Question

Divide 900 into two parts such that \[60\% \] of one part exceeds \[30\% \] of the other by 270.
A) \[800\] and \[900\]
B) \[300\] and \[500\]
C) \[600\] and \[300\]
D) \[200\] and \[700\]\[\]

Answer 1

Hint:
Here, we will assume the first part to be some variable and subtract it from 900 to get the second part. Then we will use the given information and find both the parts in terms of variable. Then we will find their difference and equate it to 270. We will solve the equation further to get the value of the variable. We will then use this value to find the other part.

Complete step by step solution:
Let us consider that first part of \[900\] to be \[x\].
So the other part of \[900\] will be \[900 - x\] .
It is given that \[60\% \] of one part is to be found.
Therefore,
\[60\% \] of first part \[ = 60\% \times x\]
Expressing the percentage as fraction, we get
\[ \Rightarrow \] \[60\% \] of first part \[ = \dfrac{{60}}{{100}} \times x\]
Now, multiplying the terms, we get
\[ \Rightarrow \] \[60\% \] of first part \[ = \dfrac{{3x}}{5}\]…………………………..\[\left( 1 \right)\]
Next, it’s given that \[30\% \] of other parts is to be found.
Therefore it will be
\[30\% \] of second part \[ = 30\% \times \left( {900 - x} \right)\]
Expressing the percentage as fraction, we get
\[ \Rightarrow \] \[30\% \] of second part \[ = \dfrac{{30}}{{100}} \times \left( {900 - x} \right)\]
Now, multiplying the terms, we get
\[ \Rightarrow \] \[30\% \] of second part \[ = \dfrac{{3\left( {900 - x} \right)}}{{10}}\] …………………………..\[\left( 2 \right)\]
So, as it is given in the question that \[60\% \] of one part exceeds \[30\% \] of the other by\[270\].
Now subtracting equation \[\left( 2 \right)\] from \[\left( 1 \right)\] and equating it to 270, we get
\[\dfrac{{3x}}{5} - \dfrac{{3\left( {900 - x} \right)}}{{10}} = 270\]
Taking LCM on RHS, we get
\[ \Rightarrow \dfrac{{3x \times 2 - 3\left( {900 - x} \right)}}{{10}} = 270\]
Simplifying the expression, we get
\[ \Rightarrow 6x - 2700 + 3x = 2700\]
Adding the like terms we get
\[ \Rightarrow 9x = 2700 + 2700\]\[\]
\[ \Rightarrow x = \dfrac{{5400}}{9}\]
Dividing the terms, we get
\[ \Rightarrow x = 600\]
Now, we will subtract the value of \[x\] from 900 to get our second part. Therefore, we get
\[900 - x = 900 - 600 = 300\]
Hence, the value of two parts in which 900 is divided is 600 and 300.

So option (C) is correct.

Note:
We can also solve this question in an alternate way. So writing the given information mathematically, we get
\[60\% \times x = 30\% \left( {900 - x} \right) + 270\]
Expressing the percentage into fraction, we get
\[ \Rightarrow \dfrac{{60}}{{100}} \times x = \dfrac{{30}}{{100}} \times \left( {900 - x} \right) + 270\]
Simplifying the expression, we get
\[ \Rightarrow \dfrac{{3x}}{5} + \dfrac{{30}}{{100}} \times x = \dfrac{{30}}{{100}} \times 900 + 270\]
Multiplying the terms, we get
\[ \Rightarrow \dfrac{{3x}}{5} + \dfrac{{3x}}{{10}} = 270 + 270\]
Taking LCM and adding the like terms, we get
\[ \Rightarrow \dfrac{{9x}}{{10}} = 540\]
On cross multiplication, we get
\[ \Rightarrow x = 540 \times \dfrac{{10}}{9} = 600\]
Now, we will subtract the value of \[x\] from 900 to get our second part. Therefore, we get
\[900 - x = 900 - 600 = 300\]
Therefore, we get the first part as 600 and second part as 300.