Question

The ratio of expenditure and savings is \(3: 2\) If the income increases by \(15 \%\) and the savings increases by \(6 \%\), then by how much per cent should his expenditure increases? (a) 25 (b) 21 (c) 12 (d) 24

Short answer

Answer: The expenditure should increase by 21%.

Step-by-step solution

Set up a ratio equation

Given the initial ratio of expenditure and savings is 3:2, let's denote the original expenditure as \(3x\) and savings as \(2x\).

Calculate the percentage increase in income and savings

If the income increases by 15%, the new total income will be \(1.15(3x + 2x)\). Similarly, if the savings increases by 6%, the new savings will be \(1.06(2x)\).

Calculate the new income based on increased savings

Since the income is the sum of expenditure and savings, we can calculate the income after the increase in savings as \(3x + 1.06(2x)\).

Set up an equation for new expenditure

Let the new expenditure be denoted as \(y\). Then we can write the equation for the new total income as \(1.15(3x + 2x) = y + 1.06(2x)\).

Solve the equation for y

Simplify and solve for \(y\): \begin{align*} 1.15(5x) &= y + 1.06(2x)\\ (1.15 \cdot 5) x &= y + (1.06 \cdot 2)x\\ 5.75x &= y + 2.12x\\ 3.63x &= y \end{align*}

Calculate the percentage increase in expenditure

The initial expenditure was \(3x\), and the final expenditure is \(3.63x\), we can find the percentage increase as follows: \[\frac{3.63x - 3x}{3x} \times 100\] This simplifies to: \[\frac{0.63x}{3x} \times 100\] Cancel out the \(x\) in the numerator and the denominator: \[\frac{0.63}{3} \times 100\]

Compute the final result

Calculate the result and find the percentage increase in expenditure: \[\frac{0.63}{3} \times 100 \approx 21 \% \] So the correct answer is (b) 21. The expenditure should increase by 21%.

Key concepts

Percentage Increase

Understanding percentage increase is crucial for analyzing how quantities change over time. A percentage increase indicates how much a value has grown, relative to its original amount. For instance, if a person’s expenditure grows from $50 to $60, the increase is $10. To find this in percentage terms, use the formula:

• Subtract the original amount from the new amount: $60 - $50 = $10
• Divide the increase by the original amount: $10 / $50 = 0.2
• Multiply by 100 to convert to a percentage: 0.2 x 100 = 20%

This means that the expenditure increased by 20%. In the context of the exercise, calculating percentage increase helps to determine how expenditure should adjust when income and savings change.

Income Calculation

Calculating income is about understanding how different elements like savings and expenditures combine. When income changes, it impacts how much you have for spending and setting aside savings. If your income increases by a certain percentage, you need to recalculate your total income.
Here's how to handle it:

• Start with your initial income (sum of expenditures and savings).
• Apply the percentage increase to find the new income: New Income = Initial Income x (1 + Percentage Increase).

For example, if an income of $500 increases by 15%, the new income becomes 500 x 1.15 = $575. In problems like the exercise, calculating changes in income is essential for understanding how expenditure and savings need to be balanced afterward.

Budget Allocations

Budget allocations require balancing between different financial needs like expenditures and savings. The key is to maintain a ratio that reflects your priorities. In the exercise, the initial ratio was 3:2, meaning for every 3 units of expenditure, 2 units are saved.
When changes occur, such as increases in income or savings, maintaining or adjusting this ratio is important. Here's how:

• Recalculate your expenditures and savings based on the new income.
• Ensure that the total (new income) equals the sum of the revised expenditures and savings.

By effectively managing budget allocations after changes in income or expenditure, you can maintain financial stability and meet your budgeting goals.