The incomes of A, B and C are in the ratio 7 ∶ 9 ∶ 12 and their expenditures are in the ratio 8 ∶ 9 ∶ 15. If A's saving is one-fourth of his income, then the ratio of savings of A, B and C is
56 ∶ 99 ∶ 69
Calculate Ratio of Savings from Income and Expenditure Ratios
This problem involves ratios of incomes and expenditures for three individuals, A, B, and C, and a condition about A's savings. We need to find the ratio of their savings.
Understanding the Given Ratios
We are given the following information:
- Ratio of incomes of A, B, and C = \(7 \ratio 9 \ratio 12\).
- Ratio of expenditures of A, B, and C = \(8 \ratio 9 \ratio 15\).
- A's saving is one-fourth of his income.
Setting up Variables for Income and Expenditure
Let's represent the incomes and expenditures using variables:
- Let the incomes of A, B, and C be \(7x\), \(9x\), and \(12x\) respectively, where \(x\) is a common factor.
- Let the expenditures of A, B, and C be \(8y\), \(9y\), and \(15y\) respectively, where \(y\) is another common factor.
Saving is calculated as Income minus Expenditure.
- A's Saving = A's Income - A's Expenditure = \(7x - 8y\)
- B's Saving = B's Income - B's Expenditure = \(9x - 9y\)
- C's Saving = C's Income - C's Expenditure = \(12x - 15y\)
Using the Condition on A's Saving
We are told that A's saving is one-fourth of his income. A's income is \(7x\).
A's Saving = \(\frac{1}{4} \times (\text{A's Income})\)
\(7x - 8y = \frac{1}{4} \times (7x)\)
\(7x - 8y = \frac{7x}{4}\)
Finding the Relationship between \(x\) and \(y\)
Now, let's solve the equation \(7x - 8y = \frac{7x}{4}\) to find a relationship between \(x\) and \(y\).
To eliminate the fraction, multiply the entire equation by 4:
\(4 \times (7x - 8y) = 4 \times \left(\frac{7x}{4}\right)\)
\(28x - 32y = 7x\)
Rearrange the terms to group \(x\) terms on one side and \(y\) terms on the other:
\(28x - 7x = 32y\)
\(21x = 32y\)
From this, we can express \(y\) in terms of \(x\): \(y = \frac{21}{32}x\). This relationship is crucial for calculating the savings.
Calculating Savings for A, B, and C
Now substitute the value of \(y\) in terms of \(x\) into the saving expressions for A, B, and C.
- A's Saving:
A's Saving = \(7x - 8y = 7x - 8\left(\frac{21}{32}x\right)\)
A's Saving = \(7x - \frac{8 \times 21}{32}x = 7x - \frac{1 \times 21}{4}x\)
A's Saving = \(7x - \frac{21}{4}x = \left(7 - \frac{21}{4}\right)x = \left(\frac{28 - 21}{4}\right)x = \frac{7}{4}x\)
(Note: This is \(\frac{1}{4}\) of \(7x\), confirming our calculation for A's saving is consistent with the given condition).
- B's Saving:
B's Saving = \(9x - 9y = 9x - 9\left(\frac{21}{32}x\right)\)
B's Saving = \(9x - \frac{9 \times 21}{32}x = 9x - \frac{189}{32}x\)
B's Saving = \(\left(9 - \frac{189}{32}\right)x = \left(\frac{9 \times 32 - 189}{32}\right)x = \left(\frac{288 - 189}{32}\right)x = \frac{99}{32}x\)
- C's Saving:
C's Saving = \(12x - 15y = 12x - 15\left(\frac{21}{32}x\right)\)
C's Saving = \(12x - \frac{15 \times 21}{32}x = 12x - \frac{315}{32}x\)
C's Saving = \(\left(12 - \frac{315}{32}\right)x = \left(\frac{12 \times 32 - 315}{32}\right)x = \left(\frac{384 - 315}{32}\right)x = \frac{69}{32}x\)
Finding the Ratio of Savings
The ratio of savings of A, B, and C is (A's Saving) : (B's Saving) : (C's Saving).
Ratio = \(\frac{7}{4}x : \frac{99}{32}x : \frac{69}{32}x\)
We can cancel out the common factor \(x\) (assuming \(x \ne 0\), which must be true for them to have income).
Ratio = \(\frac{7}{4} : \frac{99}{32} : \frac{69}{32}\)
To get rid of the fractions and express the ratio in whole numbers, we multiply each part of the ratio by the least common multiple (LCM) of the denominators (4 and 32), which is 32.
Ratio = \(32 \times \frac{7}{4} : 32 \times \frac{99}{32} : 32 \times \frac{69}{32}\)
Ratio = \((8 \times 7) : (1 \times 99) : (1 \times 69)\)
Ratio = \(56 : 99 : 69\)
So, the ratio of savings of A, B, and C is \(56 \ratio 99 \ratio 69\).
Summary of Calculations
| Individual | Income (Ratio) | Expenditure (Ratio) | Income (Variable) | Expenditure (Variable) | Saving (Income - Expenditure) | Saving (in terms of \(x\), using \(y = \frac{21}{32}x\)) |
|---|---|---|---|---|---|---|
| A | 7 | 8 | \(7x\) | \(8y\) | \(7x - 8y\) | \(\frac{7}{4}x\) |
| B | 9 | 9 | \(9x\) | \(9y\) | \(9x - 9y\) | \(\frac{99}{32}x\) |
| C | 12 | 15 | \(12x\) | \(15y\) | \(12x - 15y\) | \(\frac{69}{32}x\) |
Ratio of Savings A:B:C = \(\frac{7}{4}x : \frac{99}{32}x : \frac{69}{32}x\) = \(56 : 99 : 69\) (after multiplying by 32).
Revision Table: Income, Expenditure, and Savings Ratio
| Concept | Definition/Formula | Application in Problem |
|---|---|---|
| Income Ratio | Proportion of incomes among individuals. | A:B:C = 7:9:12, implies incomes are \(7x, 9x, 12x\). |
| Expenditure Ratio | Proportion of expenditures among individuals. | A:B:C = 8:9:15, implies expenditures are \(8y, 9y, 15y\). |
| Saving | Income minus Expenditure. | Saving = Income - Expenditure. |
| Ratio of Savings | Proportion of savings among individuals. | Goal: Find A's Saving : B's Saving : C's Saving. |
| Connecting Ratios | Use given conditions (like A's saving) to find relationship between ratio variables (\(x\) and \(y\)). | \(7x - 8y = \frac{1}{4}(7x)\), led to \(21x = 32y\). |
Additional Information: Ratio and Proportion Concepts
A ratio is a comparison of two or more quantities of the same kind. It shows how much of one quantity is in another. For example, the ratio \(7 \ratio 9\) means that for every 7 units of the first quantity, there are 9 units of the second.
A proportion is an equality of two ratios. For instance, \(\frac{a}{b} = \frac{c}{d}\).
When dealing with ratios involving multiple quantities like \(a \ratio b \ratio c\), we can represent the actual quantities as \(ka, kb, kc\) for some non-zero constant \(k\). In this problem, we used different constants (\(x\) for income and \(y\) for expenditure) because the income ratio and expenditure ratio are independent unless a specific relationship is given (like the saving condition).
Savings problems often require setting up equations based on the formula Saving = Income - Expenditure and using given conditions to solve for unknown relationships between variables.
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