Learn on PengiPengi Math (Grade 5)Chapter 2: Multi-Digit Multiplication and Division with Place Value

Lesson 6: Applying Division Algorithms and Interpreting Remainders

In this Grade 5 lesson from Pengi Math's Chapter 2, students apply standard division algorithms to solve multi-digit division problems and use estimation to select reasonable quotients. They also practice interpreting remainders in context, learning how the meaning of a remainder changes depending on the real-world situation presented in a problem.

Section 1

Standard Algorithm for Division: No Remainder

Property

When a dividend is perfectly divisible by a divisor, the result is a whole number quotient with a remainder of zero. This relationship can be expressed as:

Dividend÷Divisor=QuotientDividend \div Divisor = Quotient

or

Dividend=Divisor×QuotientDividend = Divisor \times Quotient

Examples

  • 378÷14=27378 \div 14 = 27
      2714)378‾   28‾        98     98‾        0\begin{array}{r} \ \ \ \ \ \ 27 \\ 14 \overline{)378} \\ \ \ \ \underline{28}\ \ \\ \ \ \ \ \ \ 98 \\ \ \ \ \ \ \underline{98} \\ \ \ \ \ \ \ \ \ 0 \\ \end{array}
  • 2,125÷25=852,125 \div 25 = 85
      8525)2125‾   200‾       125     125‾        0\begin{array}{r} \ \ \ \ \ \ 85 \\ 25 \overline{)2125} \\ \ \ \ \underline{200}\ \ \\ \ \ \ \ \ 125 \\ \ \ \ \ \ \underline{125} \\ \ \ \ \ \ \ \ \ 0 \\ \end{array}

Explanation

The standard algorithm for division is a systematic method for dividing multi-digit numbers. The process involves a repeating cycle of steps: divide, multiply, subtract, and bring down the next digit from the dividend. You repeat these steps until all digits of the dividend have been used and the final remainder is zero. This algorithm is an efficient way to solve division problems with large numbers.

Section 2

Standard Algorithm for Division: With Remainder

Property

When a dividend is not perfectly divisible by a divisor, the result is a quotient and a remainder. The relationship is expressed as:

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

Where the remainder rr must be greater than or equal to zero and less than the divisor bb: 0≤r<b0 \leq r < b.

Examples

  • 89÷5=1789 \div 5 = 17 with a remainder of 44. This can be written as 17 R 417 \text{ R } 4.
   175)89‾  5‾     39  35‾     4\begin{array}{r} \ \ \ 17 \\ 5 \overline{)89} \\ \ \ \underline{5}\ \ \\ \ \ \ 39 \\ \ \ \underline{35} \\ \ \ \ \ \ 4 \\ \end{array}
  • 457÷12=38457 \div 12 = 38 with a remainder of 11. This can be written as 38 R 138 \text{ R } 1.
    3812)457‾   36‾      97   96‾      1\begin{array}{r} \ \ \ \ 38 \\ 12 \overline{)457} \\ \ \ \ \underline{36}\ \ \\ \ \ \ \ 97 \\ \ \ \ \underline{96} \\ \ \ \ \ \ \ 1 \\ \end{array}
  • 5,843÷25=2335,843 \div 25 = 233 with a remainder of 1818. This can be written as 233 R 18233 \text{ R } 18.
    23325)5843‾   50‾       84     75‾       93    75‾     18\begin{array}{r} \ \ \ \ 233 \\ 25 \overline{)5843} \\ \ \ \ \underline{50}\ \ \ \ \\ \ \ \ 84 \ \ \\ \ \ \ \underline{75}\ \ \\ \ \ \ \ \ 93 \\ \ \ \ \ \underline{75} \\ \ \ \ \ \ 18 \\ \end{array}

Explanation

The standard algorithm for division is a step-by-step process used to divide multi-digit numbers. When the divisor cannot divide the dividend evenly, the amount left over is called the remainder. The remainder is always a whole number that is smaller than the divisor. This skill extends the standard division algorithm to problems that do not result in a whole number quotient.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 2: Multi-Digit Multiplication and Division with Place Value

  1. Lesson 1

    Lesson 1: Estimating Products Using Rounding and Compatible Numbers

    LearnPractice
  2. Lesson 2

    Lesson 2: Modeling Multiplication with Area Models and Partial Products

    LearnPractice
  3. Lesson 3

    Lesson 3: Connecting Partial Products to the Standard Algorithm

    LearnPractice
  4. Lesson 4

    Lesson 4: Understanding Division as Finding a Missing Factor

    LearnPractice
  5. Lesson 5

    Lesson 5: Dividing Multi-Digit Numbers Using Area Models and Partial Quotients

    LearnPractice
  6. Lesson 6Current

    Lesson 6: Applying Division Algorithms and Interpreting Remainders

    LearnPractice

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Standard Algorithm for Division: No Remainder

Property

When a dividend is perfectly divisible by a divisor, the result is a whole number quotient with a remainder of zero. This relationship can be expressed as:

Dividend÷Divisor=QuotientDividend \div Divisor = Quotient

or

Dividend=Divisor×QuotientDividend = Divisor \times Quotient

Examples

  • 378÷14=27378 \div 14 = 27
      2714)378‾   28‾        98     98‾        0\begin{array}{r} \ \ \ \ \ \ 27 \\ 14 \overline{)378} \\ \ \ \ \underline{28}\ \ \\ \ \ \ \ \ \ 98 \\ \ \ \ \ \ \underline{98} \\ \ \ \ \ \ \ \ \ 0 \\ \end{array}
  • 2,125÷25=852,125 \div 25 = 85
      8525)2125‾   200‾       125     125‾        0\begin{array}{r} \ \ \ \ \ \ 85 \\ 25 \overline{)2125} \\ \ \ \ \underline{200}\ \ \\ \ \ \ \ \ 125 \\ \ \ \ \ \ \underline{125} \\ \ \ \ \ \ \ \ \ 0 \\ \end{array}

Explanation

The standard algorithm for division is a systematic method for dividing multi-digit numbers. The process involves a repeating cycle of steps: divide, multiply, subtract, and bring down the next digit from the dividend. You repeat these steps until all digits of the dividend have been used and the final remainder is zero. This algorithm is an efficient way to solve division problems with large numbers.

Section 2

Standard Algorithm for Division: With Remainder

Property

When a dividend is not perfectly divisible by a divisor, the result is a quotient and a remainder. The relationship is expressed as:

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

Where the remainder rr must be greater than or equal to zero and less than the divisor bb: 0≤r<b0 \leq r < b.

Examples

  • 89÷5=1789 \div 5 = 17 with a remainder of 44. This can be written as 17 R 417 \text{ R } 4.
   175)89‾  5‾     39  35‾     4\begin{array}{r} \ \ \ 17 \\ 5 \overline{)89} \\ \ \ \underline{5}\ \ \\ \ \ \ 39 \\ \ \ \underline{35} \\ \ \ \ \ \ 4 \\ \end{array}
  • 457÷12=38457 \div 12 = 38 with a remainder of 11. This can be written as 38 R 138 \text{ R } 1.
    3812)457‾   36‾      97   96‾      1\begin{array}{r} \ \ \ \ 38 \\ 12 \overline{)457} \\ \ \ \ \underline{36}\ \ \\ \ \ \ \ 97 \\ \ \ \ \underline{96} \\ \ \ \ \ \ \ 1 \\ \end{array}
  • 5,843÷25=2335,843 \div 25 = 233 with a remainder of 1818. This can be written as 233 R 18233 \text{ R } 18.
    23325)5843‾   50‾       84     75‾       93    75‾     18\begin{array}{r} \ \ \ \ 233 \\ 25 \overline{)5843} \\ \ \ \ \underline{50}\ \ \ \ \\ \ \ \ 84 \ \ \\ \ \ \ \underline{75}\ \ \\ \ \ \ \ \ 93 \\ \ \ \ \ \underline{75} \\ \ \ \ \ \ 18 \\ \end{array}

Explanation

The standard algorithm for division is a step-by-step process used to divide multi-digit numbers. When the divisor cannot divide the dividend evenly, the amount left over is called the remainder. The remainder is always a whole number that is smaller than the divisor. This skill extends the standard division algorithm to problems that do not result in a whole number quotient.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 2: Multi-Digit Multiplication and Division with Place Value

  1. Lesson 1

    Lesson 1: Estimating Products Using Rounding and Compatible Numbers

    LearnPractice
  2. Lesson 2

    Lesson 2: Modeling Multiplication with Area Models and Partial Products

    LearnPractice
  3. Lesson 3

    Lesson 3: Connecting Partial Products to the Standard Algorithm

    LearnPractice
  4. Lesson 4

    Lesson 4: Understanding Division as Finding a Missing Factor

    LearnPractice
  5. Lesson 5

    Lesson 5: Dividing Multi-Digit Numbers Using Area Models and Partial Quotients

    LearnPractice
  6. Lesson 6Current

    Lesson 6: Applying Division Algorithms and Interpreting Remainders

    LearnPractice