Question

The population of town A is 96,000 more than town B. If 62,000 people move from town B to town A, the ratio between the population of town A and that of town B will be 11: 1. Find the original population of the two towns.
(a) \[A = 1,40,000\] and \[B = 90,000\]
(b) \[A = 1,50,000\] and \[B = 52,000\]
(c) \[A = 1,80,000\] and \[B = 84,000\]
(d) \[A = 1,90,000\] and \[B = 60,000\]

Answer 1

Hint: Here, we need to find the original population of both the towns. Using the given information, we can form two linear equations in two variables. We will solve these equations to find the values of the two variables and use these values to find the original population of the two towns.

Complete step by step solution:
Let the population of town A and town B be \[x\] and \[y\] respectively.
Now, it is given that the population of town A is 96,000 more than town B.
Thus, we get
\[ \Rightarrow x = y + 96000 \ldots \ldots \ldots \left( 1 \right)\]
If 62,000 people move from town B to town A, the population of town A and town B becomes \[x + 62000\] and \[y - 62000\] respectively.
The ratio of the population of town A to town B (after 62,000 people move from town B to A) is given as \[11:1\].
Thus, we get
\[\left( {x + 62000} \right):\left( {y - 62000} \right) = 11:1\]
We can write this as
\[ \Rightarrow \dfrac{{x + 62000}}{{y - 62000}} = \dfrac{{11}}{1}\]
Multiplying both sides by \[y - 62000\], we get
\[\begin{array}{l} \Rightarrow \left( {\dfrac{{x + 62000}}{{y - 62000}}} \right)\left( {y - 62000} \right) = \dfrac{{11}}{1}\left( {y - 62000} \right)\\ \Rightarrow x + 62000 = 11\left( {y - 62000} \right)\end{array}\]
Multiplying the expression using the distributive law of multiplication, we get
\[ \Rightarrow x + 62000 = 11y - 682000\]
Rewriting the equation, we get
\[ \Rightarrow x - 11y = - 682000 - 62000\]
Subtracting the terms, we get
\[ \Rightarrow x - 11y = - 744000 \ldots \ldots \ldots \left( 2 \right)\]
We can observe that the equations \[\left( 1 \right)\] and \[\left( 2 \right)\] are a pair of linear equations in two variables.
We will solve the equations to find the values of \[x\] and \[y\].
Substituting \[x = y + 96000\] from equation \[\left( 1 \right)\] in equation \[\left( 2 \right)\], we get
\[ \Rightarrow y + 96000 - 11y = - 744000\]
Subtracting the like terms, we get
\[ \Rightarrow 96000 - 10y = - 744000\]
Rewriting the equation, we get
\[ \Rightarrow 10y = 744000 + 96000\]
Adding the terms, we get
\[ \Rightarrow 10y = 840000\]
Dividing both sides of the equation by 10, we get
\[ \Rightarrow y = 84000\]
Substituting \[y = 84000\] in the equation \[\left( 1 \right)\], we get
\[ \Rightarrow x = 84000 + 96000\]
Thus, we get
\[ \Rightarrow x = 180000\]
\[\therefore \] We get the population of town A as 1,80,000 and the population of town B as 84,000.
Thus, the correct option is option (c).

Note: We have formed two linear equations in two variables and simplified them to find the fraction. A linear equation in two variables is an equation which has only one variable with the highest exponent as 1 and is of the form \[ax + by + c = 0\], where \[b\] and \[a\] are not equal to 0. For example, \[2x - 7y = 4\] is a linear equation in two variables.
We have used the distributive law of multiplication to multiply \[11\] by \[y - 62000\]. The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].
We can verify our answer by using the given information.
The population of town A is 1,80,000, which is 96,000 more than the population of 84,000.
If 62,000 people move from town B to town A, the population of town A and town B becomes \[180000 + 62000 = 242000\] and \[84000 - 62000 = 22000\] respectively.
We can observe that \[242000:22000 = 11:1\].
Hence, we have verified our answer.