Answer The amount invested in 5.75% bond is $ 20000 Explanation Given information : Total amount invested in two municipal bonds that pay 5.75% and 6.25% simple interest is $32000. Annual interest income required for the investments is $1900 Formula used: I = P ⋅ r ⋅ t whereP represents the initial investment, r represents the annual rate of interest in decimal, t represents the number of years and I represent the simple interest. Concept Involved: Assign variable for the unknown that we need to find. Write system of equations based on the given statements. Solve the system of equation either by elimination method or by substitution method. Calculation: Description Steps Assign variables for the unknown that we need to find Let x be the amount invested in municipal bond paying 5.75% as interest, and y be the amount invested in municipal bond paying 6.25% as interest Write an equation based on total amount of investment. Amount invested in municipal bond paying 5.75% interest + Amount invested in municipal bond paying 6.25% interest = Total investment x + y = 32000 Use the interest formula to find the interest amount earned due to the investment in municipal bond paying 5.75% interest in one year I 5.75 % = x ⋅ 0.0575 ⋅ 1 = 0.0575 x Calculation (Continued): Description Steps Use the interest formula to find the interest amount earned due to the investment in municipal bond paying 6.25% interest in one year I 6.25 % = y ⋅ 0.0625 ⋅ 1 = 0.0625 y Write an equation based on the information that Annual interest income required for the investments is $1900 0.0575 x + 0.0625 y = 1900 Label the system of equations x + y = 32000 → 1 s t E q u a t i o n 0.0575 x + 0.0625 y = 1900 → 2 n d E q u a t i o n Solve the 1 s t equation for y by subtracting x on both sides x + y − x = 32000 − x y = 32000 − x Substitute 32000 − x for y in the 2 n d equation 0.0575 x + 0.0625 ( 32000 − x ) = 1900 Distribute 0.0625 in the left side of the equation 0.0575 x + 0.0625 ( 32000 ) + 0.0625 ( − x ) = 1900 0.0575 x + 2000 − 0.0625 x = 1900 Group and combine like terms 0.0575 x − 0.0625 x + 2000 = 1900 − 0.005 x + 2000 = 1900 In the process of solving for x subtract 2000 on both sides of the equation and simplify − 0.005 x + 2000 − 2000 = 1900 − 2000 − 0.005 x = − 100 Dividing -0.005 on both sides and simplify fraction on both sides − 0.005 x − 0.005 = − 100 − 0.005 x = 20000 Substitute 20000 for x in y = 32000 − x y = 32000 − 20000 = 12000 Conclusion: The amount invested in municipal bond paying 5.75% as interest is $20000 and the amount invested in municipal bond paying 5.75% as interest is $12000

3rd Edition

ISBN: 9781133947202

Author: Ron Larson

Publisher: Cengage Learning

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Question

Chapter 7.2, Problem 52E

To determine

To find : the amount invested in 5.75% bond.

Expert Solution & Answer

Answer to Problem 52E

The amount invested in 5.75% bond is $20000

Explanation of Solution

Given information : Total amount invested in two municipal bonds that pay 5.75% and 6.25% simple interest is $32000. Annual interest income required for the investments is $1900

Formula used:

I = P⋅r⋅t whereP represents the initial investment, r represents the annual rate of interest in decimal, t represents the number of years and I represent the simple interest.

Concept Involved:

Assign variable for the unknown that we need to find. Write system of equations based on the given statements. Solve the system of equation either by elimination method or by substitution method.

Calculation:

Description	Steps
Assign variables for the unknown that we need to find	Let x be the amount invested in municipal bond paying 5.75% as interest, and y be the amount invested in municipal bond paying 6.25% as interest
Write an equation based on total amount of investment. Amount invested in municipal bond paying 5.75% interest + Amount invested in municipal bond paying 6.25% interest = Total investment	x+y=32000
Use the interest formula to find the interest amount earned due to the investment in municipal bond paying 5.75% interest in one year	I5.75%=x⋅0.0575⋅1 =0.0575x

Calculation (Continued):

Description	Steps
Use the interest formula to find the interest amount earned due to the investment in municipal bond paying 6.25% interest in one year	I6.25%=y⋅0.0625⋅1 =0.0625y
Write an equation based on the information that Annual interest income required for the investments is $1900	0.0575x+0.0625y=1900
Label the system of equations	x+y=32000→1st Equation0.0575x+0.0625y=1900→2nd Equation
Solve the 1^s^t equation for y by subtracting x on both sides	x+y−x=32000−xy=32000−x
Substitute 32000 − x for y in the 2ⁿ^d equation	0.0575x+0.0625(32000−x)=1900
Distribute 0.0625 in the left side of the equation	0.0575x+0.0625(32000)+0.0625(−x)=19000.0575x+2000−0.0625x=1900
Group and combine like terms	0.0575x−0.0625x+2000=1900−0.005x+2000=1900
In the process of solving for x subtract 2000 on both sides of the equation and simplify	−0.005x+2000−2000=1900−2000 −0.005x=−100
Dividing -0.005 on both sides and simplify fraction on both sides	−0.005x−0.005=−100−0.005 x=20000
Substitute 20000 for x in y=32000−x	y=32000−20000 =12000

Conclusion:

The amount invested in municipal bond paying 5.75% as interest is $20000 and the amount invested in municipal bond paying 5.75% as interest is $12000

Chapter 7.2, Problem 51E

Chapter 7.2, Problem 53E

Chapter 7 Solutions

Precalculus with Limits

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