Question

If in a proportion, the first, second, and fourth terms are 32, 112, and 217respectively, then find the third term.

Answer 1

Hint: We solve this problem by using the definition of proportion that is if four numbers are in proportion then the ratio of the first two numbers is equal to the ratio of the last two numbers. That is if a, b, c, and d are in proportion then
\[\Rightarrow a:b=c:d\]
Then we use the definition of ratio that is
\[a:b=\dfrac{a}{b}\]
We use the above two formulas to get the unknown number.

Complete step-by-step solution
We are given that the first, second, and third numbers of proportion are 32, 112, and 217.
Let us assume that the third number which we need to find as \['x'\]
We know that the definition of proportion is that if four numbers are in proportion then the ratio of the first two numbers is equal to the ratio of the last two numbers. That is if a, b, c, and d are in proportion then
\[\Rightarrow a:b=c:d\]
By using the above definition let us find the ratio of the first two numbers as
\[\Rightarrow p=32:112\]
We know that the definition of ratio that is
\[a:b=\dfrac{a}{b}\]
By using this formula we get the ratio of first two numbers as
\[\begin{align}
  & \Rightarrow p=\dfrac{32}{112} \\
 & \Rightarrow p=\dfrac{2}{7} \\
\end{align}\]
Now, let us find the ratio of last two numbers then we get
\[\begin{align}
  & \Rightarrow q=x:217 \\
 & \Rightarrow q=\dfrac{x}{217} \\
\end{align}\]
Now, by using the definition of proportion we have
\[\Rightarrow p=q\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow \dfrac{2}{7}=\dfrac{x}{217} \\
 & \Rightarrow x=\dfrac{2}{7}\times 217 \\
 & \Rightarrow x=62 \\
\end{align}\]
Therefore the third number of given proportion is 62

Note: We have a shortcut for solving this problem.
The proportion has a direct formula that if four numbers are in proportion then the product of the first and fourth numbers is equal to the product of the second and third numbers. That is if a, b, c, and d are in proportion then
\[\Rightarrow a\times d=b\times c\]
By using the above formula to given numbers we get
\[\begin{align}
  & \Rightarrow 32\times 217=x\times 112 \\
 & \Rightarrow x=62 \\
\end{align}\]
Therefore the third number of given proportions is 62.