Zero in the quotient with a remainder
Zero in the quotient with a remainder is a Grade 4 long division skill in Saxon Math Intermediate 4 Chapter 8. When a brought-down digit is smaller than the divisor, students must write a 0 in the quotient as a placeholder before continuing. For example, dividing 417 by 4: after the first division step, bringing down 1 gives a value of 1, which is less than 4, so 0 is written in the quotient. Then 7 is brought down to make 17, which divides to give 4. The result is 104 remainder 1. Skipping the zero produces the wrong answer of 14 instead of 104.
Key Concepts
Property When you bring down a digit and the new number is smaller than the divisor, you must write a zero in the quotient. The process then continues, often resulting in a remainder.
Example 1: In $3)\overline{121}$, after dividing $12$ by $3$ to get $4$, bring down the $1$. Since $1$ is less than $3$, write $0$ in the quotient. The final answer is $40 \text{ R } 1$. Example 2: In $8)\overline{241}$, after dividing $24$ by $8$ to get $3$, bring down the $1$. Since $1$ is less than $8$, write $0$ in the quotient. The final answer is $30 \text{ R } 1$.
What happens when you bring down a digit, but it's too small to be divided by the divisor? Don't panic! You simply write a big '0' in that spot in the quotient. This zero holds the place value correctly. You then multiply the divisor by zero, subtract, and the number you couldn't divide becomes your final remainder. Easy peasy!
Common Questions
When must I write a zero in the quotient during long division?
Write a zero in the quotient whenever you bring down a digit and the new number is still smaller than the divisor.
How do I solve 417 divided by 4 with a zero in the quotient?
Divide 4 by 4 equals 1. Bring down 1 to get 1—too small for 4, so write 0. Bring down 7 to make 17. Divide 17 by 4 equals 4 remainder 1. Answer: 104 R 1.
Why does skipping the zero placeholder cause a wrong answer?
Without the zero, a three-digit quotient like 104 becomes 14, which is 90 less than the correct value. The zero holds the tens place.
How do examples like 3 divided into 121 use zero in the quotient?
After dividing 12 by 3 to get 4, bring down 1. Since 1 is less than 3, write 0 in the quotient. The final answer is 40 remainder 1.
Does the remainder follow the same rules when a zero appears in the quotient?
Yes. The remainder is whatever is left after the final subtraction, regardless of whether a zero appeared earlier in the quotient.