This problem aims to familiarize us with fractions and their **ratio** and **proportion**. Basically, this problem is related to **fundamental calculus**. Ratio and Proportion are described mainly founded on **fractions**. When a fraction is expressed in the form of a:b, it is called a **ratio,** whereas a **proportion** declares that two ratios are equivalent.

Here, we have taken a and b as any two **integers**. **Ratio** and **proportion** are essential concepts, and they collectively form a foundation to comprehend the diverse concepts in **mathematics** as well as in **science**. **Proportion** can be categorized into the subsequent categories such as **Direct** Proportion, **Continued** Proportion, and **Inverse** Proportion.

## Expert Answer

Let’s say that a **proportion** in the format xy = a indicates to us that the **ratio** of x to y will consistently be a constant **digit**. With that being said, we can still have **different** **values** for x and y, but their **ratios** will always stay fixed.

We are given an **expression** $ \dfrac{a}{b} $ which is equal to $ \dfrac {8}{15} $ and we have to find out what this **fraction** $ \dfrac{a}{8} $ is equal to.

To acquire the **answer** of the fraction $ \dfrac{a}{8} $, we will first **eliminate** the variable $b$ from the given **expression** because the required expression does not have a $b$ in the **denominator**.

So, to **eliminate** $b$ we **multiply** both the sides by $ b $:

\[ b \times \dfrac{a} {b} = \dfrac{8} {15} \times b \]

\[ \cancel{b} \dfrac{a} { \cancel{b} } = \dfrac{8b} {15} \]

\[ a = \dfrac{8b} {15} \]

Since $b$ has been **eliminated**, we get $a$ on the left side and we are asked to find $ \dfrac{a} {8} $. The only thing left is the **numeral** $8$ in the **denominator**, so to obtain $ \dfrac{a} {8} $, we **divide** the expression $ a = \dfrac{8b} {15} $ by $8$ on the both sides:

\[ \dfrac{a}{8} = \dfrac{8b} {15 \times 8} \]

\[ \dfrac{a}{8} = \dfrac{ \cancel{8} b} {15 \times \cancel{8}} \]

\[ \dfrac{a}{8} = \dfrac{ b} {15} \]

## Numerical Answer

Given the **proportion** $ \dfrac{a} {b} = \dfrac{8} {15} $, the equivalent **proportion** $ \dfrac{a} {8} $ will be equal to $ \dfrac{b} {15} $.

## Example

Given the **proportion** $ \dfrac{a} {b} = \dfrac{10} {21} $, what **ratio** completes the equivalent proportion $ \dfrac{a} {5}$.

To obtain $ \dfrac{a}{5} $, firstly **eliminate** the $b$ because required **expression** does not have a $b$ in the **denominator**.

So to eliminate $b$, we **multiply** both sides by $ b $.

\[ b \times \dfrac{a} {b} = \dfrac{10} {21} \times b \]

\[ \cancel{b} \dfrac{a} { \cancel{b} } = \dfrac{10b} {21} \]

\[ a = \dfrac{10b} {21} \]

Since $b$ has been **eliminated**, we get $a$ on the **left** side and we are asked to find $ \dfrac{a} {8} $. Now obtaining $ \dfrac{a} {5} $ by **dividing** the expression $ a = \dfrac{10b} {21} $ by $5$ on the both sides:

\[ \dfrac{a}{5} = \dfrac{10b} {21 \times 5}\]

\[\dfrac{a}{5} = \dfrac{2b} {21}\]