Mastering Essential Math Concepts: Algebra to Statistics

Posted on Feb 4, 2025 in Mathematics

Multiplying Binomials

Multiply (3r – 1)(2r + 3).

Use the FOIL method, keeping the largest exponent on the left. Multiply each number on the left by both numbers on the right.

This gives you 6r² + 9r – 2r – 3. Combine like terms, 9r – 2r = 7r. This gives you (6r² + 7r – 3). This is the answer.

Dividing Fractions and Exponents

Divide:

When dividing two fractions, cross-multiply. When dividing exponents, subtract them. So:

Adding and Simplifying Algebraic Fractions

Add and simplify:

Finding X and Y Intercepts

Find the x and y intercepts for:

x-intercept =

y-intercept =

x-intercept = (6, 0); y-intercept = (0, 4)

Logarithm Rules and Examples

Logarithm Rules:

• log_e = ln
• log₁₀ = log
• log_b(ab) = log_ba + log_bb
• log_ba/b = log_ba – log_bb
• log_baⁿ = nlog_ba
• log_b1 = 0
• log_bb = 1
• log_bx = logx / logb = lnx / lnb
• log_bn = x_b^x = n

Solve the equation:

2 to what power equals 16? 2⁴. Solve for x: x = 5/2

Calculating Compound Interest

When Jason Levy was born, his grandfather deposited $10,000 into a savings account for Jason’s college education, earning 6.5% interest compounded weekly.

a. How much will there be in the account when Jason is 18?

b. How much interest will the account generate in these 18 years?

Calculating the Future Value of an Annuity

Dick Eckel recently set up an annuity to save for his retirement. He arranged to have $250 taken out of each of his monthly checks; it will earn 8% interest. He just had his 25th birthday, and his ordinary annuity comes to term when he is 65. Find the future value of his account.

r = rate; r = periodic payment; a = total amount; m = compounding coefficient; n = number of payments

Exponential Growth Model for a Population

A biology lab has 7 mice. After 10 weeks in the lab, there are 14 mice.

a. Determine the exponential model n = n₀b^t for the population of mice in this lab.

b. Using the exponential model, determine how many mice the lab will have after 30 weeks.

n₀ = 7; t = 10; n = 14

Probability with Coin Flips

A fair coin is flipped 3 times.

a. What is the sample space for this probabilistic experiment? = {HHH, HHT, HTH, THH, THT, HTT, TTH, TTT}

b. What is the probability that the coin will come up “tails” each time?

c. What is the probability that the coin will come up “tails” exactly once?

d. What are the odds that the coin will come up “tails” exactly twice?

Probability with Marbles

A marble is selected at random from a jar containing 5 red marbles, 4 yellow, and 2 green.

a. What is the probability that the marble is red?

b. What are the odds that the marble is green?

c. What is the probability that the marble is red or yellow?

d. What are the odds that the marble is not yellow?

e. What is the probability that the marble is red and green?

Probability with a Delegation

A delegation of 3 is selected randomly from a city council made up of 5 liberals and 4 conservatives.

a. What is the number of all possible delegations?

b. What is the probability that the delegation will consist of only liberals?

c. What is the probability that the delegation will consist of 2 liberals and 1 conservative?

d. What is the probability that the delegation will consist of 1 liberal and 2 conservatives?

e. What is the probability that the delegation will consist of only conservatives?

f. What is the probability distribution of the number of liberals in the delegation?

g. What is the expected number of liberals in the delegation?

Probability with Students and Courses

A UNR student is picked randomly. The UNR records indicate that the probability that the student took Math 120 is 0.4 and the probability that the student took both Math 120 and English 101 is 0.3. It is also known that the probability that the student took both Math 120 and English 101 is 0.25.

a. What is the probability that the student took Math 120 and did not take English 101?

b. What is the probability that the student took at least one of Math 120 and English 101?

c. What is the probability that the student did not take any of these two classes?

Standard Deviation and Variance

Standard Deviation:

where n = total data points; m = mean

Variance: