Topic 1 Test Review: Numbers and Algebra Jeopardy Template

Arithmetic/Geometric Sequences/Series

Finance

Standard Form/Approximation/% Error

Laws of Exponents/Logarithms/Exponentials

Solving Systems of Equations/Polynomials

100

Divide the second term by the first, 12669/12300 = 1.03 so r = 1.03

100

Paul wants to buy a car. He needs to take out a loan for $7000. The car salesman offers him a loan with an interest rate of 8%, compounded annually. Paul considers two options to repay the loan. Option 1: Pay $200 each month, until the loan is fully repaid Option 2: Make 24 equal monthly payments. Use option 1 to calculate the number of months it will take for Paul to repay the loan.

N = *alpha enter to solve*

I% = 8

PV = 7000

PMT = -200 (if PV is positive, PMT must be negative)

FV = 0

P/Y = 12

C/Y = 1 (I made a mistake during class sorry y'all)

N = about 40 months

100

A mathematics student decides to use an approximation to 𝝅 of ³⁵⁵⁄₁₁₃. Calculate his percentage error in using this value, giving your answer in standard form.

Use the percent error formula: V_A = ³⁵⁵⁄₁₁₃ and V_E = 𝝅

8.49 x 10^-6%

100

What is the relationship between logarithms and exponentials?

They are inverses.

100

Fearful of a bank failure, Norman split his life savings of $60,000 among three banks. He received 5%, 6%, and 7% on the three deposits. In the account earning 7% interest, he deposited twice as much as in the account earning 5% interest. If his total earnings were $3,760, then how much did he deposit in each account?

x = amount in 5% bank, y = amount in 6% bank, and z = the amount in 7% bank

system:

x + y + z = 60000

0.05x + 0.06y + 0.07z = 3760

2x + 0y - z = 0

(the last equation comes from the sentence: In the account earning 7% interest, he deposited twice as much as in the account earning 5% interest. So, 2x = z, but we need all the variables on one side so we subtract z to the left and need to put a 0 coefficient on y as a placeholder)

Solve using the Matrix editor

x = $16,000

y = $12,000

z = $32,000

200

Use the geometric sequence formula and plug in the first term (12300) and common ratio (1.03) from the problem

200

To find the total amount paid, multiply PMT*N

200*39.8 =

$7960

200

The speed of light is 300000 kilometres per second. The average distance from the Sun to the Earth is 149.6 million km. Calculate the time, in minutes, it takes for light from the Sun to reach the Earth.

Convert 300000 km/s to km/min by multiplying by 60 (60 seconds in 1 min) = 1.8 x 10⁷ km/min

Then, divide the distance (1.496 x 10⁸) by the speed (1.8 x 10⁷) to get the time in minutes:

8.31 minutes

200

Simplify the following expression:

First, apply the power of -2 to both the top and bottom, giving you x^3*-2/x^2*-2 which is x^-6/x^-4. Subtract the power of the top minus the bottom to get x^-6-(-4) = x^-2 which means x² is on the bottom.

200

What are the zeros of the following polynomial?

-2x³ - 12x² + 30x = -200

Add 200 to the both sides and graph -2x³ - 12x² + 30x + 200 to find the zeros (x-intercepts)

x = -5, x = 4

300

The admissions team at a new university are trying to predict the number of student applications they will receive each year. Let n be the number of years that the university has been open. The admissions team collect the following data for the first two years. In the first year there were 10380 places at the university available for applicants. The admissions team announce that the number of places available will increase by 600 every year. Let v_n represent the number of places available at the university in year n. Write down an expression for v_n.

Use the arithmetic sequence formula and plug in the first term (10380) and common difference (600) from the problem.

300

N = 24

I% = 8

PV = 7000

PMT = *alpha enter to solve*

FV = 0

P/Y = 12

C/Y = 1

PMT = 315.70

300

The speed of light is approximately 3 x 10⁸ ms^-1. The distance from the Earth to the Sun is 1.496 x 10⁸ km. The distance from Earth to Proxima Centauri, the nearest star, is 4.014 x 10¹³ km. Find the time it takes (in minutes) for the light from the sun to reach the Proxima Centauri.

First, convert the speed of light from m/s to km/s by dividing it by 1000 to get 3x10⁵ km/s. Then convert the speed from km/s to km/min by multiplying it by 60 (60 seconds in 1 min) = 1.8 x 10⁷ km/min.

To find the time, divide the total distance (4.014 x 10¹³ + 1.496 x 10⁸) by the speed (1.8 x 10⁷) to get:

2.23 x 10⁶ min

300

The pH of a solution measures its acidity and can be determined using the formula where C is the concentration of hydronium ions in the solution, measured in moles per litre. The concentration of hydronium ions in a particular type of coffee is moles per litre. Calculate the pH of the coffee.

Evaluate -log(1.3 x 10^-5) in the calculator to get 4.89

300

On Monday Headley paid $1.70 for two cups of coffee and one doughnut, including the tip. On Tuesday he paid $1.65 for two doughnuts and a cup of coffee, including the tip. On Wednesday he paid $1.30 for one coffee and one doughnut, including the tip. If he always tips the same amount, then what is the amount of each item?

x = amount for a coffee, y = amount for a doughnut, and z = the amount for a tip

system:

2x + y + z = 1.7

1x + 2y + z = 1.65

x + y + z = 1.30

solve using the Matrix editor

x = $0.40

y = $0.35

z = $0.55

400

The word TOTAL indicates that this is a sum (series) problem. The sum of the first 10 terms (n = 10) represents the total number of accepted students in the first 10 year. Each of these students paid an $80 acceptance fee which is why we multiply the series formula by 80.= $10 494 000

correct answer: $10 500 000

400

Paul wants to buy a car. He needs to take out a loan for $7000. The car salesman offers him a loan with an interest rate of 8%, compounded annually. Paul considers two options to repay the loan. Option 1: Pay $200 each month, until the loan is fully repaid Option 2: Make 24 equal monthly payments. Use option 2 to calculate the total amount that Paul has to pay.

To find the total amount paid, multiply PMT*N

315.7*24 =

$7576.80

400

The speed of light is approximately 3 x 10⁸ ms^-1. The distance from the Earth to the Sun is 1.496 x 10⁸ km. The distance from Earth to Proxima Centauri, the nearest star, is 4.014 x 10¹³ km. Find the time (in years) it takes for light from Proxima Centauri to reach the Earth. Assume that one year is equivalent to 365 days.

First, convert the speed of light from m/s to km/s by dividing it by 1000 to get 3x10⁵ km/s. Then convert the speed from km/s to km/year by multiplying it by 3.1536 x 10⁷ (60 *60*24*365, the product to convert from seconds to years) = 9.46 x 10¹² km/min.

To find the time, divide the distance (4.014 x 10¹³) by the speed (9.46 x 10¹²) to get:

4.24 years

400

The pH of a solution measures its acidity and can be determined using the formula where C is the concentration of hydronium ions in the solution, measured in moles per litre. A lower pH indicates a more acidic solution. The concentration of hydronium ions in a particular type of coffee is moles per litre. A different, unknown, liquid has 10 times the concentration of hydronium ions of the coffee in part. Determine whether the unknown liquid is more or less acidic than the coffee. Justify your answer mathematically.

Evaluate -log(1.3 x 10^-5 x 10) in the calculator to get

pH of unknown liquid is 3.89 so it is more acidic

400

A theater has 490 seats. Seats sell for $25 on the floor, $20 in the mezzanine, and $15 in the balcony. The number of seats on the floor equals the total number of seats in the mezzanine and balcony. Suppose the theater takes in $10,520 from each sold-out event. How many seats does the mezzanine section hold?

x = number of seats in floor section, y = number of seats in mezzanine section, and z = number of seats in balcony section

system:

x + y + z = 490

25x + 20y + 15z = 10520

x - y - z = 0

(the last equation comes from the sentence: The number of seats on the floor equals the total number of seats in the mezzanine and balcony. So, x = y + z, but we need all the variables on one side so we subtract y and z to the left)

Solve using the Matrix editor

y = 144

500

Set the geometric sequence formula (applications received) equal to the arithmetic sequence formula (places available).

12300(1.03)^{n - 1} = 10380 + (n - 1)600

Replace n with x and use the numeric solver in the calculator to find k (k is n).

k = 13

500

Option 1 because the monthly payment is lower or option 2 because the total amount to repay is lower.

500

The speed of light is approximately 3 x 10⁸ ms^-1. The distance from the Earth to the Sun is 1.496 x 10⁸ km. The distance from Earth to Proxima Centauri, the nearest star, is 4.014 x 10¹³ km. Given that light from the Andromeda galaxy takes approximately 2.5 million years to reach us, find how far (in km) Andromeda is from the Earth.

To find how far, multiply the time (2.5 x 10⁶) by the speed (9.46 x 10¹²) to get:

2.37 x 10¹⁹ km

500

The pH of a solution measures its acidity and can be determined using the formula where C is the concentration of hydronium ions in the solution, measured in moles per litre. Solve for C.

Use inverse operations to solve for C. First divide by -1 on both sides to get -pH = log₁₀(C). The inverse of log is exponential, so -pH becomes the power of 10 which gives us C = 10^-pH

500

Salvador’s Fruit Mart sells variety packs. The small pack contains three bananas, two apples, and one orange for $1.80. The medium pack contains four bananas, three apples, and three oranges for $3.05. The family size contains six bananas, five apples, and four oranges for $4.65. What price should Salvador charge for his lunch-box special that consists of one banana, one apple, and one orange?

x = cost of 1 banana, y = cost of 1 apple, and z = cost of 1 orange

system:

3x + 2y + 1z = 1.80

4x + 3y + 3z = 3.05

6x + 5y + 4z = 4.65

Solve using the Matrix editor

x = $0.20

y = $0.45

z = $0.30

The lunch box special is x + y + z, so 0.20 + 0.45 + 0.30 =

$0.95