Problem 6 - Technical Mathematics with Calculus

Question

Convert each fraction to a percent. Round to three significant digits. $$\frac{3}{4}$$

Step-by-Step Solution

Answer

75.0%

Step 1: Convert Fraction to Decimal

The first step is to convert the fraction \(\frac{3}{4}\) into decimal form by dividing the numerator by the denominator. Calculate 3 divided by 4.

Step 2: Multiply the Decimal by 100

After converting the fraction to a decimal, the next step is to convert this decimal to a percentage by multiplying it by 100. This conversion is based on the definition that one whole is equivalent to 100%.

Step 3: Round the Result

The final step is to round the result of the multiplication to three significant digits. The result before rounding is exact, and rounding it gives us the final percent value to three significant digits.

Key Concepts

Fraction to Decimal Conversion

Understanding how to convert fractions to decimals is a fundamental skill in mathematics, particularly when dealing with percentages. Converting a fraction into a decimal involves a simple division operation: the numerator (which is the top number of the fraction) is divided by the denominator (the bottom number).

For example, to convert the fraction \(\frac{3}{4}\) to a decimal, we divide 3 by 4. Performing this division gives us 0.75. Now, this decimal represents the same value as the fraction - they are simply different ways to express the same amount. Remember, the decimal value can range from 0 (for the fraction \(\frac{0}{any\;number}\)) to just under 1 (for fractions like \(\frac{999}{1000}\)).

Multiplying Decimals by 100

Once you have a decimal, you might wonder how to convert it to a percent. This second step is straightforward: you multiply the decimal by 100. This action shifts the decimal point two places to the right. Why multiply by 100? Because a percent represents a part out of 100, so multiplying by 100 scales the decimal to reflect that proportion.

Using our previous example with the decimal 0.75, when we multiply by 100, we arrive at 75. Voila, we have our percentage! This tells us that \(\frac{3}{4}\) is equivalent to 75%. Always remember, the process of multiplying a decimal by 100 to find the percentage is a crucial skill not just for math, but for understanding data and statistics in real life.

Rounding to Significant Digits

The last piece of the puzzle is rounding to significant digits. When we convert fractions to decimals and then to percentages, we sometimes deal with non-terminating or too many decimal places. Rounding makes these numbers more manageable and easier to work with, especially in real-world scenarios where extreme precision is not necessary.

Rounding to three significant digits means looking at the number as a whole and identifying the three most important digits that represent its value. Any digits after the third significant digit are rounded off. For instance, if we have a decimal like 0.857142857, and we need to round it to three significant digits, our result would be 0.857 or 85.7% once multiplied by 100. This method keeps the number accurate and reliable, without the unwieldy tail of digits. When rounding, if the first digit not included is 5 or higher, we round up the last significant digit; if it is 4 or lower, we leave the last significant digit as it is.