Question

exit ticket - precalculus
name: alana riston
date: 4/23/2023
find a solution to the following problems

1. ladonna borrowed a total of $10,000. part of the money was borrowed from a lending institution that charged 5.5% simple interest. the rest of the money was borrowed from a friend to whom ladonna paid 2.5% simple interest. ladonna paid the friend back after 9 months (0.75 yr) and paid the lending institution after 2 yr. if the total amount ladonna paid in interest was $735, how much was borrowed from each source?
2. how much 4% acid solution should be mixed with 200 ml of a 12% acid solution to make a 9% acid solution?

solution
4 1 = r+
1=(10,000)(5.5) · 01
goodluck!!!

Explanation:

Step1: Set up variables for problem 1

Let $x$ be the amount borrowed from the lending - institution. Then the amount borrowed from the friend is $10000 - x$.
The interest paid to the lending institution is $I_1=x\times0.055\times2$ (using the simple - interest formula $I = Prt$, where $P$ is the principal, $r$ is the annual interest rate, and $t$ is the time in years).
The interest paid to the friend is $I_2=(10000 - x)\times0.025\times0.75$.
The total interest is $I_1+I_2 = 735$. So, $x\times0.055\times2+(10000 - x)\times0.025\times0.75=735$.

Step2: Simplify the equation for problem 1

First, calculate the coefficients:
$0.055\times2 = 0.11$ and $0.025\times0.75=0.01875$.
The equation becomes $0.11x+0.01875\times(10000 - x)=735$.
Expand: $0.11x + 187.5-0.01875x=735$.
Combine like - terms: $(0.11x-0.01875x)+187.5 = 735$.
$0.09125x+187.5 = 735$.
Subtract 187.5 from both sides: $0.09125x=735 - 187.5$.
$0.09125x=547.5$.
Divide both sides by 0.09125: $x=\frac{547.5}{0.09125}=6000$.
The amount borrowed from the friend is $10000 - 6000 = 4000$.

Step3: Set up equation for problem 2

Let $V$ be the volume (in mL) of the 4% acid solution.
The amount of acid in the 4% solution is $0.04V$.
The amount of acid in the 200 mL of 12% solution is $0.12\times200 = 24$.
The total volume of the final solution is $V + 200$, and the amount of acid in the final 9% solution is $0.09\times(V + 200)$.
So, $0.04V+24=0.09\times(V + 200)$.

Step4: Solve the equation for problem 2

Expand the right - hand side: $0.04V+24=0.09V+18$.
Subtract $0.04V$ from both sides: $24=0.09V - 0.04V+18$.
$24 = 0.05V+18$.
Subtract 18 from both sides: $0.05V=24 - 18$.
$0.05V = 6$.
Divide both sides by 0.05: $V=\frac{6}{0.05}=120$.

Answer:

1. Amount borrowed from the lending institution: $\$6000$, Amount borrowed from the friend: $\$4000$.
2. Volume of 4% acid solution: $120$ mL.