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:
\[ \dfrac{a}{5} = \dfrac{10b} {21 \times 5}\]
\[\dfrac{a}{5} = \dfrac{2b} {21}\]