ASVAB Study Guide
Basic Math · Lesson 3 of 6

Long Division

Key rules and formulas, worked examples and 12 practice problems with step-by-step solutions.

1. Divide by allowed components

Basic Terminology

$$a:b=c \; (\mathrm{remainder} \; r)$$

(a) Dividend, (b) Divisor ,(c) Quotient, (d) Remainder

Example

In $$17 \div 5 = 3 \; (\mathrm{remainder} \; 2)$$, identify the components.

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$$a = 17$$ (Dividend)

$$b = 5$$ (Divisor)

$$c = 3$$ (Quotient)

$$r = 2$$ (Remainder)

Practice problems

  1. Problem 1

    In $$25 \div 4 = 6 \; (\mathrm{remainder} \; 1)$$, what is the Quotient?

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    Answer: 6

    The result of the division is 6 (the Quotient), and 1 is the Remainder.

  2. Problem 2

    If Dividend = 47, Divisor = 5, find Quotient and Remainder.

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    Answer: Quotient = 9, Remainder = 2

    $$47 = 5 \times 9 + 2$$. Quotient = 9, Remainder = 2.

  3. Problem 3

    Find the Dividend if Divisor = 12, Quotient = 8, and Remainder = 5.

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    Answer: 101

    Dividend = Divisor × Quotient + Remainder = $$12 \times 8 + 5 = 96 + 5 = 101$$

Remember

Always identify dividend, divisor, quotient, and remainder clearly. Ensure remainder r < divisor b.

2. Vertical division

Steps to Follow

Divide → Multiply → Subtract → Bring down

Repeat the process for each digit from left to right.

Example

Perform vertical division: $$\frac{432}{3}$$

Show solution

1. Divide: $$4 \div 3 = 1$$ (rem 1)

2. Bring down 3 -> $$13 \div 3 = 4$$ (rem 1)

3. Bring down 2 -> $$12 \div 3 = 4$$ (rem 0)

Result = 144

Practice problems

  1. Problem 4

    What is the second step in vertical division?

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    Answer: Multiply

    The standard order of steps is: Divide -> Multiply -> Subtract -> Bring down.

  2. Problem 5

    Calculate $$\frac{845}{5}$$ using vertical division steps.

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    Answer: 169

    $$8 \div 5 = 1$$ (rem 3); $$34 \div 5 = 6$$ (rem 4); $$45 \div 5 = 9$$ (rem 0). Result = 169.

  3. Problem 6

    Calculate $$\frac{1{,}452}{12}$$ using vertical division steps.

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    Answer: 121

    $$14 \div 12 = 1$$ (rem 2); $$25 \div 12 = 2$$ (rem 1); $$12 \div 12 = 1$$ (rem 0). Result = 121.

Remember

Ensure each subtraction gives a remainder smaller than the divisor before bringing down the next digit.

3. Divide the dividend less than the divisor

Dividing a Smaller Number by a Larger Number

a<b→$$c=0$$,...

Write 0 and a decimal point in the quotient, add 0 to the dividend to continue.

Example

Calculate: $$\frac{3}{4}$$

Show solution

1. 3 < 4, so write '0.' in quotient.

2. Add 0 to dividend -> 30.

3. $$30 \div 4 = 7$$ (rem 2).

4. Add 0 -> $$\frac{20}{4} = 5$$. Result = 0.75.

Practice problems

  1. Problem 7

    Calculate $$\frac{1}{2}$$ using decimal division.

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    Answer: 0.5

    1 < 2, write 0., then $$\frac{10}{2} = 5$$-> Result = 0.5.

  2. Problem 8

    Calculate $$\frac{3}{8}$$

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    Answer: 0.375

    3 < 8 -> 0.3 (rem 6) -> $$60 \div 8 = 7$$ (rem 4) -> $$\frac{40}{8} = 5$$-> Result = 0.375.

  3. Problem 9

    Calculate $$\frac{2}{5} + \frac{1}{4}$$ as decimal value.

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    Answer: 0.65

    $$\frac{2}{5} = 0.4$$ and $$\frac{1}{4} = 0.25$$-> $$0.4 + 0.25 = 0.65$$

Remember

When dividend < divisor, the integer part of the quotient is 0. Add decimal point to continue dividing.

4. Dividing Decimals

Handling Decimal Points

Move decimal point in divisor & dividend by the same number of places

Move the decimal point in the divisor to the right to make it a whole number, then move the decimal point in the dividend by the same number of places.

Example

Calculate: $$\frac{12.5}{0.5}$$

Show solution

Step 1: Shift decimal point 1 place right for both numbers: 12.5 -> 125, and 0.5 -> 5.

Step 2: Perform standard integer division: $$\frac{125}{5}$$.

Step 3: Calculate: $$\frac{125}{5} = 25$$

Practice problems

  1. Problem 10

    Calculate the value of: $$\frac{4.8}{0.6}$$

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    Answer: 8

    Step 1: Shift decimal point 1 place right for both numbers: 4.8 -> 48 and 0.6 -> 6.

    Step 2: Divide as whole numbers: $$\frac{48}{6}$$.

    Step 3: Result = 8.

  2. Problem 11

    Calculate the value of: $$\frac{1.44}{0.12}$$

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    Answer: 12

    Step 1: Shift decimal point 2 places right for both numbers: 1.44 -> 144 and 0.12 -> 12.

    Step 2: Divide as whole numbers: $$\frac{144}{12}$$.

    Step 3: Result = 12.

  3. Problem 12

    Calculate the value of: $$\frac{7.5}{0.25}$$

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    Answer: 30

    Step 1: Shift decimal point 2 places right: 0.25 -> 25, and 7.5 -> 750 (add a zero).

    Step 2: Divide as whole numbers: $$\frac{750}{25}$$.

    Step 3: Result = 30.

Remember

Always shift the decimal point in BOTH the divisor and dividend by the exact same number of places before performing long division