This question aims to develop an understanding of the **algebraic simplification** of the equation for the **volume of a block** using basic **arithmetic operations**.

The **volume of a block** is the product of its **length, width, and height**. It is mathematically defined by the following **formula**:

\[ \boldsymbol{ V \ = \ L \times W \times H } \]

Where $ V $ represents the **volume of the block**, $ L $ represents the **length**, $ W $ represents the **width**, and $ H $ represents the **height**. Now this **formula can be directly used** to calculate the volume **given the length, width and height** of the block, however, if we were **to evaluate** the value of $ h $ **given the volume**, then we may have to **modify** it a little bit. This **rearrangement** process is called the **algebraic simplification** process, which is further explained in the following solution.

## Expert Answer

Given the **formula of the volume** of the block:

\[ V \ = \ L \times W \times H \]

**Dividing both sides by $ W $:**

\[ \dfrac{ V }{ W } \ = \ \dfrac{ L \times W \times H }{ W } \]

\[ \Rightarrow \dfrac{ V }{ W } \ = \ L \times H \]

**Dividing both sides by $ H $:**

\[ \dfrac{ V }{ W \times H } \ = \ \dfrac{ L \times H }{ H } \]

\[ \Rightarrow \dfrac{ V }{ W \times H } \ = \ L \]

**Exchanging sides:**

\[ L \ = \ \dfrac{ V }{ W \times H } \]

Which is the required expression.

## Numerical Result

\[ L \ = \ \dfrac{ V }{ W \times H } \]

## Example

**Part (a)** – The **area of a rectangle** is given by the following formula:

\[ A \ = \ L \times W \]

**Find the value of $ L $.**

**Dividing the above equation by $ W $:**

\[ \dfrac{ A }{ W } \ = \ \dfrac{ L \times W }{ W } \]

\[ \Rightarrow \dfrac{ A }{ W } \ = \ L \]

**Exchanging sides:**

\[ L \ = \ \dfrac{ A }{ W } \]

**Part (b)** – The **area of a right triangle** is given by the following formula:

\[ A \ = \ \dfrac{ 1 }{ 2 } b \times h \]

**Find the value of $ h $.**

**Dividing the above equation by $ b $:**

\[ \dfrac{ A }{ b } \ = \ \dfrac{ 1 }{ 2 } \dfrac{ b \times h }{ b } \]

\[ \Rightarrow \dfrac{ A }{ b } \ = \ \dfrac{ 1 }{ 2 } h \]

**Multiplying the above equation with $ 2 $:**

\[ 2 \times \dfrac{ A }{ b } \ = \ 2 times \dfrac{ 1 }{ 2 } h \]

\[ \Rightarrow 2 \times \dfrac{ A }{ b } \ = \ h \]

**Exchanging sides:**

\[ h \ = \ 2 \times \dfrac{ A }{ b } \]