USING TOOLS STRATEGICALLY

Example 1 : 

The depth of Golden Trout Lake has been decreasing in recent years. Two years ago, the depth of the lake was 186.73 meters. Since then the depth has been changing at an average rate of -1 3/4 % per year. What is the depth of the lake today ?

Solution : 

Step 1 : Convert the percent to a decimal

Write the fractional part as a decimal :

-1 3/4%  =  -1.75%

Move the decimal point two places left.

-1 3/4%  =  -0.0175

Step 2 : 

Find the depth of the lake after one year. Use a calculator to simplify the computations.

Find the change in depth :

186.73 × (−0.0175) ≈ −3.27 meters

Find the new depth :

186.73 − 3.27 = 183.46 meters

Step 3 : 

Find the depth of the lake after two years.

Find the change in depth :

183.46 × (−0.0175) ≈ −3.21 meters

Find the new depth :

183.46 − 3.27 = 180.25 meters

Step 4 : 

Check the answer for reasonableness.

The original depth was about 190 meters. The depth changed by  about −2% per year. Because (−0.02)(190) = −3.8, the depth changed by about −4 meters per year or about −8 meters over two years. So, the new depth was about 182 meters. The answer is close to the estimate, so it is reasonable.

Example 2 : 

Three years ago, Jolene bought $750 worth of stock in a software company. Since then the value of her purchase has been increasing at an average rate of 12 3/5 % per year. How much is the stock worth now 

Solution : 

Step 1 : Convert the percent to a decimal

Write the fractional part as a decimal :

12 3/5%  =  12.6%

Move the decimal point two places left.

12 3/5 %  =  0.126

Step 2 : 

Find the change in value after one year. Use a calculator to simplify the computations.

Find the change in value :

750 × (0.126) ≈ $94.5

Find the value after one year :

750 + 94.5 = $844.5

Step 3 : 

Find the change in price after two years. 

Find the change in value :

844.5 × (0.126) ≈ $106.41

Find the value after two years :

844.5 + 106.41 = $950.91

Step 4 : 

Find the change in price after three years. 

Find the change in value :

950.91 × (0.126) ≈ $119.81

Find the value after three years :

950.91 + 119.81 = $1070.72

Step 5 : 

Check the answer for reasonableness.

Jolene bought $750 worth of stock in a software company. The value of her purchase has been increasing about 13% per year. Because (750)(0.13) = 97.5, the value of the stock changed by about $100 per year or about $300 ( = 100 x 3) over three years. So, the value of the stock after three years is $1050 ( = 750 + 300). The answer is close to the estimate, so it is reasonable.

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