Problem 7 - Technical Mathematics with Calculus

Question

Convert each fraction to a percent. Round to three significant digits. $$\frac{7}{10}$$

Step-by-Step Solution

Answer

The fraction \(\frac{7}{10}\) is equal to 70% when converted to percent and rounded to three significant digits.

Step 1: Convert Fraction to Decimal

To convert a fraction to a decimal, divide the numerator by the denominator. For the fraction \(\frac{7}{10}\), divide 7 by 10.

Step 2: Perform Division

Dividing 7 by 10 gives 0.7. Next, you need to convert this decimal to a percentage.

Step 3: Convert Decimal to Percent

To convert a decimal to a percent, multiply it by 100 and add the percent symbol (%). Multiplying 0.7 by 100 gives 70%.

Step 4: Round to Three Significant Digits

Since 70% is already to three significant digits, rounding is not necessary.

Key Concepts

Fraction to Decimal Conversion

Understanding how to convert fractions to decimals is a foundational math skill that's important for many higher concepts. Essentially, a fraction represents a division operation. The numerator (the top number) is divided by the denominator (the bottom number) to obtain the decimal form. For example, to convert the fraction \(\frac{7}{10}\) to a decimal, you simply divide 7 by 10.

Carrying out this division, \(\frac{7}{10}=0.7\). It's important to perform the division carefully and to understand that the resulting decimal is another form of representing the same value as the fraction. These decimals are useful because they can be easily compared and used in further calculations or converted into percentages, as we will explore in the next section.

Decimal to Percent Conversion

Once you have a decimal, converting it to a percentage is straightforward. The concept here is to understand that 'percent' means 'per one hundred.' So, to convert from a decimal to a percent, you multiply the decimal by 100.

For instance, the decimal 0.7 is converted to a percentage by multiplying it by 100, which gives you 70. This implies that 0.7 is equivalent to 70% or 70 per 100. Mathematically, it's expressed as \(0.7 \times 100 = 70%\). Always remember to add the percentage sign to indicate that the number is a percent. This conversion is helpful when comparing proportions or illustrating parts of a whole in graphs and charts.

Significant Digits Rounding

Rounding to significant digits is a method to approximate a number while maintaining its meaningful digits. It's particularly important in scientific contexts where precision matters, but excessive detail is unnecessary. The rule for rounding is to look at the digit right after the last significant digit you want to keep. If this digit is five or greater, you increase the last significant digit by one. If it's less than five, you leave the last significant digit as is.

In our exercise, we looked at 70%, which is already rounded to three significant digits — the 7, the 0 after the 7, and the 0 after the decimal are the significant digits. Since there are no additional digits, there's no need to round. However, if we had a number like 0.7465 and we wanted three significant digits, we would round to 0.747, or 74.7% when converted to a percent, because the fourth digit (6) is greater than five.