Problem 50 - Elementary Algebra

Question

In 2011 , about 593 grizzly bears lived in Wyoming, Montana, and Idaho, a reduction from the 602 bears who lived there in 2010 . Find the percent decrease. Round to the nearest tenth of a percent. (Source: www.lmtribune com, Oct. 14, 2011)

Step-by-Step Solution

Answer

The percent decrease is approximately 1.5%.

Step 1: Determine the original number

Identify the original number of grizzly bears in 2010. This is given as 602.

Step 2: Determine the new number

Identify the new number of grizzly bears in 2011. This is given as 593.

Step 3: Calculate the decrease in number

Subtract the new number from the original number: \[ 602 - 593 = 9 \]

Step 4: Calculate the decrease in decimal form

Divide the decrease by the original number to find the decrease in decimal form: \[ \frac{9}{602} \approx 0.0149 \]

Step 5: Convert the decimal to a percentage

Multiply the decimal from Step 4 by 100 to convert it to a percent: \[ 0.0149 \times 100 = 1.49 \]

Step 6: Round the percentage

Round the result to the nearest tenth of a percent: \[ 1.49 \approx 1.5 \% \]

Key Concepts

Percent Decrease

When finding the percent decrease, you're essentially trying to determine how much a value has reduced in relation to the original amount. This is especially useful in various fields such as finance, population studies, and general mathematics. To find the percent decrease, you follow these steps:

First, identify the original number and the new number. In our example, the original number of grizzly bears in 2010 was 602, and the new number in 2011 was 593.

Next, calculate the decline by subtracting the new number from the original number. In the exercise, this would result in: 602 - 593 = 9.

Once you have the decline, you move on to convert this value into a percentage. This transformation involves several critical steps, including decimal conversion and rounding.

Decimal Conversion

Decimal conversion is essential when dealing with percentages. After you determine the decrease (which is 9 in our example), you need to express this decrease as a fraction of the original number.

You do this by dividing the decrease by the original number: \( \frac{9}{602} \). Calculating this division gives us approximately 0.0149.

This decimal represents the ratio of the decrease to the original number. Decimal conversion helps simplify complex fractions and prepares the value for the next step: converting it to a percentage.

Arithmetic Subtraction

Subtraction is one of the fundamental operations in arithmetic. In the context of percent decrease, subtraction helps us find the difference between the original and new values.

Let's go through the subtraction step-by-step. We were given 602 grizzly bears in 2010 and 593 in 2011. First, you write down the original value (602) and subtract the new value (593): \( 602 - 593 = 9 \).

This result, 9, is the decrease in the number of bears. Subtraction is straightforward but crucial, as it lays the groundwork for further calculations like converting to a decimal and then to a percentage.

Rounding Percentages

Rounding percentages is the final step in this exercise. After converting the decrease into a decimal, we multiply the decimal by 100 to obtain the percentage: \( 0.0149 \times 100 = 1.49 \).

Percentages often require rounding to make them easier to understand or to fit specific criteria (e.g., to the nearest tenth). In our problem, rounding 1.49 to the nearest tenth means examining the digit in the hundredths place. If this digit is 5 or more, round the tenths digit up.

Since 1.49 has a 9 in the hundredths place, we round up the 4 to get 1.5. Therefore, the approximate percent decrease in the grizzly bear population is 1.5%.