A certain sum of money was distributed among Darshana, Swati and Nivriti. Nivriti has Rs. 539 with her. If the ratio of the money distributed among Darshana, Swati and Nivriti is 5 : 6 : 7, what is the total sum of money that was distributed?
Rs. 1,386
Solving Money Distribution Ratio Problems
This problem asks us to find the total sum of money distributed among three people, Darshana, Swati, and Nivriti, given the ratio of their shares and the exact amount received by one of them, Nivriti.
The ratio of the money distributed among Darshana, Swati, and Nivriti is given as 5 : 6 : 7.
This ratio means that for every 5 parts Darshana receives, Swati receives 6 parts, and Nivriti receives 7 parts. We can represent these parts using a common multiplier, let's call it \(x\).
- Darshana's share = \(5x\)
- Swati's share = \(6x\)
- Nivriti's share = \(7x\)
We are given that Nivriti has Rs. 539. Using our representation, Nivriti's share is \(7x\).
So, we can set up the following equation:
$$7x = 539$$
To find the value of \(x\), we need to divide Nivriti's amount by her share in the ratio:
$$x = \frac{539}{7}$$
Now, we calculate the value of \(x\):
$$x = 77$$
The value of the common multiplier \(x\) is 77.
The total sum of money distributed is the sum of the shares of Darshana, Swati, and Nivriti.
Total sum = Darshana's share + Swati's share + Nivriti's share
Total sum = \(5x + 6x + 7x\)
Total sum = \((5 + 6 + 7)x\)
Total sum = \(18x\)
Now, we substitute the value of \(x\) (which is 77) into the expression for the total sum:
$$\text{Total sum} = 18 \times 77$$
Let's perform the multiplication:
$$18 \times 77 = 18 \times (70 + 7)$$
$$18 \times 77 = (18 \times 70) + (18 \times 7)$$
$$18 \times 70 = 1260$$
$$18 \times 7 = 126$$
$$1260 + 126 = 1386$$
So, the total sum of money that was distributed is Rs. 1386.
The final answer is Rs. 1,386.
| Person | Ratio Part | Share (\(x=77\)) |
|---|---|---|
| Darshana | 5 | \(5 \times 77 = 385\) |
| Swati | 6 | \(6 \times 77 = 462\) |
| Nivriti | 7 | \(7 \times 77 = 539\) |
| Total | 18 | \(385 + 462 + 539 = 1386\) |
Revision Table: Key Concepts
| Concept | Explanation | Application in Problem |
|---|---|---|
| Ratio | A way to compare quantities; shows relative sizes. | The distribution ratio 5:6:7 guides how money is split. |
| Common Multiplier (\(x\)) | A variable used to represent the actual value of one 'part' in a ratio. | We used \(x\) to find the real amounts (5x, 6x, 7x). |
| Solving for \(x\) | Equating the known quantity to its ratio representation to find the value of \(x\). | \(7x = 539\) helped us find \(x=77\). |
| Total Sum Calculation | Adding all the individual shares or multiplying the total ratio parts by \(x\). | Total sum = \(18x = 18 \times 77\). |
Additional Information on Ratios and Distribution
Ratios are fundamental in mathematics and are used in many real-life situations, such as mixing ingredients, scaling recipes, or distributing resources like money or assets.
When a quantity is distributed in a ratio \(a : b : c\), it means the total quantity is divided into \(a+b+c\) equal parts. The first person gets \(a\) parts, the second gets \(b\) parts, and the third gets \(c\) parts.
If you know the value of one person's share and their corresponding part in the ratio, you can always find the value of one 'part' (our \(x\)). Once you know the value of one part, you can calculate the share of anyone else or the total sum by multiplying the total ratio parts by the value of one part.
For example, if the ratio was 2:3 and the first person received 10 units, then 2 parts = 10 units, so 1 part = 5 units. The second person would receive 3 parts = \(3 \times 5 = 15\) units. The total distributed would be \(2+3=5\) parts = \(5 \times 5 = 25\) units.
Understanding ratios and how to work with a common multiplier is key to solving these types of distribution problems efficiently.
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