Question

D)) The ratio of the monthly income and expenditure of Sunayana is 5 : 3. If she

Answer 1

Answer:

Sunayana's income is ₹25000 and expenditures are ₹15000.

Step-by-step explanation:

Let i denote the monthly income and e denote the expenditures.

We know that the ratio of the monthly income to expenditure is 5:3. So, we can write the following proportion:  

$\frac{i}{e}=\frac{5}{3}$

Let's multiply both sides by e. This yields:

$i=\frac{5}{3}e$

We know that when the income is increased by 5000 and the expenditures are decreased by 3000, the new ratio is 5:2. So, we can write the following proportion:

$\frac{i+5000}{e-3000}=\frac{5}{2}$

Let's multiply both sides by $(e-3000)$:

$i+5000=\frac{5}{2}(e-3000)$

Since we know that $i=\frac{5}{3}e$, substitute:

$\frac{5}{3}e+5000=\frac{5}{2}(e-3000)$

So, let's solve for the expenditures. Distribute the right:

$\frac{5}{3}e+5000=\frac{5}{2}e-7500$

Subtract $\frac{5}{2}e$ from both sides:

$-\frac{5}{6}e+5000=-7500$

Subtract 5000 from both sides:

$-\frac{5}{6}e=-12500$

Multiply both sides by -6/5. So, the expenditures are:

$e=\text{Rs }15000$

We can use the original ratio to find Sunayana's income:

$i=\frac{5}{3}e$

Substitute 15000 for e. Evaluate:

$i=\frac{5}{3}(15000)=\text{Rs }25000$

So, Sunayana's income is ₹25000 and expenditures are ₹15000.

And we're done!

Edit: Wrong currency, sorry about that!

Answer 2

Answer: Income = 25000, Expenditures = 15000

Step-by-step explanation:

A ratio of 5:3 can also be written in fraction form as $\dfrac{5}{3}$ (which is in simplified format from $\dfrac{5x}{3x}$).

Adding 5000 to 5x results in 5x + 5000

Subtracting 3000 from 3x results in 3x - 3000

--> $\dfrac{5x+5000}{3x-3000}$

The new ratio of 5:2 can be written in simplified fraction form of $\dfrac{5}{2}$.

Set the fractions equal to each. Then cross multiply and solve for x.

   $\dfrac{5x+5000}{3x-3000}=\dfrac{5}{2}$

5(3x - 3000) = 2(5x + 5000)             Cross multiplied

15x - 15000 = 10x + 10000                Distributed

5x - 15000 =            10000                Subtracted 0x from both sides

5x               =            25000               Added 15000 to both sides

               x = 5000                            Divided 5 from both sides

The original ratio of 5 : 3     -->  5x:3x

where 5x is the income and 3x is the expenditures results in:

income = 5x                            expenditures = 3x

             = 5(5000)                                         = 3(5000)

             = 25000                                          = 15000