Question

name:

1. the sum of the ages of ryan and his father is 66. his father is 10 years more than 3 times as old as ryan. how old are ryan and his father?

a. ryan is 14 and his father is 52
b. ryan is 15 and his father is 53
c. ryan is 12 and his father is 50
d. ryan is 10 and his father is 66

1. a total of $10,000 is invested in two funds, fund a and fund b. fund a pays 5% annual interest and fund b pays 7% annual interest. the combined annual interest is $630. how much of the $10,000 is invested in each fund?

a. fund a is 3500 and fund b is 6500
b. fund a is 6500 and fund b is 3500
c. fund a is 3000 and fund b is 6000
d. fund a is 6000 and fund b is 3000

1. we need to rent a large truck for one week. rental companies charge an initial cost plus an additional cost for each mile driven. one company, paenz, will rent a 27 - foot truck for us for $525 plus $0.21 per mile. another company, opan, will rent us the same size truck for $585 plus $0.13 per mile. create two equations in terms of x for miles and y for the total cost.

a. equation 1: $y = 525x + 0.21$ equation 2: $y = 585x + 0.13$
b. equation 1: $y = 27x + 525$ equation 2: $y = 27x + 585$
c. equation 1: $y = 0.21x + 525$ equation 2: $y = 0.13x + 585$
d. equation 1: $y = 0.21x + 585$ equation 2: $y = 0.13x + 525$

1. the larger of two numbers is 7 less than 8 times the smaller. if the larger number is decreased by twice the smaller, the result is 329. what equations should we use to solve the problem?

a. equation 1: $y = 7x - 8$ equation 2: $y - 2x = 329$
b. equation 1: $y = 8 - 7x$ equation 2: $y = 329 - 2x$
c. equation 1: $y = 8x - 7$ equation 2: $y = 2x - 329$
d. equation 1: $y = 8x - 7$ equation 2: $y - 2x = 329$

Explanation:

Problem 1

Step1: Define variables

Let $r$ = Ryan's age, $f$ = father's age.

Step2: Set up equations

Sum of ages: $r + f = 66$
Father's age relation: $f = 3r + 10$

Step3: Substitute and solve

Substitute $f$ into first equation:
$r + (3r + 10) = 66$
$4r + 10 = 66$
$4r = 56$
$r = 14$

Step4: Find father's age

$f = 3(14) + 10 = 52$

Problem 2

Step1: Define variables

Let $a$ = amount in Fund A, $b$ = amount in Fund B.

Step2: Set up equations

Total investment: $a + b = 10000$
Total interest: $0.05a + 0.07b = 630$

Step3: Isolate and substitute

From first equation: $a = 10000 - b$
Substitute into interest equation:
$0.05(10000 - b) + 0.07b = 630$
$500 - 0.05b + 0.07b = 630$
$0.02b = 130$
$b = 6500$

Step4: Find Fund A amount

$a = 10000 - 6500 = 3500$

Problem 3

Step1: Identify cost structure

Total cost $y$ = fixed cost + (cost per mile $\times$ miles $x$)

Step2: Write Paenz equation

Fixed cost = $525$, cost per mile = $0.21$:
$y = 0.21x + 525$

Step3: Write Opan equation

Fixed cost = $585$, cost per mile = $0.13$:
$y = 0.13x + 585$

Problem 4

Step1: Define variables

Let $x$ = smaller number, $y$ = larger number.

Step2: Set up first equation

Larger number relation: $y = 8x - 7$

Step3: Set up second equation

Larger minus twice smaller = 329:
$y - 2x = 329$

Answer:

1. a. Ryan is 14 and his father is 52
2. a. Fund A is 3500 and Fund B is 6500
3. c. Equation 1: $y = 0.21x + 525$ Equation 2: $y = 0.13x + 585$
4. d. Equation 1: $y = 8x - 7$ Equation 2: $y - 2x = 329$