Question

Mr. Seth inherits 2505 gold coins and divides them among his three sons: Brij,Purab and Mohan in a certain ratio.Out of the total coins received by each of them,Brij sells 30 coins,Purab donates his 30 coins and Mohan looses 25 coins.Now,the ratio of gold coins with them is 46: 41: 34 respectively.How many coins did Purab receive from his father?

• A. 705
• B. 950
• C. 800
• D. 850

Answer 1

The Correct Option is D

To solve this problem, let's follow these steps:

Let the initial number of coins received by Brij, Purab, and Mohan be \(B\), \(P\), and \(M\) respectively. Initially, we know that after selling, donating, and losing coins, the ratio of coins left with them is \(46:41:34\).

After events: 

• Brij sells 30 coins, so he has \(B - 30\) coins left.
• Purab donates 30 coins, so he has \(P - 30\) coins left.
• Mohan loses 25 coins, so he has \(M - 25\) coins left.

The ratio of remaining coins is given as 46:41:34. Therefore, we have:

• \(\frac{B - 30}{P - 30} = \frac{46}{41}\)
• \(\frac{P - 30}{M - 25} = \frac{41}{34}\)
• \(\frac{B - 30}{M - 25} = \frac{46}{34}\)

Another constraint given is that the total number of coins inherited is 2505:

1. \(B + P + M = 2505\)

First, consider the equation \(\frac{P - 30}{M - 25} = \frac{41}{34}\):

This gives us:

1. \(34(P - 30) = 41(M - 25)\)

Simplifying, we have:

1. \(34P - 1020 = 41M - 1025\)

Rearranging terms, we have:

1. \(34P = 41M - 5\) ... (i)

Consider equation \(\frac{B - 30}{P - 30} = \frac{46}{41}\):

This gives us:

1. \(41(B - 30) = 46(P - 30)\)

After simplification:

1. \(41B - 1230 = 46P - 1380\)

Rearranging terms, we have:

1. \(41B = 46P - 150\) ... (ii)

We solve equations (i), (ii) along with \(B + P + M = 2505\):

• From (i): \(34P = 41M - 5 \Rightarrow 41M = 34P + 5\)
• Substitute this into \(B + P + M = 2505\):
• \(B + P + \frac{34P + 5}{41} = 2505\)
• Simplifying allows us to find values compared to equal simplifications aligning with ratio values.

Using ratios and solving above expression, we find Purab's coins:

1. \(P = 850\).

 

Purab received 850 coins from his father, which corresponds to the correct option.

Answer 2

Let the number of coins received by Brij, Purab, and Mohan be \( x \), \( y \), and \( z \) respectively. According to the problem, the total number of coins is 2505:

\[ x + y + z = 2505 \]

After Brij sells 30 coins, Purab donates 30 coins, and Mohan loses 25 coins, the remaining coins with them are \( x - 30 \), \( y - 30 \), and \( z - 25 \), and the ratio of their coins is given as:

\[ \frac{x - 30}{y - 30} = \frac{46}{41}, \quad \frac{y - 30}{z - 25} = \frac{41}{34} \]

Step 1: Solving the First Ratio Equation

From the first ratio:

\[ \frac{x - 30}{y - 30} = \frac{46}{41} \]

Cross-multiply:

\[ 41(x - 30) = 46(y - 30) \]

Simplifying:

\[ 41x - 1230 = 46y - 1380 \]

\[ 41x - 46y = -150 \]

This is equation (1).

Step 2: Solving the Second Ratio Equation

From the second ratio:

\[ \frac{y - 30}{z - 25} = \frac{41}{34} \]

Cross-multiply:

\[ 34(y - 30) = 41(z - 25) \]

Simplifying:

\[ 34y - 1020 = 41z - 1025 \]

\[ 34y - 41z = -5 \]

This is equation (2).

Step 3: Solving the System of Equations

We now solve the system of equations:

\[ 41x - 46y = -150 \quad \text{(1)} \]

\[ 34y - 41z = -5 \quad \text{(2)} \]

From equation (1), solve for \( x \) in terms of \( y \):

\[ x = \frac{46y - 150}{41} \]

Substitute this value of \( x \) into the total number of coins equation:

\[ \frac{46y - 150}{41} + y + z = 2505 \]

Multiply through by 41 to eliminate the denominator:

\[ 46y - 150 + 41y + 41z = 2505 \times 41 \]

Simplify:

\[ 87y + 41z = 102855 \]

Use equation (2) to express \( z \) in terms of \( y \):

\[ 41z = 34y + 5 \quad \Rightarrow \quad z = \frac{34y + 5}{41} \]

Substitute \( z \) into the equation \( 87y + 41z = 102855 \):

\[ 87y + 41\left(\frac{34y + 5}{41}\right) = 102855 \]

Simplify:

\[ 87y + 34y + 5 = 102855 \]

\[ 121y + 5 = 102855 \]

Subtract 5 from both sides:

\[ 121y = 102850 \]

Solve for \( y \):

\[ y = \frac{102850}{121} = 850 \]

Thus, Purab received 850 coins from his father.