Calculating the Value of a Complex Expression

We need to find the value of the given mathematical expression:

\((0.5)^2\div(0.125)+(0.12)^2\div(0.02)+(0.18)^2\div(0.04)+(0.22)^2\div(0.02)^2+(0.9)^3\div(0.03)^2\)

Let's break down the expression into individual terms and calculate each one separately.

Step 1: Evaluate the first term

The first term is \((0.5)^2 \div (0.125)\).

• Calculate the square: \((0.5)^2 = 0.5 \times 0.5 = 0.25\)
• Convert division to multiplication by the reciprocal: \(0.25 \div 0.125 = 0.25 \div \frac{125}{1000} = 0.25 \div \frac{1}{8}\)
• Perform the multiplication: \(0.25 \times 8\)
• \(0.25 \times 8 = 2\)

So, the value of the first term is \(2\).

Step 2: Evaluate the second term

The second term is \((0.12)^2 \div (0.02)\).

• Calculate the square: \((0.12)^2 = 0.12 \times 0.12 = 0.0144\)
• Perform the division: \(0.0144 \div 0.02 = \frac{0.0144}{0.02}\)
• To simplify, multiply numerator and denominator by \(10000\) to remove decimals: \(\frac{0.0144 \times 10000}{0.02 \times 10000} = \frac{144}{200}\)
• Simplify the fraction: \(\frac{144}{200} = \frac{72}{100} = 0.72\)

So, the value of the second term is \(0.72\).

Step 3: Evaluate the third term

The third term is \((0.18)^2 \div (0.04)\).

• Calculate the square: \((0.18)^2 = 0.18 \times 0.18 = 0.0324\)
• Perform the division: \(0.0324 \div 0.04 = \frac{0.0324}{0.04}\)
• To simplify, multiply numerator and denominator by \(10000\): \(\frac{0.0324 \times 10000}{0.04 \times 10000} = \frac{324}{400}\)
• Simplify the fraction: \(\frac{324}{400} = \frac{81}{100} = 0.81\)

So, the value of the third term is \(0.81\).

Step 4: Evaluate the fourth term

The fourth term is \((0.22)^2 \div (0.02)^2\).

• Calculate the squares: \((0.22)^2 = 0.22 \times 0.22 = 0.0484\) and \((0.02)^2 = 0.02 \times 0.02 = 0.0004\)
• Perform the division: \(0.0484 \div 0.0004 = \frac{0.0484}{0.0004}\)
• To simplify, multiply numerator and denominator by \(10000\): \(\frac{0.0484 \times 10000}{0.0004 \times 10000} = \frac{484}{4}\)
• Perform the division: \(\frac{484}{4} = 121\)

So, the value of the fourth term is \(121\).

Step 5: Evaluate the fifth term

The fifth term is \((0.9)^3 \div (0.03)^2\).

• Calculate the cube: \((0.9)^3 = 0.9 \times 0.9 \times 0.9 = 0.81 \times 0.9 = 0.729\)
• Calculate the square: \((0.03)^2 = 0.03 \times 0.03 = 0.0009\)
• Perform the division: \(0.729 \div 0.0009 = \frac{0.729}{0.0009}\)
• To simplify, multiply numerator and denominator by \(10000\): \(\frac{0.729 \times 10000}{0.0009 \times 10000} = \frac{7290}{9}\)
• Perform the division: \(\frac{7290}{9} = 810\)

So, the value of the fifth term is \(810\).

Step 6: Sum all the evaluated terms

Now, we add the values of all five terms:

Total Value = (Value of Term 1) + (Value of Term 2) + (Value of Term 3) + (Value of Term 4) + (Value of Term 5)

Total Value = \(2 + 0.72 + 0.81 + 121 + 810\)

Let's add these values:

• \(2 + 0.72 = 2.72\)
• \(2.72 + 0.81 = 3.53\)
• \(3.53 + 121 = 124.53\)
• \(124.53 + 810 = 934.53\)

The total value of the expression is \(934.53\).

Let's summarize the calculation steps in a table.

Sum of values = \(2 + 0.72 + 0.81 + 121 + 810 = 934.53\).

The final calculated value matches one of the provided options.

Revision Table: Key Calculations

Reviewing the core calculations involved in solving this problem:

• Decimal squaring: \((0.5)^2 = 0.25\), \((0.12)^2 = 0.0144\), \((0.18)^2 = 0.0324\), \((0.22)^2 = 0.0484\), \((0.02)^2 = 0.0004\), \((0.03)^2 = 0.0009\)
• Decimal cubing: \((0.9)^3 = 0.729\)
• Decimal division: Performing division involving decimals often requires converting the divisor to a whole number by multiplying both numerator and denominator by a power of 10. For example, \(\frac{0.0144}{0.02} = \frac{0.0144 \times 10000}{0.02 \times 10000} = \frac{144}{200}\).
• Addition of decimals and whole numbers: Summing the final results accurately.

Additional Information: Operations with Decimals

Working with decimals requires careful attention to place values.

• Squaring decimals: To square a decimal, multiply the number by itself. Count the total number of decimal places in the original number; the result will have double that number of decimal places. E.g., \(0.5\) has one decimal place, \((0.5)^2 = 0.25\) has two. \(0.12\) has two decimal places, \((0.12)^2 = 0.0144\) has four.
• Cubing decimals: To cube a decimal, multiply the number by itself three times. Count the total number of decimal places in the original number; the result will have triple that number of decimal places. E.g., \(0.9\) has one decimal place, \((0.9)^3 = 0.729\) has three.
• Dividing decimals: When dividing by a decimal, it's often easiest to convert the divisor into a whole number. Multiply both the divisor and the dividend by the same power of 10 that makes the divisor a whole number. For example, in \(0.25 \div 0.125\), multiply both by \(1000\) to get \(250 \div 125\). In \(0.0144 \div 0.02\), multiply both by \(100\) to get \(1.44 \div 2\), or by \(10000\) to get \(144 \div 200\). Both approaches are valid.
• Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). In this case, calculate powers first, then divisions, and finally additions.

Understanding these basic rules for decimal operations is crucial for solving complex expressions accurately.