Lesson 68: Division with Two-Digit Answers and a Remainder

New Concept

The pencil-and-paper method we use for dividing has four steps: divide, multiply, subtract, and bring down.

Why it matters

Long division is your first encounter with a powerful mathematical tool called an algorithm, a step-by-step procedure for calculations. Mastering this systematic process of breaking down problems is essential for tackling more complex topics like fractions, ratios, and algebraic equations.

What’s next

Next, you'll apply this four-step cycle to solve division problems that result in two-digit answers and a remainder.

Property: The method for dividing has four steps that are repeated until the division is complete: 1. Divide, 2. Multiply, 3. Subtract, and 4. Bring down. After the last subtraction, the number left over is the remainder, which is written with an uppercase 'R' in front of it.

For $6\overline{)147}$: First, divide 14 by 6 to get 2. Multiply $2 \times 6 = 12$. Subtract $14 - 12 = 2$. Bring down 7 to make 27. Then, divide 27 by 6 to get 4. Multiply $4 \times 6 = 24$. Subtract $27 - 24 = 3$. The answer is 24 R 3.
For $97 \div 4$: First, divide 9 by 4 to get 2. Multiply $2 \times 4 = 8$. Subtract $9 - 8 = 1$. Bring down 7 to make 17. Then, divide 17 by 4 to get 4. Multiply $4 \times 4 = 16$. Subtract $17-16=1$. The answer is 24 R 1.

Think of long division as a repeating four-step dance routine for numbers. First, you divide a part of the dividend. Then, you multiply that result back. Next, you subtract to find the leftover amount. Finally, you bring down the next digit to start the dance all over again until you run out of partners to dance with.

Property: Why do we write the first digit of the quotient in the tens place? Dividing 13 tens by 5 is 2 tens with 3 tens left over. We write the 2 in the tens place of the quotient. The placement of the quotient digit is determined by the place value of the digits being divided.

In $5\overline{)137}$, we first divide 13 tens by 5. The answer, 2, represents 2 tens and is placed in the tens spot above the 3.
In $7\overline{)240}$, we first divide 24 tens by 7. The answer, 3, represents 3 tens and is placed in the tens spot above the 4.
In $8\overline{)259}$, we first divide 25 tens by 8. The answer, 3, represents 3 tens and is placed in the tens spot above the 5.

Don't just throw numbers anywhere! When you divide the tens part of your big number, your answer is also in tens. So, the first digit of your quotient must live in the tens place, right above the last digit you used from the dividend. It’s all about keeping the place values lined up correctly and not misplacing your digits.

Property: To check a division answer that has a remainder, we multiply the quotient (without the remainder) by the divisor and then add the remainder. The result should be the original number you started with, also known as the dividend. This confirms your calculation is correct. For example: (quotient $\times$ divisor) + remainder = dividend.

To check $137 \div 5 = 27 \text{ R } 2$, we calculate $(27 \times 5) + 2 = 135 + 2 = 137$. It matches!
To check $240 \div 7 = 34 \text{ R } 2$, we calculate $(34 \times 7) + 2 = 238 + 2 = 240$. It matches!
To check $95 \div 4 = 23 \text{ R } 3$, we calculate $(23 \times 4) + 3 = 92 + 3 = 95$. It matches!

How do you know you got it right? It's like reverse engineering your math! Multiply your main answer (the quotient) by the number you divided by (the divisor), then add the little leftover part (the remainder). If you get back to the number you started with, you've nailed it! It is the ultimate proof of your division skills.

Property: To find an approximate answer for a division problem, you can replace the numbers with 'compatible numbers' that are close to the original numbers but much easier to divide in your head. This method helps you get a reasonable estimate quickly without performing the full calculation. This is particularly useful for checking if your final answer is sensible.

To estimate $375 \div 8$, you can round 375 to a nearby multiple of 8, like 400. Then, $400 \div 8 = 50$.
To estimate $175 \div 9$, you can round 175 to a nearby multiple of 9, like 180. Then, $180 \div 9 = 20$.
To estimate $259 \div 8$, you can round 259 to a nearby multiple of 8, like 240. Then, $240 \div 8 = 30$.

Why wrestle with tricky numbers when you can use friendly ones that get along? For a problem like $375 \div 8$, find a nearby number that 8 loves to divide, like 400. This mental shortcut gives you a quick, 'about right' answer without the headache of long division. It's the lazy genius way to check your work.