Kate and Nora each have a sum of money. The ratio of the amount of money Kate has to that of Nora is 3:53:5. After Nora gives Rs. 150 to Kate, the ratio of the amount of money Kate has to that of Nora becomes 7:97:9 then the sum of money Kate had initially is ____.

Question

Kate and Nora each have a sum of money. The ratio of the amount of money Kate has to that of Nora is 3:53:5. After Nora gives Rs. 150 to Kate, the ratio of the amount of money Kate has to that of Nora becomes 7:97:9 then the sum of money Kate had initially is ____.

Accepted Answer

Concept: The sum of money Kate had initially is 900. The initial amount of money that Kate and Nora have will be given variables. We will create an equation using the provided ratio. To obtain the second equation, we will then change Kate and Nora's respective financial situations. We also know how much money each of them has in their modified sum. We can determine the starting amount of money that both Kate and Nora had by using this ratio.
Let's assume that Kate starts out with x rupees and Nora starts out with y rupees. Kate has three times as much money as Nora does, or a 3:5 ratio. As a result, we have

\frac{x}{y}

=

\frac{3}{5}

.
So, x=

\frac{3 y}{5}

. Kate now receives 150 rupees from Nora. The updated amounts of money that Kate and Nora each have are x+150 Rs. and y-150 Rs., respectively. Following Nora's gift of Rs. 150 to Kate, the ratio of Kate's to Nora's wealth is 7:9. As a result, the new ratio leads to the equation

\frac{x + 150}{y - 150}

=

\frac{7}{9}

.
This equation is simplified to give us

9 (x + 150) = 7 (y - 150) .

9 x + 1350 = 7 y - 1050

The terms will now be rearranged to create the following linear equation with two variables :

9 x - 7 y = - 2400

When x = \frac{3 y}{5} is substituted in the equation above, we get 9 (\frac{3 y}{5}) - 7 y = - 2400 .

To simplify the equation above, write it as follows :

\frac{27 y}{5} - 7 y = - 2400

\frac{27 y - 35 y}{5} = - 2400

- 8 y = - 2400 \times 5

When we solve for y, we obtain y=

\frac{- 2400 \times 5}{- 8}

y=300

\times

5
y=1500.
Knowing that x=

\frac{3 y}{5}

, The value of x will be as follows when y=1500 is substituted in this equation: x=

\frac{3 \times 1500}{5}

x=3

\times

300
x=900
Hence, the correct answer is Rs.900

Kate and Nora each have a sum of money. The ratio of the amount of money Kate has to that of Nora is 3:53:5. After Nora gives Rs. 150 to Kate, the ratio of the amount of money Kate has to that of Nora becomes 7:97:9 then the sum of money Kate had initially is ____.

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