A complete course on ACT Math | ACT Exam Prep made easy

Explore core percentage concepts, convert between percent and fraction, and apply shortcuts for percent increases or decreases to compare quantities across arithmetic topics.

Solve a percentage word problem by setting up x from 58% minus 37% of x equals 399. Then compute 72% of the number to obtain 1368.

Practice calculating percent values by finding 38% of 1850 to determine the number of employees transferred. The calculation yields 703, reinforcing percent problems in ACT Math.

Illustration3 teaches solving a percentage problem by converting percentage to a fraction of 100, setting up an equation with an unknown x, and simplifying to find x.

Solve a proportional fraction problem by increasing the numerator by 250% and the denominator by 300% to obtain 7/9, then find the original fraction x/y = 8/9 via cross-multiplication.

Solve a percent problem by setting x as the number and equating 58% of x minus 39% of x to 247. Compute 82% of x to obtain 1066.

Learn to solve a percentage word problem by letting the monthly income be x, equating 20% of x to 3818, and solving for x.

Solve a percent problem by setting up x where 56% of x equals 463.68, then compute 25% of 828, practicing percentage equations and numeric reasoning.

Master calculating percent of a total: determine 25% of 1156 to find transferred employees, working through the steps to choose the correct option.

Calculate the approximate percentage from marks by dividing 555 by 850, showing that 555 out of 850 equals about 65%, as in the example.

Illustration 10 from a complete course on act math demonstrates percent based reasoning, computing 64% of 900 (576) and 25% of that total (144) for Hindi and English.

Calculate the two-year population by applying 7% growth then 5% decline to an initial 198,000, yielding 201,267 via a direct percent-multiplication approach.

Compute 26% of 55% of ten divided by 13th of 6100, simplifying percent notation along the way. Cancel zeros and factors to arrive at 671, the correct option D.

Determine the maximum aggregate marks by setting 36% of the aggregate equal to 261 marks, given a 40% passing threshold, resulting in 725.

Determine that one-eighth of a number equals 17.25, find the number as 138, then compute 73% of 138 to obtain 100.74 and confirm that option A is correct.

Understand how to compare like quantities using ratios with the same units, form proportions with extremes and means, apply cross-multiplication, and use the componendo-dividendo property to find mean proportion.

Multiply the given ratios to form the compound ratio, then simplify to show the final result is two to one, illustrating the correct option.

compute incomes from ratios: set a=5x and b=4x, expenses a=3y and b=2y, then use income minus expenses equals 1600 to form and solve linear equations, yielding a's income as 4000.

Subtract x from 6 and 7 to form a ratio less than 16:21, then solve 21(6-x) < 16(7-x) to get x > 2.8, so the least integer is 3.

Solve a ratio problem where the terms differ by 40, and the ratio equals 2/7 by setting x/(x+40) = 2/7, yielding x = 16 and 16/56 = 2/7.

Apply the method of key to solve proportional equations in ACT math, deriving A, B, and C in terms of k from E/3 = B/4 = C/7, then evaluate (A+B+C)/C.

Use ratio reasoning with E, B, C, and A and the method of K to equate E+B over six, B+C over seven, and C+A over eight, with A+B+C=14 deduce C=6.

Derive the ratio 4:5 in an ACT math percent increase problem by setting the third number to x and calculating 20% and 50% increases.

Set up a proportion with x to 1/27 equal to 3/11 to 5/9, then use product of extremes equals means to get x = 1/55, so option B is correct.

Compute zinc mass when copper and zinc are in a 9:4 ratio, with 24 g of copper; zinc equals 24 × 4/9 ≈ 10 2/3 g, illustrating ratio reasoning.

Solve for two numbers in a 3:4 ratio with lcm 180 by setting them as 3x and 4x, compute x as 15, giving the first number 45.

Compute third proportional to 12 and 30 via 12:30 = 30:X, obtaining X; then find mean proportional between 9 and 25 using B^2 = AC and simplify the ratio.

Determine the third proportional E for x^2 minus y^2 and x minus y using the extremes and means property, yielding E equals (x minus y) over (x plus y).

Learn to evaluate (ab+bc+ca)/(a^2+b^2+c^2) by substituting a=3x, b=4x, c=5x, simplifying to 47/50 in ACT math.

Divide by y^2 to get (x/y)^2 - 10(x/y) - 119 = 0; let t = x/y and take t = 17 since x/y have the same sign.

Learn to divide rupees 3400 among three sons using a simplified ratio of 9:6:2, identify the highest share as 1800, and verify the totals with a consistency check.

Learn the basic concepts of average, or mean, including the formula and example. Understand when harmonic mean applies to average speed with 2xy/(x+y), not (x+y)/2.

explain that the average of twenty numbers is zero, so their sum is zero, and at most nineteen numbers can be greater than zero, making option d correct.

Compute the average of the first 20 natural numbers using the sum formula, expressed with sigma as n(n+1)/2, then divide by 20.

Apply the age ratio 8:3 and the average of 33 years to solve for the man's age. Conclude the man is 48 years old (option C).

Factor 47 from 47A+47B=5452 to get A+B=5452/47=116, then compute the average as (A+B)/2=58, identifying option C as correct.

Analyze five consecutive even numbers with average 34 to determine B and D, yielding B equals 32 and D equals 36, and compute their product.

Demonstrate how to find the sixth number by using the total sum of 11 numbers and the sums of the first five and last five, resulting in 50.

Determine the weight of the new person when six people average 60 kg, and adding one person raises the average to 62 kg. The new person weighs 74 kg.

Solve an ACT math problem using averages: six numbers average seven, three numbers average five, so the remaining three have average nine.

Explore logarithms, including base ten and base two, and apply product, quotient, and power rules, change of base, and the special case log 1 = 0.

Master logarithms in ACT math through worked examples, applying the logarithm definition to solve for x in log base 2 of 16, yielding x equals 4.

Apply the definition of logarithms to compute base-ten values, as shown for 0.001 and 0.0001, concluding x equals minus four.

Solve log base 2√3 of 1728 by equating powers of 2 and 3, yielding x = 6.

Analyze a base-ten logarithm exercise, using log ten equals one and log one equals zero, while applying log properties to determine the not-correct option.

Solve a logarithmic equation by applying log laws and base-change rules, converting log base 3 and log base 2 to powers, until x is found, concluding with option D.

Apply the log power rule to simplify log_3(27^3), using 27=3^3, and compute log_3(9)=2 and log_2(2)=1.

Apply the change-of-base rule to combine log x terms, use power and base properties, and obtain log base 2 of x equals 4, hence x equals 16.

Apply logarithmic properties to simplify a multi-base expression, using log base ten equals one and the base e property, and conclude the value equals one.

Solve a base five logarithmic equation by applying log properties to simplify (x^2 + x)/(x + 1), convert to exponential form, and conclude x = 25.

Explore the change-of-base property for logarithms using a chain of bases, showing how the expressions cancel to yield one and identifying option B as correct.

Apply log properties to a product to prove x = 1 using log m^n = n log m, log(m/n) = log m - log n, and log one = zero.

Apply log properties to show that the sum log x base a plus log x^2 base a^2 plus … plus log x^n base a^n equals log x^n base e.

Use log properties: replace log m + log n with log(mn) and apply n log m = log(m^n); equate and square to get (m−n)^2=0, hence m=n.

Apply log identities to simplify an expression with log x^2, log y^2, and log(xy), showing the left-hand side reduces to zero by a difference-of-squares cancellation.

Solve log base 5 of (x+12) minus log base 5 of x equals 1 by using the log subtraction rule, then solve (x+12)/x = 5 to get x = 3.

Master the basics of linear equations, from one-variable to two-variable forms, and learn solving rules: add, subtract, multiply, divide, transposition, and cross multiplication, to find x and y.

Explore methods for solving linear equations in two variables, within the ACT Math course, including substitution, elimination, and graphical approaches, and learn how to identify unique, inconsistent, or parallel-line cases.

Understand when a two-variable linear system is consistent or inconsistent, and identify conditions for a unique solution, infinitely many solutions, or no solution, through coefficient comparisons.

Convert 1/x and 1/y to u and v, solve the resulting linear system u+2v=4 and -u+3v=11 to get u=-2, v=3, hence x=-1/2 and y=1/3.

Solve a system of two linear equations by cross-multiplication to find x and y, as demonstrated with step-by-step comparisons of expressions in ACT math prep.

Learn how to determine a unique solution for a system of linear equations using the ratio condition e1/e2 ≠ b1/b2, applied to x−y=2 and 6x−2y=3, identifying option B.

Analyze a two-equation linear system to determine no-solution conditions in ACT math exam prep, convert to standard form, and identify parameter values such as k.

Solve a parallel lines problem by cross multiplying to find k, conclude k = 15.4, and select option B.

Express triangle angles a, b, and c in terms of x, apply the 180-degree angle sum, solve x = 120, and find b = 40 degrees.

discover how to solve act math two-digit number questions by using options to test digit interchanges that yield a difference of nine, and by forming equations from digit sums.

Solve a classic age problem where five years ago one was twice the other and ten years later the same; find B's present age: 20 (option B).

Apply a two-equation work-rate model with men and boys to determine X and Y for one man and one boy, showing that one man finishes the work in 70 days.

Solve a two-equation algebra problem to find the fraction x/y that satisfies the given conditions. Verify the result by substitution, confirming the fraction is 15/26.

determine that opposite angles of a cyclic quadrilateral are supplementary, form equations from A, B, C, D to solve for x and y, then compute angles A, B, C, D and identify largest and smallest.

Analyze two linear equations in standard form to determine whether they are parallel, coincident, or intersecting, using ratios a1/a2, b1/b2, and c1/c2 to identify inconsistency and parallelism.

Apply the a1/a2 not equal to b1/b2 condition to kx minus 2y equals 3 and 3x plus y equals 5, yielding k not equal to minus six for unique intersection.

Solve a pair of equations by substituting u=1/(x+y) and v=1/(x−y), reduce to a linear system, find x=5/2 and y=1/2.

Explains how to form equations and inequalities, identifying left-hand and right-hand sides, and distinguishing linear from quadratic forms. Shows properties: add, subtract, multiply, divide, and sign changes when moving terms.

Solve the inequality 2x minus 4 ≤ 0 to get x ≤ 2, express the solution as an interval on the number line, with closed or open brackets.

Demonstrates solving a linear inequality by multiplying by minus one and flipping the sign, yielding x > 4. Visualize the solution on a number line as the interval (4, ∞).

solve the inequality 3x+17 <= 2(1 - x) to obtain x <= -3, and illustrate the solution on a number line with interval notation and equality at -3.

Solve a linear inequality to find the solution set for x, showing that x is greater than or equal to 4, represented as the interval [4, ∞).

Solve the linear inequality with fractional terms using the LCM and cross multiplication, concluding the solution set is x > 4 with an open boundary on the number line.

Master the basic concepts of quadratic equations for act math, including standard form ax^2+bx+c=0, solving by factoring, completing the square, and the quadratic formula, with discriminant insights.

Compute the discriminant D = B^2 - 4AC for x^2 + 4x + K = 0, and show that D > 0 implies K < 4.

Determine C in a quadratic with equal roots by setting the discriminant to zero, yielding C = B^2/4, and identify option d as correct.

Substitute x=2 into the first equation to get b=-8, then set the discriminant to zero for the second equation, yielding c=16 and option 16.

Solve an infinite nested radical by setting x = sqrt(6 + x), derive x^2 - x - 6 = 0, and take the positive root x = 3.

In this ACT math lecture, convert the equation to a standard quadratic via x^2 = y, analyze the discriminant to show it has imaginary roots and no real solutions.

Explore solving quadratic equations in act math by completing the square, adding and subtracting terms to form a perfect square and identify the correct option.

Compute the sum of reciprocals of the roots for a quadratic using the sum and product of roots, alpha and beta, to obtain 1/alpha+1/beta equals minus four.

Discriminant shows real distinct roots of 5x^2 - kx + 1 = 0 when k^2 > 20, i.e., k > two root five or k < minus two root five.

Solve a rectangle problem where the length exceeds the breadth by eight and the area is 240 metres. Solve the quadratic x^2+8x-240=0 to find breadth x=12 and select option A.

Determine lambda using the reciprocal roots condition for the quadratic equation 4x^2 - 2x + lambda - 4 = 0, yielding product of roots equals one and lambda equals eight.

Set the sum of the roots equal to the product for the quadratic kx^2+6x+4k=0, using alpha+beta = -b/a and alpha beta = c/a; this gives k = -3/2.

Eliminate alpha and relate the roots sin alpha and cos alpha to the quadratic a x^2 + b x + c = 0. Use the sum and product of roots with sin^2 alpha + cos^2 alpha = 1 to derive b^2 = a^2 + 2ac.

Explore complex numbers, including imaginary unit i, their real and imaginary parts, and operations—addition, subtraction, multiplication, division by conjugates—plus modulus and equality insights.

Practice solving a sequence sum of powers of two with alternating signs to evaluate the expression and identify the correct option.

Compute the product of sqrt(-2) and sqrt(-3) by writing each as i times the root, then apply i^2 = -1 to get - sqrt(6), confirming option B.

Demonstrate how to simplify a complex fraction by multiplying by conjugates, applying i^2=-1, and rationalizing the denominator, culminating in the value 9i/4.

Identify the additive inverse by finding a value that sums to zero with the original number. For 1 minus ETA, the inverse is minus 1 plus ETA.

Illustration demonstrates evaluating the imaginary part of a complex quotient by multiplying by the conjugate, expanding, and simplifying to obtain imag(z) = 4/5.

Tackle a seventh ACT math question by simplifying a complex expression using conjugates, powers of eta, and exponent rules to identify the correct option.

Explore solving a complex-number power equation by equating moduli: only n = 0 satisfies (1−i)^n = 2^n, with step-by-step option testing and modulus reasoning.

Apply binomial expansion to compute (1+η)^8+(1−η)^8. Use η^2 = -1 to simplify (1±η)^2 to 2η and -2η, yielding (2η)^4 for both terms and a total of 32.

Explore the basics of arithmetic progression (AP), including sequences, common difference, the general term, and how to compute sums and means between numbers.

Identify the first term and common difference in the progression. Apply t_n = a + (n-1)d to determine that -39 is the 20th term.

Solve for x so that 8x+4, 6x-2, and 2x+7 form an arithmetic progression, yielding x = 15/2.

Compute the arithmetic mean of 33 and 77 using the formula M = (A + B) / 2; the result is 55, so option C is correct.

Learn to sum a series with first term 9, common difference -4, and 200 terms using the formula S_n = n/2[2a+(n−1)d], yielding a negative total.

Determine the number of terms in the AP with first term -8 and sum 52 by using the sum formula and factoring to obtain n = 13 (option B).

Explore solving arithmetic progression problems by using the sum formula s_n and the relation s_n - s_{n-1} to find the next term, leading to option C.

Find the first term of an arithmetic progression given the seventh term equals four and the common difference is minus four; using a7 = a + 6d, a = 28.

Solve for the first term and common difference of an arithmetic progression from the second and fifth terms, then compute the 17th term.

Identify the arithmetic progression 5, 13, 21, …, 181, determine n with d = 8, then apply the sum formula s_n = n/2(a1 + an) to obtain 2139.

Calculate the 20th term of this arithmetic progression by reversing the series from 53 to 3 and applying t_n = a + (n−1)d to get 158.

Identify the arithmetic progression with a1 = 25 and d = -5, where terms decrease by five; the first negative term is the seventh term, so option A is correct.

Treat the first 40 numbers divisible by six as an arithmetic progression with a = 6 and d = 6; the last term is 240 and the sum is 4920.

Explore an ACT math AP problem by linking the first term a, last term l, and sum S; derive d and apply AP sums to solve for k.