Calculating the Value of P and Q

The problem asks us to find the value of the ratio \( \frac{P}{Q} \), where P is given by a specific mathematical expression involving decimal multiplication and subtraction, and Q is a given decimal value.

First, let's calculate the value of P using the given expression:

\( P = 0.3 \times 0.3 + 0.03 \times 0.03 - 0.6 \times 0.03 \)

Step-by-Step Calculation of P

We need to perform the multiplications first, following the order of operations.

• Calculate the first term: \( 0.3 \times 0.3 \)
• \( 0.3 \times 0.3 = 0.09 \)

• Calculate the second term: \( 0.03 \times 0.03 \)
• \( 0.03 \times 0.03 = 0.0009 \)

• Calculate the third term: \( 0.6 \times 0.03 \)
• \( 0.6 \times 0.03 = 0.018 \)

Now, substitute these values back into the expression for P:

\( P = 0.09 + 0.0009 - 0.018 \)

Perform the addition and subtraction:

• Add the first two terms: \( 0.09 + 0.0009 \)
• Aligning the decimal points:

• So, \( 0.09 + 0.0009 = 0.0909 \)

Now, subtract the third term from this sum:

\( P = 0.0909 - 0.018 \)

• Aligning the decimal points:

• So, \( 0.0909 - 0.018 = 0.0729 \)

Thus, the value of P is \( 0.0729 \).

Calculating the Ratio P/Q

We are given that \( Q = 0.54 \). We need to calculate \( \frac{P}{Q} \).

\( \frac{P}{Q} = \frac{0.0729}{0.54} \)

To divide decimals, we can remove the decimals by multiplying both the numerator and the denominator by a power of 10. The numerator (0.0729) has 4 decimal places, and the denominator (0.54) has 2 decimal places. We should multiply by \( 10^4 = 10000 \) to clear the decimal in the numerator (which will also clear it in the denominator).

\( \frac{0.0729 \times 10000}{0.54 \times 10000} = \frac{729}{5400} \)

Now, we can simplify this fraction or perform the division directly.

Let's simplify the fraction \( \frac{729}{5400} \).

• Both 729 and 5400 are divisible by 9 (sum of digits of 729 is \( 7+2+9=18 \), which is divisible by 9; sum of digits of 5400 is \( 5+4+0+0=9 \), which is divisible by 9).
• \( 729 \div 9 = 81 \)
• \( 5400 \div 9 = 600 \)

The fraction becomes \( \frac{81}{600} \).

• Both 81 and 600 are divisible by 3.
• \( 81 \div 3 = 27 \)
• \( 600 \div 3 = 200 \)

The fraction becomes \( \frac{27}{200} \).

Now, convert the fraction \( \frac{27}{200} \) to a decimal.

We can do this by performing the division \( 27 \div 200 \) or by converting the denominator to a power of 10.

To convert the denominator to 1000, multiply both numerator and denominator by 5:

\( \frac{27 \times 5}{200 \times 5} = \frac{135}{1000} \)

\( \frac{135}{1000} = 0.135 \)

So, the value of \( \frac{P}{Q} \) is \( 0.135 \).

Comparing with Options

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

• Option 1: 0.45
• Option 2: 4.05
• Option 3: 0.135
• Option 4: 4.5

Our calculated value, 0.135, matches Option 3.

Summary of Calculation

• Calculated \( P = 0.0729 \)
• Given \( Q = 0.54 \)
• Calculated \( \frac{P}{Q} = \frac{0.0729}{0.54} = 0.135 \)

Revision Table - Decimal Calculations

Additional Information - Working with Decimals

Working with decimals requires careful attention to place values and decimal points. When multiplying decimals, the number of decimal places in the result is the sum of the number of decimal places in the numbers being multiplied. When adding or subtracting decimals, it is crucial to align the decimal points before performing the operation. This can be helped by adding trailing zeros to ensure both numbers have the same number of decimal places after the point. Decimal division can be simplified by converting the divisor into a whole number, applying the same multiplication factor to the dividend.

Converting decimals to fractions or vice versa can also be a useful technique for simplifying calculations, as shown in the steps where we converted the decimal division into a fraction simplification problem.