Section 1.B Percents and Estimation
ΒΆObjectives
Subsection 1.B.1 Percents
Percent means "parts per hundred." For example, the figure below represents \(23\%\text{.}\)
The large square is partitioned into \(10\times 10 = 100\) equal parts. We have shaded \(23\) of the \(100\) little squares, or \(23\%\) of the total number of little squares.
To change a percent to a decimal, we need only remember that percent means "parts per hundred," \(23\%=\dfrac{23}{100}\) or \(23\) parts out of \(100\text{.}\) Dividing \(23\) by \(100\) yields \(\dfrac{23}{100}\) or \(0.23\) or \(23\%\text{.}\) (Recall that in the decimal representation, \(0.23,\) \(2\) is in the tenths position and \(3\) is in the hundredths position.
Subsection 1.B.2 Converting Between Percents and Decimals
Example 1.B.1. Convert a Percent to a Decimal.
Convert \(37.5\%\) into a decimal.
WeBWorK: Entering Decimals.
Type .375 for \(.375\)
This last example motivates the following simple rule. To change a percent to a decimal, drop the percent symbol and move the decimal point two places to the left.
Example 1.B.2. Convert a Percent to a Decimal.
Convert \(2\frac{1}{2}\%\) into a decimal.
Checkpoint 1.B.3. Convert Percent to Decimal Form.
Changing a decimal to a percent is the exact opposite of changing a percent to a decimal: move the decimal point two places to the right and add a percent symbol.
Example 1.B.4. Convert a Decimal to a Percent.
Convert \(0.0725\) to a percent.
WeBWorK: Entering Percents.
Type 7.25% for \(7.25\%\text{.}\)
Example 1.B.5. Convert a Decimal to a Percent.
Convert \(1.025\) to a percent.
Checkpoint 1.B.6. Convert Decimal to Percent Form.
Example 1.B.7. Determine a Percent.
Suppose that results of a survey revealed that \(172\) out of \(325\) respondents had brown eyes. Give the percent of respondents having brown eyes.
\(52.923\%\) of the respondents have brown eyes.\(\checkmark\)
WeBWorK: Decimal Accuracy.
It is generally a good idea to maintain \(3\) to \(5\) decimals of accuracy when submitting answers.
Subsection 1.B.3 Determining a Percent of a Number
To find a specific percent of a given number, we multiply the number by the decimal form of the percent.
Example 1.B.8. Determine a Percent of a Number.
Example 1.B.9. Determine a Percent of a Number.
Checkpoint 1.B.10. Determine a Percent of a Number.
Subsection 1.B.4 Rounding Decimals
In rounding a number we consider the digit to the right of the digit we will round. This digit determines how to round the number. We refer to it as the target digit. It determines whether the number is closer to the next bigger digit or not. If the target digit is \(6\text{,}\) \(7\text{,}\) \(8\) or \(9\text{,}\) the number is closer to the next larger number and we round our digit up. If the target digit is \(0\text{,}\) \(1\text{,}\) \(2\text{,}\) \(3\) or \(4\text{,}\) then we do not round up. If the target digit is \(5\text{,}\) the number is half way between the smaller and larger values. Because there are five digits for which we round down, we round up for the target digit \(5\) so that there are the same number of digits for which we round up as round down.
Example 1.B.11. Rounding to Tenths.
Round \(6.7291\) to the nearest tenth.
Find the tenths place: one place after the decimal.
Example 1.B.12. Rounding to Hundredths.
Round \(6.7291\) to the nearest hundredth.
Find the hundredths place: two places after the decimal.
In summary, round up if the trigger digit is \(5\) or greater. Do not round up if the trigger is less than \(5.\)
Checkpoint 1.B.13. Round Decimals.
Subsection 1.B.5 Estimation
Estimation can be useful for determining the reasonableness of a result.
Example 1.B.14. Estimating.
Estimate the product by rounding each of the given values to the nearest tenth.