Question

A fraction becomes 4÷5 if 1 is added to both numerator and denominator. If, however, 5 is subtracted from both numerator and denominator , the fraction becomes 1÷2. Find tge fraction by using crammer rule.

Answer 1

`(7)/(9)`

Explanation:

`(x+1)/(y+1)=(4)/(5)\\\Rightarrow 5x-4y=-1`

`(x-5)/(y-5)=(1)/(2)\\\Rightarrow 2x-y=5`

Putting it in matrix form

`\begin{bmatrix}a_(1)&b_(1)\\a_(2)&b_(2)\end{bmatrix}{\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}{c_(1)}\\{c_(2)}\end{bmatrix}\\\Rightarrow\begin{bmatrix}5 & -4\\2 & -1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}-1\\ 5\end{bmatrix}`

From Cramer's rule we have

`x=\frac{\begin{vmatrix}c_1 &b_1 \\ c_2 & b_2\end{vmatrix}}{\begin{vmatrix}a_1 &b_1 \\ a_2& b_2\end{vmatrix}}\\\Rightarrow x=\frac{\begin{vmatrix}-1 &-4 \\ 5 & -1\end{vmatrix}}{\begin{vmatrix}5 &-4 \\ 2& -1\end{vmatrix}}\\\Rightarrow x=(1+20)/(-5+8)\\\Rightarrow x=7`

`y=\frac{\begin{vmatrix}a_1 &c_1 \\ a_2 & c_1\end{vmatrix}}{\begin{vmatrix}a_1 &b_1 \\ a_2& b_2\end{vmatrix}}\\\Rightarrow y=\frac{\begin{vmatrix}5 &-1 \\ 2 & 5\end{vmatrix}}{\begin{vmatrix}5 &-4 \\ 2& -1 \end{vmatrix}}\\\Rightarrow y=(25+2)/(-5+8)\\\Rightarrow y=9`

Verifying the results

`(7+1)/(9+1)=(8)/(10)=(4)/(5)`

`(7-5)/(9-5)=(2)/(4)=(1)/(2)`

Hence, the fraction is
`(7)/(9)`.