Question

Three numbers \[A\] , \[B\] and \[C\] are in the ratio \[1:2:3\]. Their average is 600 If \[A\] is increased by \[10\% \] and \[B\] is decreased by \[20\% \], then by how much will \[C\] be increased to get the average increased by \[5\% \] ?

Answer 1

Hint:
Here we will first assume these three numbers to be any variable in such a way when we find their ratio, we get the same ratio as given. Then we will find the average of these three using the formula of average. We will write all the given conditions mathematically and solve it further to get the value of the variables. From there we will get the value by which the third number will increase.

Formula Used:
We will use the formula Average \[ = \] Sum of all numbers \[ \div \] total number of observations.

Complete Step by Step Solution:
It is given that there are three numbers \[A\] , \[B\] and \[C\] whose ratios are \[1:2:3\].
As the ratio between the three numbers is \[1:2:3\], we will assume the three numbers as variables such that we will get the same ratio.
Let the number \[A\] be \[x\], number \[B\] be \[2x\] and number \[C\] be \[3x\].
We know that the average of numbers is equal to the sum of the numbers to the number of numbers.
Mathematically, Average \[ = \] Sum of all numbers \[ \div \] total number of observations
Therefore,
\[{\rm{Average}} = \dfrac{{x + 2x + 3x}}{3}\]
It is given that the average between these numbers is 600.
Now, substituting 600 for the average in the above equation, we get
 \[ \Rightarrow 600 = \dfrac{{x + 2x + 3x}}{3}\]
On adding the numbers in the numerator, we get
\[ \Rightarrow 600 = \dfrac{{6x}}{3}\]
On further simplification, we get
\[ \Rightarrow 600 = 2x\]
Dividing both sides by 2, we get
\[\begin{array}{l} \Rightarrow \dfrac{{600}}{2} = x\\ \Rightarrow x = 300\end{array}\]
Now using the value of \[x\], we get
Number \[A\] \[ = x = 300\]
Number \[B\] \[ = 2x = 2 \times 300 = 600\]
Number \[C\] \[ = 3x = 3 \times 300 = 900\]
Now we will calculate value of number \[A\] after increment ,
Value of number \[A\] after increment of \[10\% \] \[ = 300 + 10\% \times 300\]
On simplifying the terms, we get
\[ \Rightarrow \] Value of number \[A\] after increment of \[10\% \] \[ = 300 + \dfrac{{10}}{{100}} \times 300\]
On multiplying the numbers, we get
 \[ \Rightarrow \] Value of number \[A\] after increment of \[10\% \] \[ = 300 + 30\]
On adding the numbers, we get
\[ \Rightarrow \] Value of number \[A\] after increment of \[10\% \] \[ = 330\] ……….. \[\left( 1 \right)\]
Now we will calculate the value of number \[B\] after decrement.
Value of number \[B\] after decrement of \[20\% \] \[ = 600 - 20\% \times 600\]
On simplifying the terms, we get
\[ \Rightarrow \] Value of number \[B\] after decrement of \[20\% \]\[ = 600 - \dfrac{{20}}{{100}} \times 600\]
On multiplying the numbers, we get
\[ \Rightarrow \] Value of number \[B\] after decrement of \[20\% \]\[ = 600 - 120\]
On subtracting the numbers, we get
\[ \Rightarrow \] Value of number \[B\] after decrement of \[20\% \]\[ = 480\] ……….. \[\left( 2 \right)\]
We will now calculate the value of average after increment .
Value of average after increment of \[5\% \] \[ = 600 + 5\% \times 600\]
On simplifying the terms, we get
\[ \Rightarrow \] Value of average after increment of \[5\% \] \[ = 600 + \dfrac{5}{{100}} \times 600\]
On multiplying the numbers, we get
 \[ \Rightarrow \] Value of average after increment of \[5\% \] \[ = 600 + 30\]
On adding the numbers, we get
\[ \Rightarrow \] Value of average after increment of \[5\% \] \[ = 630\] ……….. \[\left( 3 \right)\]
The new average becomes 630.
Now, we will apply the formula of average for these new numbers.
\[ \Rightarrow {\rm{Average}} = \dfrac{{A + B + C}}{3}\]
Substituting \[A = 330\], \[B = 480\] and \[{\rm{Average}} = 630\] in the above equation, we get
\[ \Rightarrow 630 = \dfrac{{330 + 480 + C}}{3}\]
On adding the terms, we get
\[ \Rightarrow 630 = \dfrac{{810 + C}}{3}\]
Cross multiplying the terms, we get
\[\begin{array}{l} \Rightarrow 630 \times 3 = 810 + C\\ \Rightarrow 1890 = 810 + C\end{array}\]
Now, we will subtract 810 from both sides.
\[\begin{array}{l} \Rightarrow 1890 - 810 = 810 + C - 810\\ \Rightarrow 1080 = C\end{array}\]
Hence, the number \[C\] becomes 1080.
We can see that the value of the number \[C\] has increased. Now, we will calculate the percent increase in the number \[C\], which will be equal to the multiplication of the ratio of the difference between the new and old value to the old number with the number 100.
Mathematically, it is equal to
\[ \Rightarrow \] increase in \[C = \dfrac{{1080 - 900}}{{900}} \times 100\]
On subtracting the numbers, we get
 \[ \Rightarrow \] increase in \[C = \dfrac{{180}}{{900}} \times 100\]
On further simplification, we get
\[ \Rightarrow \] increase in \[C = 20\% \]

Hence, the value of the number \[C\] will increase by \[20\% \] or increase by the value 180.

Note:
Here, we need to know the meaning of the term ‘average’ and its formula. Here the average of the numbers is defined as the ratio of the sum of the numbers to the number of numbers. If we increase the value of any of the numbers, then the value of the average of the numbers will also get increased and if we decrease the value of any of the numbers, then the value of the average of the numbers will also get decreased.