decimals
Dividing numbers is straightforward when you are dealing with whole integers, but introducing a decimal point can often make the process feel intimidating. Whether you are balancing a budget, converting units, or helping a student with homework, understanding how to handle decimals in long division is a critical math skill.
Terminology
- The Dividend: This is the number you want to divide up. It sits inside the division bracket (or “house”).
- The Divisor: This is the number you are dividing by. It sits outside the bracket.
- The Quotient: This is your answer. It sits on top of the bracket.
The Golden Rule: Eliminate the Decimal in the Divisor
The single most important rule in decimal division is this: Never divide by a decimal.
If your divisor (the outside number) is not a whole number, you cannot easily proceed with standard long division. You must transform it into an integer first. Here is the standard “Slide Method” used by mathematicians:
- Shift the Divisor: Move the decimal point in the divisor to the right until it becomes a whole number. Count how many “jumps” you made.
- Match the Dividend: Move the decimal point in the dividend (the inside number) to the right by the exact same number of jumps.
- Place the Decimal: Immediately place a decimal point in the quotient (answer line) directly above its new position in the dividend.
Example: If you are calculating 15.6 ÷ 0.3:
- Move the decimal in 0.3 one spot to the right to make it 3.
- Move the decimal in 15.6 one spot to the right to make it 156.
- The problem is now simply 156 ÷ 3.
Step-by-Step
1. Divide Look at the first digit (or first two digits) of the dividend. Determine how many times the divisor fits into that number without going over. Write that digit on the quotient line.
2. Multiply Multiply the digit you just wrote on top by the divisor. Write the result underneath the portion of the dividend you are currently working on.
3. Subtract Subtract the result of your multiplication from the dividend digits. This gives you a remainder. Note: This remainder must always be smaller than your divisor.
4. Bring Down Bring down the next digit from the dividend and place it next to your remainder. This forms a new number to divide into.
5. Repeat Start the cycle over from step 1 with your new number. Continue until you have no digits left to bring down or you reach a remainder of zero.
Handling Remainders and Precision
In the world of decimals, we rarely use “R” for Remainder (e.g., “5 R 2”). Instead, we continue the division to achieve precision.
- Adding Zeros: If you run out of digits in the dividend but still have a remainder, simply add a zero to the end of the dividend and bring it down. You can continue doing this to reach as many decimal places as required.
- Rounding: Sometimes, a division problem results in a repeating decimal (like 0.333…). In these cases, you typically calculate one digit beyond your required precision and then round up or down.
Applications
Why does this matter outside the classroom?
- Finance: Splitting a dinner bill of $85.50 among 4 people requires decimal division.
- Construction: If you have a 12.5-meter plank and need to cut it into 0.75-meter sections, you need this calculation to know how many pieces you will get.
- Unit Conversion: Converting grams to kilograms or centimeters to inches often involves dividing by decimal factors.
FAQs
Q1. What if the dividend is a whole number but the divisor is a decimal?
A: You still move the decimal on the divisor. For the whole number dividend, imagine a decimal point at the end of it. Move that invisible decimal to the right, filling the empty “jumps” with zeros. For example, 10 ÷ 0.5 becomes 100 ÷ 5.
Q2. How do I know where to put the decimal in the answer?
A: This is the most common error source. Always place the decimal in the quotient line before you start dividing numbers. It goes directly above the decimal’s final position in the dividend. Once placed, you can ignore it and divide as if you were dealing with whole numbers.
Q3. Why does dividing by a decimal strictly less than 1 result in a larger number?
A: It can feel counterintuitive that 10 ÷ 0.5 = 20. However, division asks “How many times does X fit into Y?” Since 0.5 is a half, it fits into 10 twice as many times as 1 would. Therefore, dividing by a small fraction increases the total count.
Sources: Khan Academy, Calculator Soup, DreamBox Learning, IXL Learning, LibreTexts (Mathematics), Mad for Math, AllMath, Symbolab, Free Online Calculator Use, Study.com.