Question #8dedb

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Question #8dedb

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Answer 1 · 2015-11-07T14:23:43

Last Years prices are exactly: Rs 57600 and Rs 82800
Used a more difficult approach when 1st attempted, Stefan's is much simpler. I have explained his in full. Virtually every step shown.

Explanation:

Let first house be x
Let the second house be y
Let unknown portion of ratio be $z$ $z$

Then year 1 $\to x_{1}; y_{1}$ $-> x_1 ; y_1$
Then year 2 $\to x_{2}; y_{2}$ $-> x_2; y_2$

$Expressing ratios in fractional format$ $color(blue)("Expressing ratios in fractional format")$

$Do not confuse this format with fractions of the whole!$ $color(red)("Do not confuse this format with fractions of the whole!")$

Year 1 $\frac{x_{1}}{y_{1}} = \frac{16}{23}$ $x_1/y_1 = 16/23$ ....................( 1 )

Year 2 $\frac{x_{2}}{y_{2}} = \frac{9}{11}$ $x_2/y_2=9/11$ ........................( 2 )

$Ratios expressed as fractions of the whole$ $color(blue)("Ratios expressed as fractions of the whole")$
$x_{1} \to \frac{16}{16 + 23}$ $x_1 -> 16/(16+23)$

$y_{1} \to \frac{23}{16 + 23}$ $y_1 -> 23/(16+23)$

.==============================================
$Note that keeping fractional values reduces rounding error$ $color(blue)("Note that keeping fractional values reduces rounding error")$

Let year 1 be $t_{1}$ $t_1$
Let year 2 be $t_{2}$ $t_2$
Where house 1 and 2 are $x and y$ $x " and "y$

Let $t_{1} \to \frac{x_{1}}{y_{1}} \equiv \frac{16}{23}$ $t_1 -> x_1/y_1 -= 16/23$ as a ratio
Let $t_{2} \to \frac{x_{2}}{y_{2}} \equiv \frac{9}{11}$ $t_2 -> x_2/y_2 -= 9/11$ as a ratio
.==============================================

$Taking it a step at a time and expressing mathematically:$ $color(green)("Taking it a step at a time and expressing mathematically:")$

$It is given that x_{2} = x_{1} + \frac{25}{100} x_{1}$ $color(green)("It is given that " x_2 = x_1 + 25/100 x_1)$

$x_{2} = x_{1} (1 + \frac{1}{4})$ $color(green)(x_2 =x_1(1+1/4))$

$That is: x_{2} = \frac{5}{4} x_{1}$ $color(green)("That is: "x_2 =5/4x_1)$

$It is also given that y_{2} = y_{1} + 5200$ $color(green)("It is also given that "y_2= y_1 + 5200)$

$So \frac{x_{2}}{y_{2}} = \frac{\frac{5}{4} x_{1}}{y_{1} + 5200} = \frac{9}{11}$ $color(green)("So " x_2/y_2 =(5/4 x_1)/(y_1+5200) = 9/11)$

$11 (\frac{5}{4} x_{1}) = 9 (y_{1} + 5200)$ $color(green)(11(5/4x_1) = 9(y_1+5200))$ .....................( 3 )

.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Changing two unknowns to just 1, making it solvable.$ $color(brown)("Changing two unknowns to just 1, making it solvable.")$

$But from (1) y_{1} = \frac{23}{16} x_{1}$ $color(brown)("But from (1) "y_1 = 23/16 x_1)$ ............( 4 )

$Substitute (4) into (3) giving:$ $color(brown)("Substitute (4) into (3) giving:")$

$\frac{55}{4} x_{1} = 9 (\frac{23}{16} x_{1}) + 9 (5200)$ $color(brown)(55/4x_1 = 9(23/16x_1) + 9(5200))$

$Collecting like terms$ $color(brown)("Collecting like terms")$

$\frac{55}{4} x_{1} - 9 (\frac{23}{16} x_{1}) = 9 (5200)$ $color(brown)(55/4x_1 -9(23/16x_1)= 9(5200))$

$x_{1} (\frac{55}{4} - \frac{207}{16}) = 46800$ $color(brown)(x_1(55/4 - 207/16)=46800)$

$\frac{13}{16} x_{1} = 46800$ $color(brown)(13/16x_1= 46800)$
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$x_{1} = \frac{16}{13} \times 46800 = R s 57600$ $color(red)(x_1 =16/13 times 46800 =Rs57600)$ ...... (5)

so at $t_{1} \frac{x_{1}}{y_{1}} = \frac{16}{23} = \frac{57600}{y_{1}}$ $t_1 x_1/y_1 = 16/23 = 57600/y_1$

$y_{1} = \frac{23 \times 57600}{16} = R s 82800$ $color(red)(y_1 = (23 times 57600)/16= Rs82800)$

Check: $\frac{57600}{82800} = \frac{16}{23}$ $57600/82800 = 16/23$

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Question #8dedb

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