Master HSE/GED/TASC/HISET Math Concepts & Techniques 2023

Introduction to ratios and proportions, distinguishing ratios from fractions, and understanding part-to-whole relationships while seeing why combining class ratios can yield different results.

Learn to divide a total value into shares using ratios, as in splitting twenty four thousand among A, B, and C, and to combine successive ratios by aligning common values.

Explore proportion as a relational concept, using the chocolate example to show how the product of extremes equals the product of means and how quantities scale from prices to counts.

Explore the three types of proportion: mean, third, and fourth, along with direct and inverse proportions, with examples like speed and distance, and heat and water level.

Calculate percentages by dividing the obtained value by the total value and multiplying by 100. Use examples like 400 out of 500 equals 80% to compare scores.

Learn percentage change using the formula change in value over original value times 100, with an example from 350 to 450, and convert percentages to fractions by dividing by 100.

solve percentage problems by calculating values such as 20 percent of 600 and 40 percent of 350, using right-to-left approaches and percent-to-number conversions.

Compute percent off and percent of amounts using successive discounts and price changes. Apply these concepts to determine final prices for items like shirts and televisions.

Explore percent change through practical examples, calculating final values from initial 100x, handling increases and decreases, and solving reversals like 25% up with 20% down and multi-step population changes.

Explore the relationships among cost price, selling price, and mark price (mrp), and learn to compute profit percentage, loss percentage, and discount percentage from these values.

Learn to solve a discount and profit problem by graphically relating cost price, selling price, and market price, using a $600 mark price and 20% discount to find the cost.

Calculate profit or loss percentage using cost price and selling price, including gold price versus selling price example and two mobiles with varying gains and losses to determine overall profit.

Apply successive discounts of 30% and 40% to determine the car’s initial price when the final price is 1,026,000 dollars.

Understand the difference between simple interest and compound interest, and learn formulas for principal, rate, time, and amount, including how interest on interest grows the total.

Learn how to compute averages for linear data using the first and last numbers divided by two, and distinguish linear from nonlinear data with examples on even and odd numbers.

Explain how nonlinear data has no inherent order and how the weighted average is computed, illustrated by 250 boys at 12, 120 girls at 8, 30 teachers at 60, 14.4.

Solve average problems by combining data, recognizing duplicates, and computing totals. Apply these techniques to class scenarios, such as a new student changing the average.

Define mixtures as two or more substances in a single vessel and illustrate with two vessels. Use common multiples to align quantities and compute the final milk-to-water ratio.

Learn to solve ratio problems in two vessels with volumes in a 2:1 relationship, using multiples to align milk to water to salt and condensed milk to water to salt.

Balance a milk-to-water mixture from 4:1 to 1:4 by adding water. With 64 liters of milk and 16 liters of water, solving 64:(16+x) = 1:4 yields x = 240 liters.

The lecture defines speed, average speed, and relative speed, showing how to compute them with distance and time, and using opposite- and same-direction motion plus a thief and police chase.

Solve a relative-speed problem where two trains travel at 60 mph and 80 mph from opposite stations 110 miles apart, meeting after five and a half hours, 440 miles from station B.

Compute the average speed by dividing total distance by total time for a multi-leg trip. Use speeds of 40, 50, and 60 mph.

Calculate the distance between house and office using 20 m/s and 30 m/s with six seconds late and four seconds early, via D = S × t.

Use proportion and cross-multiplying to determine how many light bulbs would be inspected: seven of 400 scales to 20,000, yielding 350.

Apply direct variation to earnings: since $120 corresponds to eight attendees, earnings scale to $300 for twenty attendees by cross-multiplying 120×20/8.

Calculate the profit from a performance: with eight attendees and total revenue of $120, subtract 43% costs to obtain about $68.4 in profit.

Determine the atomic weight of the unknown element by subtracting 20% of calcium's 40 amu, yielding about 32 amu.

Explore lines and angles, define line, ray, and line segment, and study parallel and perpendicular lines, supplementary and complementary angles, vertical opposite angles, and angle relationships on transversals.

Master triangle basics: angles sum to 180 degrees. Observe side opposite the largest angle is longest; two sides exceed the third; exterior angles equal the sum of opposite interior angles.

Learn triangle formulae, including base times height over two and the inradius with semiperimeter for area. Derive equilateral triangle height as h = (√3/2) a to obtain area (√3/4) a^2.

Apply the Pythagorean theorem to find the hypotenuse using c^2 = a^2 + b^2, explore 3-4-5 triplets, and relate similar and congruent triangles through proportional sides.

Discover how special triangles reveal clear side ratios in 30-60-90 and 45-45-90 right triangles, using an equilateral triangle with a dropped perpendicular and the Pythagorean relation to find the hypotenuse.

Identify quadrilaterals such as parallelograms, squares, rectangles, rhombuses, and trapeziums, and apply properties like opposite sides and angles equality, diagonals bisecting, and relevant area formulas.

Explore circle basics, including radius, diameter, center, chord, tangent, and segments. Distinguish major and minor arcs and segments, and learn to compute arc length and sector area.

Explore three-dimensional geometry by examining cuboids and cubes, calculate lateral and total surface areas, and determine volumes and the longest rod that fits inside a cuboid.

Explore 3-d geometry by calculating cylinder, cone, sphere, hemisphere, prism, and pyramid properties, including lateral surface area, total surface area, and volume from base area, height, and radius.

Explore coordinate geometry fundamentals, including distance between points, slope and rise over run, midpoint, perpendicular slopes, and deriving line equations from intercepts or given points.

Compute the slope of a linear function from its intercepts using the slope formula −B/A, given A+B=0 and A≠B; substitution shows a positive slope, so option A is correct.

Determine the equation of the line through points B and D in the square by analyzing slope and the x-intercept, and identify the line with x-intercept 1 (option B).

Divide the hexagon into six equilateral triangles of side a, set hexagon area to 384√3, solve for a, then area of square is a^2 = 256.

Learn algebra basics by examining expressions with variables and constants, such as 3x + 4y + 3, and identify monomials, binomials, trinomials, including 3x^2 + 2 + 3 = 0.

Factor quadratic x^2 + x - 6 to reveal roots 2 and -3, then apply quadratic formula and discriminant to explain distinct roots and sum and product of roots.

Explore exponents basics and key formulas, including power rules, and expanded forms for (x+y)^2, (x+y)^3, and identities such as x^2-y^2 and x^2+y^2+2xy.

Study functions and algebraic expressions with one known variable, and solve simple equations by elimination to find x and y, noting parallel lines and solution types.

Explore inequalities, including greater than, less than, and their equal forms, on number lines, and learn how addition, subtraction, multiplication, and division, especially with negative numbers, affect inequality signs.

Explore absolute values and the modulus function, solving inequalities like |x| ≥ 5 and |x| < 5, and after a shift with |x-3|, identify the solution ranges.

Factor x^2-4x+3 into (x-3)(x-1), apply zero-product, and use the number line to solve the inequalities. For >=0, x <= 1 or x >= 3; for <=0, 1 <= x <= 3.

Solve the quadratic X^2 - 4X + 3 by locating roots at 1 and 3, and determine where the expression is greater than, less than, or equal to zero.

Compute the musician's earnings as $0.09 per download and $0.002 per stream, expressed as 0.09 D + 0.002 S, the solution to quiz Q.No.1.

Solve a spring-mass relation by substituting l = 73 cm into l = 24 + 3.5 m to find m = 14.

Solve for x-intercepts of the parabola by setting y to zero and factoring x^2 - 6x + 8 as (x-4)(x-2), yielding x = 4 or x = 2; intercepts are (4,0) and (2,0).

Explains a game scoring problem: starting with K points, lose two per failed task for 100 failures, equate K minus 200 to 200 to find K equals 400.

Model a forklift load problem with x and y for 40-pound and 65-pound boxes, using x + y ≤ 45 and 40x + 65y ≤ 2400 to find feasible loads.

Compute the composition f(g(3)) by first finding g(3)=2, then applying f to 2 to get 3; therefore f(g(3))=3.

Model a landfill capacity problem with initial 75,000 tons, annual increase 7,500 tons, and capacity of 75,000 tons (three 25,000 tons), to identify years at or above capacity.

Isolate r squared to connect distance with signal intensity and power from a ridge radio antenna, yielding r squared equals pi times a.

Use the inverse-square relation to compare two observers; a sixteen-to-one intensity ratio yields r_A : r_B = 1:4. Therefore, observer A is one quarter the distance from the antenna.

Determine A and B from y=3 and y=x^2+b, apply discriminant tests to the resulting quadratics, and conclude that option B yields exactly two real solutions.

Explore the difference between permutations and combinations, focusing on when order matters versus when it does not. Learn through simple examples with two-item arrangements, three-item selections, and football team picks.

Identify the target, decide whether order matters, draw slots, and determine if you use a permutation or a combination. Compute the total handshakes for 30 students as a combination, 435.

Compute the total diagonals of an octagon by selecting vertex pairs that form diagonals and avoiding sides, using conditional slot reasoning and the selection principle to yield 20.

Use selection and counting to pick three boys from five and two girls from six, then count rectangles and squares on a five by five chessboard.

Describe probability as the ratio of favorable to total outcomes, between zero and one; use not occurring events to find it, as shown with coin tosses and rain days.

Identify independent and dependent events, apply multiplication for 'and' and addition for 'or', and analyze replacement versus non-replacement scenarios with ball and card examples.

Master arithmetic, geometric, and harmonic progressions, learn sums of sequences including first n natural numbers, squares, and cubes, and apply formulas for S_n and harmonic relations.

Explore the difference between lists and sets, and visualize unions, intersections, and complements with Venn diagrams to solve subset and disjoint-set problems.

Use a Venn diagram and the union formula to solve two-set problems, determining cricket and football participation, their intersection, and the number who neither play cricket nor football.

Explore three-set Venn diagrams, identify A, B, and C intersections, and apply the union and intersection formulas to compute totals, including none, using a cricket football tennis example.

This lecture demonstrates solving a four-way group grid problem by classifying students into boys/girls and vegetarian/non-vegetarian, then calculating non-vegetarians as 320.

Apply the line of best fit to the scatterplot of miles traveled by air passengers for Country X to estimate the year below 2005, using values around 50 billion.

Explore statistics basics, including mean, median, and mode, plus standard deviation, normal distribution, bell curve, and percentiles, and distinguish linear and nonlinear data and data ordering.

Compute the standard deviation of the data 1, 3, 5, 7, 9 by finding the mean, squaring deviations, averaging, and taking the square root; it also shows the sigma formula.

Understand normal distribution as a statistical, bell-shaped curve centered at the mean, using standard deviations to describe data spread and noting its applicability to real-time observations.

Using a normal distribution with mean 300 and standard deviation 13 1/3, determine that scores above 313 1/3 account for 16% of students, i.e., 800 of 5,000.

Differentiate percentage from percentile using the total vs rank within a group. Arrange data ascending, find median to determine quartiles, then Q1 and Q3, with Q3−Q1 as the interquartile range.

Arrange the data in ascending order, locate the median as Q2, and find Q1 and Q3 from the halves to compute the interquartile range and understand percentiles.

Examine quiz solution steps to compute the reading time for a 349,168-word novel using 250 wpm and 3 hours per day, via minutes-to-hours-to-days conversions.

Evaluate how sampling from restaurant-goers on a Saturday biases the sample, undermining representativeness and reliability for generalizing townwide sports preferences.

Q.No.3 presents a bar exam probability problem. Among those who passed, 25 form the denominator and 7 did not take the review course, so the answer is B.

This lecture explains how mean and median differ, shows that a mean of 165,000 and a median of 125,000 indicate high-valued data pulling the mean, and contrasts symmetric distributions.

derive the median number of siblings from 600 survey responses across two schools using cumulative counts, concluding the median is one.

Compare the expected counts of students with four siblings at Lincoln and Washington by applying the 10/300 proportion to larger populations using the survey data, yielding 80 and 110.