Division and Fraction Bars (Vinculum)

When we study the order of mathematical operations we come across the terms division bar and fraction bar, but what do they mean and why are they so special?

First of all, we must remember that the fraction bar—or vinculum—is exactly the same as a division. $10:2$ is the same as ${\ {10 \over 2}}$ and ${\ {10/ 2}}$ .

Two things to remember:

You cannot divide by $0$ . To prove this, let's look at the following example: ${\ {3:0=}}$ .
To solve this, we must be able to do the following: ${\ {0 \cdot ?=3}}$ . However, since there is no number that can be multiplied by $0$ to give the result $3$ , there is also therefore no number that can be divided by $0$ .
When we have a fraction bar, it is as if there are parentheses in the numerator. We solve the numerator first and then continue with the exercise. For example:

${\ {{10-2 \over 2}= {8 \over 2} = 4}}$

Detailed explanation

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Test yourself on special cases (0 and 1, inverse, fraction line)!

$(3\times5-15\times1)+3-2=$

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If you liked this article, you may also be interested in the following:

Fractions

How to reduce fractions?

Division of powers of equal base

The Distributive Property in the Case of Divisions

Division of integers between parentheses in which there is a multiplication

Division of integers between parentheses in which there is a division

In the Tutorela blog, you can find a wide variety of helpful mathematics articles!

Exerces with Divisions and Fraction lines

Write the following expressions with fraction bars and solve them:

$(17-7):(55-20)=\frac{10}{35}$

$(9+7):(24+7)=\frac{16}{31}$

$(6+1):(X\times7)=\frac{7}{7X}$

$(2:6):(49:7)=\frac{\frac{1}{3}}{7}$

$(8\times X):(22-8)=\frac{8X}{14}$

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Test your knowledge

Question 1

$(5\times4-10\times2)\times(3-5)=$

Question 2

$(5+4-3)^2:(5\times2-10\times1)=$

Question 3

$8\times(5\times1)=$

Solve the following exercises that include parentheses:

$\ {{20-5} \over {7+3}}=$
$\ 18:3=$
$\ 11-3:4=$
$\ (85+5):10=$
$\ 11:2+4{1 \over 2}=$
$\ 0.5-0.1:0.2=$
$\ {18 \over {18+36}}=$
$\ {{0.18+0.37} \over {99+13-{2 \over 1}}}=$

Division and Fraction Bar Exercises

Exercise 1

Task:

Solve the following equation:

$[(3-2+4)^2-2^2]:\frac{(\sqrt{9}\cdot7)}{3}=$

Solution:

In the first step, we solve the brackets starting with the addition and subtraction operations inside the inner parentheses, followed by the powers.

$[5^2-4]:\frac{(\sqrt{9}\cdot7)}{3}=$

In the second step, we solve the root in the additional parenthesis in the fraction.

$[5^2-4]:\frac{(3\cdot7)}{3}=$

Continue to solve according to the order of operations.

$[25-4]:\frac{21}{3}=$

$21:7=3$

Answer:

$3$

Do you know what the answer is?

Question 1

$7\times1+\frac{1}{2}=$

Question 2

$\frac{6}{3}\times1=$

Question 3

$12+3\times0=$

Exercise 2

Task:

Solve the following equation:

$\frac{(44-3\cdot0)}{11}:4-\frac{3\cdot4+5}{17}=$

Solution:

First we solve the parentheses in the first fraction. After that, we solve the equation in the second fraction using the order of arithmetic operations.

$\frac{44}{11}:4-\frac{12+5}{17}=$

We continue solving:

$4:4-\frac{17}{17}=$

$1-1=0$

Answer:

$0$

Exercise 3

Question:

Solve the following equation:

$\frac{7+8-3}{2}:3+4=$

Solution:

We start solving the equation according to the order of operations.

$\frac{12}{2}:3+4=$

$6:3+4=$

$2+4=6$

Answer:

$6$

Check your understanding

Question 1

$$ 2+0:3= $$

Question 2

$$ 19+1-0= $$

Question 3

$$ 9-0+0.5= $$

Exercise 4

Question:

Solve the following equation:

$\frac{36-(4\cdot5)}{8}-3\cdot2=$

Solution:

We begin by solving the parentheses that appear in the fraction.

$\frac{36-20}{8}-3\cdot2=$

Then we continue solving according to the order of the operations.

$\frac{16}{8}-6=$

$2-6=-4$

Answer:

$-4$

Exercise 5

Question:

Calculate the correct answer.

$\frac{25+3-2}{13}+5\cdot4=$

Solution:

We solve the equation according to the order of the operations.

$\frac{26}{13}+5\cdot4=$

We then perform the division operation of the fraction before the multiplication.

$2+20=22$

Answer:

$22$

Do you think you will be able to solve it?

Question 1

$\frac{1}{2}+0+\frac{1}{2}=$ ?

Question 2

$$ 0+0.2+0.6= $$

Question 3

$$ 12+1+0= $$

Examples with solutions for Division and Fraction Bars (Vinculum)

Exercise #1

$8\times(5\times1)=$

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the expression in parentheses:

$5\times1=5$

Now we multiply:

$8\times5=40$

Answer

Exercise #2

$7\times1+\frac{1}{2}=$

Video Solution

Step-by-Step Solution

According to the order of operations rules, we first insert the multiplication exercise into parentheses:

$(7\times1)+\frac{1}{2}=$

Let's solve the exercise inside the parentheses:

$7\times1=7$

And now we get the exercise:

$7+\frac{1}{2}=7\frac{1}{2}$

Answer

$7\frac{1}{2}$

Exercise #3

$\frac{6}{3}\times1=$

Video Solution

Step-by-Step Solution

According to the order of operations rules, we will solve the exercise from left to right, since there are only multiplication and division operations:

$\frac{6}{3}=2$

$2\times1=2$

Answer

$2$

Exercise #4

$(3\times5-15\times1)+3-2=$

Video Solution

Step-by-Step Solution

This simple rule is the order of operations which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations enclosed in parentheses precede all others,

Following the simple rule, multiplication comes before division and subtraction, therefore we calculate the values of the multiplications and then proceed with the operations of division and subtraction

$3\cdot5-15\cdot1+3-2= \\ 15-15+3-2= \\ 1$ Therefore, the correct answer is answer B.

Answer

$1$

Exercise #5

$(5\times4-10\times2)\times(3-5)=$

Video Solution

Step-by-Step Solution

This simple rule is the order of operations which states that multiplication precedes addition and subtraction, and division precedes all of them,

In the given example, a multiplication occurs between two sets of parentheses, thus we simplify the expressions within each pair of parentheses separately,

We start with simplifying the expression within the parentheses on the left, this is done in accordance with the order of operations mentioned above, meaning that multiplication comes before subtraction, we perform the multiplications in this expression first and then proceed with the subtraction operations within it, in reverse we simplify the expression within the parentheses on the right and perform the subtraction operation within them:

What remains for us is to perform the last multiplication that was deferred, it is the multiplication that occurred between the expressions within the parentheses in the original expression, we perform it while remembering that multiplying any number by 0 will result in 0:

Therefore, the correct answer is answer d.

Answer

$0$

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