# Division property of equality

Jane has two cakes of equal sizes. She has 8 guests at the party. She decides to divide both the cakes evenly throughout. How will she be able to do this?

Thus dividing both the cakes evenly to get the balance while using them is the division property of equality. Let's learn more about it in detail.

In this mini lesson let us learn about the division property of equality formula, division property of equality in geometry, division property of equality with fractions, division property of equality proof, division property of equality calculator and division property of equality angles.

**Lesson Plan**

**What Is Division Property of Equality?**

If \(a\), \(b\), \(c\) are the real numbers, such that, then the division property of equality formula is given by:

If \(a = b\) and \(c \neq 0\), then \(\dfrac{a}{c} = \dfrac{b}{c}\) |

Consider the equation \(18 = 18\)

Divide both sides by 2

\[\begin{align}\dfrac{18}{2} &= \dfrac{18}{2}\\ 9 &= 9\\\text{LHS} &= \text{RHS}\end{align}\]

The equation still remains balanced.

**Definition**

**Division property states that when we divide one side of an equation by a number, we should divide the other side of the equation by the same number so that the equation remains balanced.**

**Division Property of Equality Proof**

The following picture illustrates the division property of equality in Algebra in solving linear equations.

If \(5x = 25\), then \(x = 5\) on dividing by 5 on both sides.

Thus, we apply the division property of equality formula here to solve for \(x\)

You can verify if \(x = 5\) is the solution of the given equation by substituting \(x = 5\) in the equation.

\[\begin{align}\text{LHS} &=5 x\\ &= 5\times x\\ &= 5\times 5\\ &= 25\\ &= \text{RHS}\end{align}\]

Thus we obtain the division property of equality proof.

**Formula of Division Property of Equality**

The formula of division property of equality is given as:

If \(a = b\) and \(c = d\), then \(\dfrac{a}{c}= \dfrac{b}{d}\) |

**Example**

1. To find the value of the unknown variable, we use this division property along with the other properties of equality.

Consider solving this linear equation, \(3x - 2 = 7\)

\[\begin{align*}3x - 2 + 2 &= 7 + 2 \text{ (addition property)}\\\\ 3x &= 9\\\\\dfrac{3x}{3} &= \dfrac{9}{3}\,\text{(division proerty})\\\\x &= 3\end{align*}\]

2. Consider equation \(2x = 1\)

To solve for \(x\), divide both sides by 2.

\[\begin{align}\dfrac{2x}{2} &= \dfrac{1}{2}\,\text{(division property)}\\\\ x &= \dfrac{1}{2}\end{align}\]

**Division Property of Equality Calculator**

In the following simulation, enter the equation, try to solve the equations and check your answers with the step-by-step procedure in solving the equation.

- When the equations to be solved are in the form of multiplication, we need to use the division property of equality. \(10 x = 100\), here is the hint for us to use the division property. Divide both sides by 10.

**Division Property of Equality in Geometry**

Congruent angles have equal measures.

1. Division Property of Equality in Geometry is used along with the properties of congruence.

Given are two congruent angles. We need to find the measures of the angles.

\(5x - 8\) and \(3x + 4\) are the two congruent angles.

\(\therefore 5x-8 = 3x +4\)

We need to solve for \(x\) and then the angles by using the division property of equality angles.

Thus we find that both the angles are equal to \(22^\circ\)

2. Division Property of Equality Angles is used to find the unknown angles.

\(\angle AMN + \angle BMN = 180^\circ\) \(\because\) the angles form a linear pair and are supplementary.

\[\begin{align}x + 2x &= 180\\\\3x &= 180 (\text{division property)}\\\\ x &= 60\end{align}\]

**Division Property of Equality with Fractions**

\[\begin{align}10\text{v} &= 2\\\\\dfrac{10 \text{ v}}{10} &= \dfrac{2}{10}\text{(divide both sides by 10)}\\\\\text{v} &= \dfrac{1}{5}\end{align}\]

You can verify if \(\text{ v} = \dfrac{1}{5}\) by substituting in the equation.

\[\begin{align}\text{LHS} &= 10 \text{v}\\\\ &= 10\times \dfrac{1}{5}\\ &= 2\\\\ &=\text{RHS} \end{align}\]

Thus verified.

**What Are the 8 Properties of Equality?**

The following are the 8 properties of equality:

Property | Application |
---|---|

Substitution | If \(a = b\), then "b" can replace "a" in any expression. |

Addition | If \(a = b\), then \(a + c = b + c\) |

Subtraction | If \(a = b\), then \(a - c = b - c\) |

Multiplication | If \(a = b\), then \(ac = bc\) |

Division | If \(a = b\)and \(c \neq 0\), then \(\dfrac{a}{c}= \dfrac{b}{c}\) |

Symmetric | if \(a= b\), then \(b = a\) |

Reflex | \(a = a\) i.e. the number is equal to itself |

Transitive | If \(a = b\) and \(b = c\) , then \(a = c\) |

- Division property of equality along with the other properties allows us to solve the equations.
- Remember to divide by the same number on both sides of the equation to balance the equation.

**Solved Examples**

Example 1 |

The girls went to the canteen and bought three muffins and two coffees that together cost $17. If the cost of one coffee is $4, What is the cost of 1 muffin?

**Solution**

First, let us convert the given statement into an equation.

Let \(a\) be the cost of 1 muffin and b be the cost of 1 coffee.

Then we have \(3a + 2b = 17\)

Given that \(b = 4\), then we have 2 coffees for $8

Thus we have,

\[\begin{align}3a + 8 &= 17\\\\ 3a + 8 - 8 &\!=\! 17 - 8\text{(subtraction property)}\\\\3 a &= 9\\\\\dfrac{3a}{3} &= \dfrac{9}{3}\text{(division property)}\\\\a&=3\end{align}\]

\(\therefore\) The cost of a muffin is $3 |

Example 2 |

Andrew is about to make a recipe that requires equal amount of flour and sugar.

He knows the flour measures 144 pounds. His sugar pack after the last usage has been marked as \(12 (x+4)\). He places the packs of flour and sugar on the pans. They get balanced. What could be the value of \(x\)?

**Solution**

Since the pans get balanced, the quantities of sugar and flour should be equal.

i.e. \(12 (x+4) = 144\)

We find that this involves multiplication and hence division property could be used.

\[\begin{align}\dfrac{12 (x+4)}{12} &= \dfrac{144}{12}\\\\ x+4 &= 12\\\\\text{By subtraction property,}\\ x+ 4 -4 &= 12 - 4 \\\\ x &= 8\end{align}\]

\(\therefore x = 8\) |

Example 3 |

a) The amount earned by Shawn per hour is $25. He has earned $100. How many hours did he work?

b) Shawn completes one-eighth of his work in 4 hours. How much of work will he complete in an hour?

**Solution**

Let us convert the statements into equations and then solve them using the known properties.

a) The total amount earned by Shawn is $25

Let his working hours be denoted by \(x\).

Then, we arrive at the equation \(25 x = 100\)

\[\begin{align}\dfrac{25 x}{5} &= \dfrac{100}{5} \text{(division property)}\\\\5 x &= 20\\\\\dfrac{5 x}{5} &= \dfrac{20}{5}\,\text{(division property)}\\\\x &= 4\end{align}\]

b. Let us assume the amount of work done = \(w\)

Work done by Shawn in 4 hours is \(4\times w\)

Work completed by Shawn in 4 hours is \(\dfrac{1}{8}\)

\[\begin{align} 4 w &= \dfrac{1}{8}\\\\\dfrac{4 w}{4} &=\dfrac{1}{8}\div 4 \,\text{(division property)}\\\\ w &= \dfrac{1}{8}\times \dfrac{1}{4}\\\\&=\dfrac{1}{32} \end{align}\]

\(\therefore\) a) Shawn has worked 4 hours. b) Shawn has completed \(\dfrac{1}{32}\) of his work in an hour. |

**Interactive Questions**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result. **

**Let's Summarize**

The mini-lesson targeted the fascinating concept of division property of equality. The math journey around division property of equality starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions (FAQs)**

## 1. What is the distributive property of equality?

\(a(b + c) = ab + ac\) , where a, b and c are the real numbers. This is the distribution of multiplication over addition.

For example,

Left hand side is \(3 (4+5) = 3 \times 9 = 27\)

Right hand side is \( 3 \times 4 = 12\) and \(3 \times 5=15\)

and \(12 + 15 = 27\)

Thus LHS = RHS

## 2. What is the multiplication property of equality?

If \(a = b\), then \(ac = bc\) where a, b and c are the real numbers. In an equation, where \(a = b\), we can multiply both sides by c to keep the equation still balanced. This is the multiplication property of equality.

## 3. What are the properties of congruence?

We have three properties of congruence: the reflexive property of congruence, the symmetric property of congruence, and the transitive property of congruence.