Question

How do you solve the proportion \[\dfrac{5}{9} = \dfrac{x}{{63}}\]?

Answer 1

Hint: We use the concepts of ratios, relative ratios and proportions to solve this problem. A ratio is generally defined as comparison between two or more quantities of the same units. It also tells us how much the first quantity is present in the second quantity. And proportion is the comparison between two ratios.

Complete step by step solution:
In mathematics, Ratios and proportion concepts are related to comparison of units.
Generally, proportion is referred to as part, share, or number considered in comparative relation to a whole. When two ratios are equivalent, then they are said to be in proportion. Its symbol is “ \[::\] ” or “ \[ = \] ”.
Consider four numbers \[a,b,c{\text{ and }}d\] .
So, if \[a:b\] and \[c:d\] are equivalent, then these are said to be in proportion.
\[ \Rightarrow a:b::c:d\]
A ratio can also be written in fraction form i.e., \[a:b = \dfrac{a}{b}\]. So, if two fractions are equal, then they are said to be proportional.
If \[\dfrac{a}{b} = \dfrac{c}{d}\] , then we can conclude that those are in proportion and we can further extend it as \[ad = bc\] , by cross multiplying the fractions
So, the given question is \[\dfrac{5}{9} = \dfrac{x}{{63}}\]
On cross multiplying, we get,
\[ \Rightarrow 9x = 5 \times 63\]
Dividing the whole equation by 9, we get,
\[ \Rightarrow x = \dfrac{{5 \times 63}}{9} = \dfrac{{315}}{9} = 35\]
Therefore, $x=35$.

Note:
We use this proportions concept while solving many mathematical and scientific problems. We use these in geometry and symmetry too. If a ratio is equal to a constant i.e., if \[\dfrac{a}{b} = \dfrac{k}{1} = k\], then we can say that,
If \[a\] increases, \[k\] also increase and we can say that, \[a\] is directly proportional to \[k\] \[\left( {a \propto k} \right)\]
If \[b\] increases, \[k\] value decreases and we can say that, \[b\] is inversely proportional to \[k\] \[\left( {\dfrac{1}{b} \propto k} \right)\].