The incomes of A and B are in the ratio 5:3. The expenditure of A, B, and C are in the ratio 8:5:2. If C spends ₹2000 and B saves ₹700, then what amount does A save?
₹1000
Understanding Ratio Problems: Income, Expenditure, and Saving
This problem involves ratios of incomes and expenditures for different individuals and requires us to find the saving of one individual given information about others. We will use the fundamental relationship between income, expenditure, and saving.
The relationship is:
\(\text{Saving} = \text{Income} - \text{Expenditure}\)
We are given the following information:
- Income ratio of A and B: 5:3
- Expenditure ratio of A, B, and C: 8:5:2
- C's expenditure: ₹2000
- B's saving: ₹700
Our goal is to find the amount A saves.
Step-by-Step Calculation of Savings
1. Determine Actual Expenditures
The expenditure ratio of A, B, and C is 8:5:2. We know that C spends ₹2000, which corresponds to 2 parts of the expenditure ratio.
Let the value of one part in the expenditure ratio be \(k\).
\(2k = ₹2000\)
\(k = \frac{₹2000}{2} = ₹1000\)
Now we can find the actual expenditures of A and B:
- A's expenditure = 8 parts \(= 8 \times ₹1000 = ₹8000\)
- B's expenditure = 5 parts \(= 5 \times ₹1000 = ₹5000\)
2. Calculate B's Income
We know B's saving and B's expenditure. Using the formula \(\text{Income} = \text{Expenditure} + \text{Saving}\), we can find B's income.
B's income = B's expenditure + B's saving
B's income = \(₹5000 + ₹700 = ₹5700\)
3. Determine A's Income
The income ratio of A and B is 5:3. This means that if B's income is 3 parts, A's income is 5 parts. We know B's actual income is ₹5700.
Let the value of one part in the income ratio be \(m\).
\(3m = ₹5700\)
\(m = \frac{₹5700}{3} = ₹1900\)
Now we can find A's actual income:
A's income = 5 parts \(= 5 \times ₹1900 = ₹9500\)
4. Calculate A's Saving
We now have A's income and A's expenditure. We can find A's saving using the formula \(\text{Saving} = \text{Income} - \text{Expenditure}\).
A's saving = A's income - A's expenditure
A's saving = \(₹9500 - ₹8000 = ₹1500\)
Therefore, A saves ₹1500.
| Person | Ratio (Income) | Ratio (Expenditure) | Actual Income | Actual Expenditure | Actual Saving |
|---|---|---|---|---|---|
| A | 5 | 8 | \(5 \times ₹1900 = ₹9500\) | \(8 \times ₹1000 = ₹8000\) | \(₹9500 - ₹8000 = ₹1500\) |
| B | 3 | 5 | \(3 \times ₹1900 = ₹5700\) | \(5 \times ₹1000 = ₹5000\) | \(₹5700 - ₹5000 = ₹700\) (Given) |
| C | - | 2 | - | \(2 \times ₹1000 = ₹2000\) (Given) | - |
Revision Table: Key Concepts in Ratio Problems
| Concept | Description | Formula/Example |
|---|---|---|
| Ratio | A comparison of two or more quantities. Represents parts of a whole or proportions between quantities. | Income ratio A:B = 5:3 means A's income is 5 parts for every 3 parts of B's income. |
| Income | The total money received by an individual or entity. | Often given in total value or as a ratio. |
| Expenditure | The total money spent by an individual or entity. | Often given in total value or as a ratio. |
| Saving | The portion of income that is not spent. Income minus expenditure. | \( \text{Saving} = \text{Income} - \text{Expenditure} \) |
| Finding Actual Values from Ratios | If a ratio \(a:b\) corresponds to total value \(V\), the values are \( \frac{a}{a+b} \times V \) and \( \frac{b}{a+b} \times V \). If one part of the ratio is known (\(a\) corresponds to \(X\)), find the unit value (\(X/a\)) and multiply by other ratio parts. | If expenditure ratio 8:5:2 and C's expenditure (2 parts) is ₹2000, 1 part = ₹1000. A's expenditure (8 parts) = \(8 \times 1000\). |
Additional Information on Income and Expenditure
Understanding the relationship between income, expenditure, and saving is crucial in personal finance and economics. These ratio problems help build skills in proportional reasoning and algebraic thinking.
- Income can come from various sources like salary, business profit, investments, etc.
- Expenditure includes spending on necessities (food, rent) and discretionary items (entertainment, luxuries).
- Saving is essential for future goals, emergencies, and investments. A positive saving means income is greater than expenditure. A negative saving (or dissaving) means expenditure is greater than income.
- Problems like these often involve using information about one person or quantity to find the value of a unit in a ratio and then using that unit value to find other unknown quantities.
- Always ensure you are using the correct ratio (income or expenditure) for the corresponding values.
Important Questions from Ratio and Proportion
Choose the correct option for the missing term: 27 : 18 :: 102 : ?
Find the missing term in the given pattern: 12 : 36 :: 15 : ?
Divide 243 kg weight into three parts such that half of the first part, one-third of the second part, and one-fourth of the third part are equal.
If 5A = 4B, 7B = 3C, and 2C = 7D, then A:D is:
The ratio between two numbers is 2:3. If each number is increased by 2, then the ratio becomes 3:4. Find the sum of the original numbers.
