# Algebraic word problems

1.Samantha's student loans total $20,000. Part was a personal loan at 12% interest; the other was a Stafford loan at 10%. After one year the loans accumulated $2160 in interest. What was the amount of each loan?

a)Develop an equation to represent the sum of the loans, use 'x' for the personal loan, use 'y' for the Stafford loan.

b)Develop an equation to represent the interest accumulated in the first year. Use 'x' for the personal loan, use 'y' for the Stafford loan.

c)Use the equations from part a and b of this exercise as a system of equations. Use substitution to determine the amount of each loan. Show all your work from problem statement through the final result.

d)What are the intercepts of the equation from part a of this problem? Where would the lines intersect if you solved the system by graphing?

2.Samantha's parents have invested for many years. They invested a total of $20,000 one year. They invested part of the money in HP stock which returned 70% and another part in US Treasury Bonds which returned 5%. Their total return on their investments was $9,580.

a)Develop an equation to represent the sum of the investments, use 'x' for the HP stock, use 'y' for the Treasury bonds.

b)Develop an equation to represent the interest accumulated in the first year. Use 'x' for the HP stock, use 'y' for the Treasury bonds.

c)Use the equations from part a and b of this exercise as a system of equations Use elimination to determine how much was invested in each Stock/Treasury bond. Show all your work from problem statement through the final result.

3.Samantha's parents have decided to find an agency that can provide home services for Samantha's grandparents. They searched online and are interested in contacting two agencies, "At Home Again" and "Comfort Cares", to discuss their rates. "At Home Again" charges a $45 fee plus $22 per hour. "Comfort Cares" charges $37 per hour with no additional fees.

a)Which agency provides a better deal for 2 hours of work? Show your work upon which the answer is based.

b)Which agency provides a better deal for 8 hours of work? Show your work upon which the answer is based.

c)For what values h (hours) does Comfort Cares offer the better deal? Express your answer as an inequality. Show all your work.

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#### Solution Preview

1.Samantha's student loans total $20,000. Part was a personal loan at 12% interest; the other was a Stafford loan at 10%. After one year the loans accumulated $2160 in interest. What was the amount of each loan?

a)Develop an equation to represent the sum of the loans, use 'x' for the personal loan, use 'y' for the Stafford loan.

Let x denotes the amount of personal loan and y denotes the amount of Stafford loan.

Samantha's student loans total $20000

x+y=20000 -------------------------(1)

b)Develop an equation to represent the interest accumulated in the first year. Use 'x' for the personal loan, use 'y' for the Stafford loan.

Interest on personal loan in one year=12% of x=12x/100=3x/25

Interest on Stafford loan in one year=10% of y=10y/100=y/10

Total interest in one year is $2160.

(3x/25)+(y/10)=2160

Multiply both sides by 50, we get

6x+5y=108000 -----------------------(2)

c)Use the equations from part a and b of this exercise as a system of equations. Use substitution to determine the amount of each loan. Show all your work from problem statement through the final result.

From equation (1), we get

x+y=20000

Subtract x from both sides

y=20000-x

Put y=20000-x in equation ...

#### Solution Summary

There are three word problems. Solution to each problem depicts the methodology and calculations to find out the asked parameters by converting it into the algebraic expressions.