Work out the amount that Arjun paid in rent in 2019

[chwala]

Homework Statement

See attached.

Relevant Equations

Ratio

I am interested in a more direct linear equations approach...to solve, which i know is possible...

A. My initial thinking was along the lines,

let $x$ be the amount that Arjun paid in 2018 and let $y$ be the amount that Gretal paid in 2018 where A was the total amount paid in 2018 ... this gives me the equation,

In 2018,

Arjun paid: $\dfrac{5}{12} A = x$ and Gretal paid: $\dfrac{7}{12} A = y$. Therefore, my first equation is,

$\dfrac{5}{12} A + \dfrac{7}{12} A= x +y$

I need time to finish up on this...




B. My alternative approach which was more direct is

In 2018,
Arjun : Gretal =$45 : 63$

in 2019,
Arjun : Gretal =$45 : 65$

Therefore,

$63 = x$ and $65 = 290 +x$

$63(290 +x) = 65x$

$18,270 +63 x = 65x$

$18270 = 2x$

$x= 9135$

Therefore in 2019, Arjun paid $\dfrac{9135}{7}=1305× 5 = 6525$.

There could be a better approach. Cheers.


This is the ms approach. Not quite clear to me.

 

[Hill]

You have four unknowns, one for each person for each year: $A_8, A_9, G_8, G_9$.
You have four statements, which give you four equations:
$\frac {A_8} {G_8} = \frac 5 7$
$\frac {A_9} {G_9} = \frac 9 {13}$
$A_8=A_9$
$G_9-G_8=290$
Solve the equations.

 

[chwala]

You have four unknowns, one for each person for each year: $A_8, A_9, G_8, G_9$.
You have four statements, which give you four equations:
$\frac {A_8} {G_8} = \frac 5 7$
$\frac {A_9} {G_9} = \frac 9 {13}$
$A_8=A_9$
$G_9-G_8=290$
Solve the equations.

Smart

Hope your equations are correct...will post my working later.

...I can confirm that equations are correct will post working later. Thanks man!

 

[chwala]

...
We have

$A_8 = \dfrac{5}{7} G_9 - \dfrac{1450}{7}$

and

$A_8 = \dfrac{9}{13} G_9$


$\left[\dfrac{9}{13} G_9 = \dfrac{5}{7} G_9 - \dfrac{1450}{7}\right]$

$G_9 = \left(\dfrac {1450}{7} × \dfrac {91}{2}\right) = \left(\dfrac{131,950}{14} \right)= 9,425$

$⇒A_8 =\dfrac{(9 × 9,425)}{13} = 6,525$.

 

[gmax137]

I get the same ($6525). That's pretty pricey digs!

Edit, oops I was thinking it was 6525 per month. For the whole year that's not so bad, near $550/month.

 

[SammyS]

Your results in Post# 4 look fine.
In this Post, I will show a somewhat more direct way to get to the quantity asked for in the OP, Arjun's rent in 2019. Like you, I will be using @Hill 's variable definitions and set of equations.

You have four unknowns, one for each person for each year: $A_8, A_9, G_8, G_9$.
You have four statements, which give you four equations:
$\dfrac {A_8} {G_8} = \dfrac 5 7$
$\dfrac {A_9} {G_9} = \dfrac 9 {13}$
$A_8=A_9$
$G_9-G_8=290$
Solve the equations.

Since $\displaystyle A_8=A_9\, ,$ I will choose to use $\displaystyle A_9$ to refer to either.

Use the two ratio equations to express $\displaystyle G_8 \text{ and } G_9$ in terms of $A_9$.

$\displaystyle \quad G_8=\dfrac 7 5 A_9 \text{ and } G_9=\dfrac {13} 9 A_9 \$.

Plugging those into the equation $G_9-G_8=290$ we get the following.

$\displaystyle \quad \dfrac {13} 9 A_9-\dfrac 7 5 A_9 ==290$

Multiply both sides of the equation by $9\cdot 5$ to eliminate fractions.

$\displaystyle \quad (5\cdot 13 - 9\cdot 7 ) A_9 = 9\cdot 5\cdot 290$

 

[chwala]

Your results in Post# 4 look fine.
In this Post, I will show a somewhat more direct way to get to the quantity asked for in the OP, Arjun's rent in 2019. Like you, I will be using @Hill 's variable definitions and set of equations.


Since $\displaystyle A_8=A_9\, ,$ I will choose to use $\displaystyle A_9$ to refer to either.

Use the two ratio equations to express $\displaystyle G_8 \text{ and } G_9$ in terms of $A_9$.

$\displaystyle \quad G_8=\dfrac 7 5 A_9 \text{ and } G_9=\dfrac {13} 9 A_9 \$.

Plugging those into the equation $G_9-G_8=290$ we get the following.

$\displaystyle \quad \dfrac {13} 9 A_9-\dfrac 7 5 A_9 ==290$

Multiply both sides of the equation by $9\cdot 5$ to eliminate fractions.

$\displaystyle \quad (5\cdot 13 - 9\cdot 7 ) A_9 = 9\cdot 5\cdot 290$

Aarrgh that's the ms method on my attached post $1$. Cheers.