A certain sum is to be divided between A, B, and C in the ratio 2 ∶ 5 ∶ 7. But by mistake, it was divided in the ratio 5 ∶ 2 ∶ 7. Thus B loses by Rs. 150. Find the sum.
Rs. 700
Solving the Sum Division Ratio Problem
This problem involves dividing a total sum of money among three people, A, B, and C, based on given ratios. We are given two scenarios: the correct division ratio and a mistaken division ratio. The difference in B's share between these two scenarios is provided, and we need to find the total sum.
Understanding the Ratios
The ratios represent how the total sum is split into parts. The total number of parts is the sum of the numbers in the ratio.
- Correct Ratio: A ∶ B ∶ C = 2 ∶ 5 ∶ 7
- Mistake Ratio: A ∶ B ∶ C = 5 ∶ 2 ∶ 7
Calculating Total Parts in Each Ratio
Let's find the total number of parts for both ratios:
- Correct Ratio: Total parts = \(2 + 5 + 7 = 14\)
- Mistake Ratio: Total parts = \(5 + 2 + 7 = 14\)
Interestingly, the total number of parts remains the same in both ratios, which simplifies the calculation.
Representing the Shares
Let the total sum be \(S\). The share of each person can be represented as a fraction of the total sum, where the numerator is their part in the ratio and the denominator is the total number of parts.
| Scenario | A's Share | B's Share | C's Share |
|---|---|---|---|
| Correct Division (Ratio 2:5:7) | \(\frac{2}{14}S\) | \(\frac{5}{14}S\) | \(\frac{7}{14}S\) |
| Mistaken Division (Ratio 5:2:7) | \(\frac{5}{14}S\) | \(\frac{2}{14}S\) | \(\frac{7}{14}S\) |
Setting Up the Equation Based on B's Loss
We are told that B loses Rs. 150 due to the mistake. This means B's share in the correct division was Rs. 150 more than B's share in the mistaken division.
Difference in B's share = Correct B's share - Mistaken B's share
\(\frac{5}{14}S - \frac{2}{14}S = 150\)
Solving for the Total Sum (S)
Now, we solve the equation to find the value of \(S\):
\(\left(\frac{5 - 2}{14}\right)S = 150\)
\(\frac{3}{14}S = 150\)
To find \(S\), multiply both sides by \(\frac{14}{3}\):
\(S = 150 \times \frac{14}{3}\)
\(S = \frac{150 \times 14}{3}\)
We can simplify the calculation by dividing 150 by 3:
\(S = 50 \times 14\)
\(S = 700\)
The total sum is Rs. 700.
Verification
Let's check if B's loss is indeed Rs. 150 with the total sum of Rs. 700.
- Correct B's share: \(\frac{5}{14} \times 700 = 5 \times \frac{700}{14} = 5 \times 50 = 250\)
- Mistaken B's share: \(\frac{2}{14} \times 700 = 2 \times \frac{700}{14} = 2 \times 50 = 100\)
Difference in B's share = Rs. 250 - Rs. 100 = Rs. 150.
This matches the information given in the problem, confirming our calculated sum is correct.
Revision Table: Ratio and Sum Problems
| Concept | Description | How it applies here |
|---|---|---|
| Ratio | A comparison of two or more quantities indicating their relative sizes. E.g., a:b:c. | Given ratios for sum division (2:5:7 and 5:2:7). |
| Sum of Ratio Parts | Adding the numbers in a ratio to find the total number of relative units. | Used to find the denominator for calculating fractional shares (2+5+7=14). |
| Fractional Share | Representing a part of the total as a fraction based on the ratio (\(\frac{\text{Individual Part}}{\text{Total Parts}}\)). | Calculated B's share as \(\frac{5}{14}S\) and \(\frac{2}{14}S\). |
| Difference in Shares | The monetary difference between a person's share in two different scenarios. | Used to set up the main equation (\(\frac{5}{14}S - \frac{2}{14}S = 150\)). |
Additional Information: Ratio and Proportion Basics
Ratio and proportion are fundamental concepts in mathematics used to compare quantities. A ratio expresses the relationship between quantities, while proportion states that two ratios are equal.
- Ratio: Written as a:b or a/b. It shows how many times one quantity is contained within another. Ratios can be simplified like fractions.
- Proportion: An equality between two ratios, e.g., a:b = c:d or a/b = c/d. If four quantities are in proportion, the product of the extremes (a and d) equals the product of the means (b and c).
- Dividing a Sum in a Given Ratio: If a sum \(S\) is to be divided among A, B, C in the ratio a:b:c, the total number of parts is \(a+b+c\). A's share = \(\frac{a}{a+b+c} \times S\), B's share = \(\frac{b}{a+b+c} \times S\), and C's share = \(\frac{c}{a+b+c} \times S\).
Understanding these basics is key to solving problems involving distribution of quantities based on ratios, like the sum division problem we just solved.
Important Questions from Compound Ratios
- What is the compound ratio of $45 : 75$, $3 : 5$, $51 : 68$ and $256 : 81$?
If A : B = 2 : 5, B : C = 4 : 3 and C : D = 2 : 1, then what is value of A : C : D?
The following ratios are given:
\(a : b = 2 : 3\)
\(b : c = 4 : 5\)
\(c : d = 3 : 2\)
\(d : e = 5 : 4\)
What is the value of the ratio a : e?
If (x + 2) is a common factor of x 2+ ax + b and x 2+ bx + a, then the ratio of a ∶ b is equal to
Rs. 2420 were divided among A, B, C so that A : B = 5 : 4 and B : C = 9 : 10 then C gets:
