Understanding the Problem: Dividing a Sum Based on Ratios

The question asks us to divide a total sum of Rs. 1,180 among three individuals, A, B, and C, based on a specific condition involving their shares. The condition given is that two times A's share, five times B's share, and seven times C's share are all equal.

We need to find the individual share of A based on this division.

Setting Up the Equations for Share Division

Let A's share be represented by $S_A$, B's share by $S_B$, and C's share by $S_C$.

The total sum is Rs. 1,180. So, the first equation based on the total amount is:

$S_A + S_B + S_C = 1180$

The condition given is that 2 times A's share, 5 times B's share, and 7 times C's share are equal. This can be written as:

$2 \times S_A = 5 \times S_B = 7 \times S_C$

Solving for the Shares Using the Equal Ratio Condition

Let's assume that the equal value from the condition $2 S_A = 5 S_B = 7 S_C$ is equal to a constant, say $k$.

So, we have:

• $2 S_A = k \implies S_A = \frac{k}{2}$
• $5 S_B = k \implies S_B = \frac{k}{5}$
• $7 S_C = k \implies S_C = \frac{k}{7}$

Now, we can substitute these expressions for $S_A$, $S_B$, and $S_C$ into the equation for the total sum:

$\frac{k}{2} + \frac{k}{5} + \frac{k}{7} = 1180$

To solve for $k$, we need to find a common denominator for the fractions. The least common multiple (LCM) of 2, 5, and 7 is $2 \times 5 \times 7 = 70$.

Multiply each term by the appropriate factor to get the denominator 70:

$\frac{k}{2} \times \frac{35}{35} + \frac{k}{5} \times \frac{14}{14} + \frac{k}{7} \times \frac{10}{10} = 1180$

$\frac{35k}{70} + \frac{14k}{70} + \frac{10k}{70} = 1180$

Combine the terms on the left side:

$\frac{35k + 14k + 10k}{70} = 1180$

$\frac{59k}{70} = 1180$

Now, solve for $k$ by multiplying both sides by 70 and dividing by 59:

$59k = 1180 \times 70$

$k = \frac{1180 \times 70}{59}$

We can simplify the calculation by noticing that 1180 is a multiple of 59 ($1180 \div 59 = 20$).

$k = 20 \times 70$

$k = 1400$

Calculating A's Share

Now that we have the value of $k$, we can find the individual shares using the expressions we derived earlier:

• A's share, $S_A = \frac{k}{2} = \frac{1400}{2} = 700$
• B's share, $S_B = \frac{k}{5} = \frac{1400}{5} = 280$
• C's share, $S_C = \frac{k}{7} = \frac{1400}{7} = 200$

The question asks specifically for A's share, which is Rs. 700.

Verification of Shares and Conditions

Let's check if the calculated shares meet the conditions:

• Total sum: $S_A + S_B + S_C = 700 + 280 + 200 = 1180$. This matches the given total sum.
• Equal shares condition:
  • $2 \times S_A = 2 \times 700 = 1400$
  • $5 \times S_B = 5 \times 280 = 1400$
  • $7 \times S_C = 7 \times 200 = 1400$
  The condition $2 S_A = 5 S_B = 7 S_C$ holds true as all three values are equal to 1400.

The calculated shares are correct based on the problem statement.

Summary of Shares

A's share is Rs. 700.

Revision Table: Key Concepts for Share Division

Additional Information: Ratios from Equal Multiples

When you have a condition like $2S_A = 5S_B = 7S_C$, you can find the ratio of the shares $S_A : S_B : S_C$.

If $2S_A = 5S_B$, then $S_A/S_B = 5/2$, so $S_A : S_B = 5 : 2$.

If $5S_B = 7S_C$, then $S_B/S_C = 7/5$, so $S_B : S_C = 7 : 5$.

To combine these ratios $S_A : S_B = 5 : 2$ and $S_B : S_C = 7 : 5$, we need to make the $S_B$ term common. We can multiply the first ratio by 7 and the second ratio by 2:

• $S_A : S_B = (5 \times 7) : (2 \times 7) = 35 : 14$
• $S_B : S_C = (7 \times 2) : (5 \times 2) = 14 : 10$

Now, we have a combined ratio $S_A : S_B : S_C = 35 : 14 : 10$.

The total number of ratio parts is $35 + 14 + 10 = 59$.

The total sum is Rs. 1180. Each ratio part is worth $\frac{1180}{59} = 20$.

A's share is $35$ parts, so $S_A = 35 \times 20 = 700$.

B's share is $14$ parts, so $S_B = 14 \times 20 = 280$.

C's share is $10$ parts, so $S_C = 10 \times 20 = 200$.

This ratio method gives the same result and is another common way to solve such problems.