Question

The ratio of land to water for the whole earth is 1:2 and 2:3 for the northern hemisphere. What is the ratio of land to water in the southern hemisphere?
(a) 4:11
(b) 1:3
(c) 1:4
(d) 4:7

Answer 1

Hint: First we will assign a constant value to fractions for the whole earth and northern hemisphere. Then we will subtract the northern hemisphere values from the whole earth values. The fraction after simplification would give us the final answer.

Complete step-by-step answer:
Let the constant value x be assigned to the ratio of land and water for the whole earth. Thus the ratio 1:2 can be written as x:2x. Similarly let us assign y for the northern hemisphere. Thus the ratio becomes 2y:3y.
Let us now construct a table for reference.

	Land	Water	Total
Whole earth	x	2x	3x
Northern hemisphere	2y	3y	5y
Southern hemisphere	x-2y	2x-3y

We got the values for the southern hemisphere by subtracting the corresponding values of the whole earth and northern hemisphere.
Now let us take the ratio of the land and water in the southern hemisphere.
Therefore,
\[Ratio=\dfrac{x-2y}{2x-3y}\]
Now dividing both the numerator and the denominator by y we get,
\[Ratio=\dfrac{\dfrac{x}{y}-2}{\dfrac{2x}{y}-3}..........(i)\]
Now we know that the northern hemisphere is 50% of the total earth. Thus
Whole earth: Northern hemisphere = 2:1
Thus, now referring to the last column of the table,
\[\dfrac{3x}{5y}=\dfrac{2}{1}\]
Cross multiplying we get,
\[\dfrac{x}{y}=\dfrac{10}{3}..........(ii)\]
Now substituting (ii) in (i), we get,
\[Ratio=\dfrac{\dfrac{10}{3}-2}{\dfrac{2(10)}{3}-3}\]
Taking LCM and cancelling the common 3 in both numerator and the denominator we get,
\[\begin{align}
  & Ratio=\dfrac{10-6}{2(10)-9} \\
 & Ratio=\dfrac{4}{11} \\
\end{align}\]
Thus the correct option is option(a).

Note: We may commit a mistake by thinking that the ratio in the option(b) is the correct one because adding the corresponding numbers in 2:3 and 1:3 will give a ratio of 1:2 when simplified. But this is not the way ratios work and it will result in the wrong answer.