Question

Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion.
39 litres : 65 litres and 6 bottles : 10 bottles

Answer 1

Hint: Consider 39 litres and 65 litres and divide 39 with 65. Cancel the common factors and reduce them into simplest form. Now, consider 6 bottles and 10 bottles and divide 6 with 10. Reduce them in their simplest form. Check if the simplified form of \[\dfrac{39}{65}\] and \[\dfrac{6}{10}\] are equal or not. If they are equal then the ratios will form a proportion otherwise not. If they are in proportion, write them in the form of a: b:: c: d where ‘a’ and ‘d’ are extreme terms while ‘b’ and ‘c’ are middle terms.

Complete step by step answer:
Here, we have been provided with the ratios 39 litres : 65 litres and 6 bottles : 10 bottles and we have to determine if they are in proportion or not.
Let us consider 39 litres : 65 litres. This can be written as:
\[\begin{align}
  & \Rightarrow 39:65 \\
 & \Rightarrow \dfrac{39}{65} \\
 & \Rightarrow \dfrac{13\times 3}{13\times 5} \\
\end{align}\]
Cancelling the common factors, we get,
\[\Rightarrow 39:65=\dfrac{3}{5}=3:5\]
Therefore, 39 : 65 can be written as 3 : 5 in its simplified form.
Now, let us consider 68 bottles : 10 bottles. This can be written as:
\[\Rightarrow \] 6 bottles : 10 bottles
\[\Rightarrow 6:10=\dfrac{3}{5}=3:5\]
Therefore, 6:10 can be written as 3:5 in its simplified form.
Now, clearly we can see that the given ratios, 39 litres : 65 litres and 6 bottles : 10 bottles, are equal in their simplified form. So, we can say that they are in proportion. Therefore, mathematically it can be written as: -
\[\Rightarrow 39:65::6:10\]

Now, we know that if a, b, c and d are in proportion with the given relation: a:b::c:d then ‘a’ and ‘d’ are the extreme terms and ‘b’ and ‘c’ are the middle terms. So, for 39 : 65 :: 6 : 10, we have 39 and 10 as the extreme terms and 65 and 6 are the middle terms.

Note: One may note that we must convert the ratios into its simplified form to check if they are in proportion or not. Here, in the above solution we have divided 39 with 65 and 6 with 10. One can also reverse the numbers, that means divide 65 with 39 and 10 with 6. The main thing to remember is that if we reverse the ratio in expression 1 then we must reverse the ratio in expression 2 accordingly, otherwise we will get the wrong answer.