Solving the Percentage Calculation Problem

This problem requires us to calculate the value of a fraction where both the numerator and the denominator involve sequential percentage calculations.

To solve this, we need to understand how to calculate a percentage of a number and how to work with fractions.

Remember that a percentage can be converted to a decimal by dividing by 100. For example, \(25\% = \frac{25}{100} = 0.25\).

Calculating the Numerator Value

The numerator is given as \(25\% \;of\;(50\% \;of\;30\% \;of\;150)\). We will calculate this from the innermost percentage outwards.

1. First, calculate \(30\% \;of\;150\):
   \(30\% \;of\;150 = \frac{30}{100} \times 150 = 0.30 \times 150\)
2. Performing the multiplication: \(0.30 \times 150 = 45\).
   So, \(30\% \;of\;150 = 45\).
3. Next, calculate \(50\% \;of\;(\text{the result from step 2})\), which is \(50\% \;of\;45\):
   \(50\% \;of\;45 = \frac{50}{100} \times 45 = 0.50 \times 45\)
4. Performing the multiplication: \(0.50 \times 45 = 22.5\).
   So, \(50\% \;of\;(30\% \;of\;150) = 22.5\).
5. Finally, calculate \(25\% \;of\;(\text{the result from step 4})\), which is \(25\% \;of\;22.5\):
   \(25\% \;of\;22.5 = \frac{25}{100} \times 22.5 = 0.25 \times 22.5\)
6. Performing the multiplication: \(0.25 \times 22.5 = 5.625\).
   So, the numerator value is \(5.625\).

Calculating the Denominator Value

The denominator is given as \(40\% \;of\;2250\). We calculate this directly:

1. Calculate \(40\% \;of\;2250\):
   \(40\% \;of\;2250 = \frac{40}{100} \times 2250 = 0.40 \times 2250\)
2. Performing the multiplication: \(0.40 \times 2250 = 900\).
   So, the denominator value is \(900\).

Calculating the Final Fraction Value

Now we need to calculate the value of the fraction using the numerator and denominator values we found:

\(\frac{{25\% \;of\;(50\% \;of\;30\% \;of\;150)}}{{40\% \;of\;2250}} = \frac{5.625}{900}\)

Performing the division:

\(\frac{5.625}{900} = 0.00625\)

Converting the Result to a Percentage

The question options are given in percentage form. To convert the decimal result \(0.00625\) to a percentage, we multiply by 100 and add the percentage symbol:

\(0.00625 \times 100\% = 0.625\%\)

Thus, the value of the given expression is \(0.625\%\).

Comparing with Options

Let's compare our calculated value with the given options:

• Option 1: \(0.225\%\)
• Option 2: \(0.825\%\)
• Option 3: \(0.25\%\)
• Option 4: \(0.625\%\)

Our calculated value matches Option 4.

Revision Table: Key Percentage Concepts

Additional Information on Percentage Calculations

Percentage calculations are widely used in various fields, including finance, statistics, and everyday life. Understanding how to correctly calculate percentages, especially sequential percentages like in the numerator of this problem, is important.

When you calculate "p% of q% of r% of a number N", you are essentially performing the following calculation:

\(\frac{p}{100} \times \frac{q}{100} \times \frac{r}{100} \times N\)

In our numerator calculation, this was:

\(25\% \;of\;50\% \;of\;30\% \;of\;150 = \frac{25}{100} \times \frac{50}{100} \times \frac{30}{100} \times 150\)

\(= 0.25 \times 0.50 \times 0.30 \times 150\)

\(= 0.125 \times 0.30 \times 150\)

\(= 0.0375 \times 150\)

\(= 5.625\)

This confirms our step-by-step calculation for the numerator.

The denominator calculation was straightforward: \(40\% \;of\;2250 = \frac{40}{100} \times 2250 = 0.40 \times 2250 = 900\).

Finally, the fraction is \(\frac{5.625}{900}\). When dividing decimals, it can sometimes be helpful to make the divisor a whole number, but in this case, direct division is simple:

\(5.625 \div 900 = 0.00625\)

Converting to percentage: \(0.00625 \times 100\% = 0.625\%\).

This detailed breakdown shows how each part of the expression contributes to the final result and reinforces the understanding of percentage calculations.