A saves 35% of his income. If his income increases by 20.1% and his expenditure increases by 20%, then by what percentage do his savings increase or decrease? (correct to one decimal place)
20.3% of increase
Understanding Income, Savings, and Expenditure
This problem involves calculating the percentage change in savings when there are changes in income and expenditure. The fundamental relationship is:
\$\$ \text{Income} = \text{Expenditure} + \text{Savings} \$\$
We are given the initial percentage of savings and the percentage increase in both income and expenditure. We need to find the percentage increase or decrease in savings.
Step-by-Step Calculation
1. Assume an Initial Income
To make the calculations easier, let's assume a starting income. A common practice is to assume the initial income is \$100.
\$\$ \text{Initial Income} = \$100 \$\$
2. Calculate Initial Savings and Expenditure
A saves 35% of his income. So, the initial savings are:
\$\$ \text{Initial Savings} = 35\% \text{ of } \$100 = \frac{35}{100} \times \$100 = \$35 \$\$
Using the relationship \(\text{Income} = \text{Expenditure} + \text{Savings}\), we can find the initial expenditure:
\$\$ \text{Initial Expenditure} = \text{Initial Income} - \text{Initial Savings} = \$100 - \$35 = \$65 \$\$
3. Calculate the New Income
The income increases by 20.1%. We calculate the new income:
\$\$ \text{Increase in Income} = 20.1\% \text{ of } \$100 = \frac{20.1}{100} \times \$100 = \$20.1 \$\$
\$\$ \text{New Income} = \text{Initial Income} + \text{Increase in Income} = \$100 + \$20.1 = \$120.1 \$\$
4. Calculate the New Expenditure
The expenditure increases by 20%. We calculate the new expenditure:
\$\$ \text{Increase in Expenditure} = 20\% \text{ of } \$65 = \frac{20}{100} \times \$65 = 0.20 \times \$65 = \$13 \$\$
\$\$ \text{New Expenditure} = \text{Initial Expenditure} + \text{Increase in Expenditure} = \$65 + \$13 = \$78 \$\$
5. Calculate the New Savings
Using the fundamental relationship with the new values:
\$\$ \text{New Savings} = \text{New Income} - \text{New Expenditure} \$\$
\$\$ \text{New Savings} = \$120.1 - \$78 = \$42.1 \$\$
6. Calculate the Change in Savings
The change in savings is the difference between the new savings and the initial savings:
\$\$ \text{Change in Savings} = \text{New Savings} - \text{Initial Savings} = \$42.1 - \$35 = \$7.1 \$\$
Since the change is positive (\$7.1), the savings have increased.
7. Calculate the Percentage Change in Savings
The percentage change in savings is calculated with respect to the initial savings:
\$\$ \text{Percentage Change in Savings} = \frac{\text{Change in Savings}}{\text{Initial Savings}} \times 100 \$\$
\$\$ \text{Percentage Change in Savings} = \frac{\$7.1}{\$35} \times 100 \$\$
\$\$ \text{Percentage Change in Savings} = \frac{7.1}{35} \times 100 \approx 0.202857 \times 100 \approx 20.2857 \$\$
8. Round to One Decimal Place
Rounding the percentage change to one decimal place, we get:
\$\$ \text{Percentage Change in Savings} \approx 20.3\% \$\$
Since the savings increased, the final answer is a 20.3% increase in savings.
Revision Table: Income, Expenditure, and Savings
| Item | Initial Value | Change | New Value |
|---|---|---|---|
| Income | \$100 | +20.1% | \$120.1 |
| Expenditure | \$65 | +20% | \$78 |
| Savings | \$35 | + \$7.1 (+20.3%) | \$42.1 |
Additional Information on Percentage Changes
When dealing with percentage changes, it's crucial to remember the base amount for the percentage calculation. For example, a 20% increase in expenditure is 20% of the *initial* expenditure, not 20% of the income.
The formula for calculating a new value after a percentage increase is:
\$\$ \text{New Value} = \text{Original Value} \times (1 + \frac{\text{Percentage Increase}}{100}) \$\$
For a percentage decrease, it would be:
\$\$ \text{New Value} = \text{Original Value} \times (1 - \frac{\text{Percentage Decrease}}{100}) \$\$
In this problem, both income and expenditure increased, so we used the addition formula. The percentage change in savings was calculated relative to the initial savings.
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