ACT Math Missed Questions

The first and second terms of a geometric sequence are a and ab, in that order. What is the 643rd term of the sequence? A. (ab)^642 B. (ab)^643 C. a^642b D. a^643b E. ab^642

E

A shoe is $39 this price has a 30% tax. What was paid for it.

$30. Tax, Delivery Fee, Service Charge,

If y = z, what are the real values of x that make the following inequality true? xy − xz/3y − 3z < 0 A. All negative real numbers B. All positive real numbers C. −1/3 only D.1/3 only E. 3 only

A

Three years ago, the population of a certain species of bird was calculated at 20 birds per acre. This year, a biologist recorded a total of 47 birds in an area equal to 3.25 acres. By about what percentage has the bird population in the biologist's sample decreased over the last 3 years, to the nearest tenth? A. 14.7% B. 27.7% C. 38.3% D. 42.6% E. 72.3%

B

What is the equation of the circle in the standard (x, y) coordinate plane that has a radius of 4 units and the same center as the circle determined by x2 +y2 −6y+4 = 0? A. x^2 + y^2 = −4 B. (x + 3)^2 + y^2 = 16 C. (x − 3)^2 + y^2 = 16 D. x^2 + (y + 3)^2 = 16 E. x^2 + (y − 3)^2 = 16

E

If x and y are positive integers such that the greatest common factor of x2y2 and xy3 is 27, then which of the following could y equal? A. 81 B. 27 C. 18 D. 9 E. 3

E; GCF of x^2y^2 and xy^3

A triangle, ABD, is reflected across the line y = x to have the image A'B'D' in the standard (x, y) coordinate plane: thus A reflects to A'. The coordinates of point A are (m, n). What are the coordinates of point A'? F. (−m, n) G. (m, −n) H. (−m, −n) J. (n, m) K. Cannot be determined from the given information.

The correct answer is J. The image of a point (x, y) reflected across the line y = x will have coordinates (y, x). If the coordinates of point A are (m, n), then the coordinates of point A are (n, m).

For the area of a square to triple, the new side lengths must be the length of the old sides multiplied by: A. √3 B. 3 C. 4 D. 2√3 E. 9

The correct answer is A. Since the area of a square is equal to the square of its sides, multiplying the sides by √3 will have the effect of multiplying the area by (√3)² = 3.

In the (x, y) coordinate plane, what is the y-intercept of the line 5x + 3y = 8? A. 8/3 B. 3 C. 5/3 D.3/5 E. −5/3

The correct answer is A. To solve this problem, convert 5x + 3y = 8 to the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. To do so, first subtract 5x from both sides to get 3y = −5x + 8. Dividing the entire equation by 3 yields y = −5x/3 + 8/3. Thus 8/3 is the y-intercept.

The average of 7 consecutive numbers is 16. What is the sum of the least and greatest of the 7 integers? A. 13 B. 14 C. 16 D. 19 E. 32

The correct answer is E. You can apply common sense to solve this problem. If the average of 7 consecutive integers is 16, it would make sense that the middle number is 16 (this assumption only holds because there are an odd number of integers and because the integers are consecutive). Thus, the list of consecutive integers is 13, 14, 15, 16, 17, 18, 19. The sum of the first and last integers is 13 + 19 = 32.

Given that y − 5 = 1/2x + 1 is the equation of a line, at what point does the line cross the x axis? F. −15 G. −12 H. 1 J. 4 K. 6

The correct answer is G. A graph crosses the x-axis at the point when y = 0. Given that y − 5 = 12x +1, let y = 0 such that −5 = 12x +1. Subtracting 1 from both sides yields −6 = 12x. Multiplying by 2 yields −12 = x.

Which of the following is a value of n that satisfies logn64 = 2? F. 4 G. 6 H. 8 J. 12 K. 32

The correct answer is H. Logarithms are used to indicate exponents of certain numbers called bases. By definition, loga b = c, if ac = b. Therefore, logn 64 = 2 implies that 64 = n2. Thus n = 8.

Which of the following expressions is a polynomial factor of a^16 − 16? F. a⁴ − 4 G. a⁴ + 4 H. a⁴+ 2 J. a + 2 K. a − 2

The correct answer is H. Remember that a differ- ence of squares factors easily, such as: a^2 − b^2 = (a + b)(a − b). Using the same technique, you can factor a^16 − 16 into (a8 + 4)(a8 − 4). The factor (a8 − 4) is another difference of squares, so it can be factored further into itself: (a8 − 4) = (a4 + 2)(a4 − 2). Of these factors, only (a4 + 2) is an answer choice.

Points A, B, and C are three distinct points that lie on the same line. If the length of AB is 19 meters and the length of BC is 13 meters, then what are all the possible lengths, in meters, for AC? F. 6 only G. 32 only H. 6 and 32 only J. Any number less than 32 or greater that 6 K. Any number greater than 32 or less than 6

The correct answer is H. Since AB is longer than BC, there are only two possible configurations: B is between A and C or C is between A and B. In the case that B is between A and C, AC = AB + BC = 19 + 13 = 32. In the case that C is between A and B, AC = AB − BC = 19 − 13 = 6. Therefore, AC can be 6 and 32 only.

A shoe store charges $39 for a certain type of sneaker. This price is 30% more than the amount it costs the shoe store to buy one pair of these sneakers. At an end-of-the year sale, sales associates can purchase any remaining sneakers at 20% off the shoe store's cost. How much would it cost an employee to purchase a pair of sneakers of this type during the sale (excluding sales tax)? F. $31.20 G. $25.00 H. $24.00 J. $21.84 K. $19.50

The correct answer is H. To solve this problem, recognize that the $39 price of the sneakers is 30 percent more than the amount it costs the store to purchase one pair of the sneakers. This can be represented as 130%, or 1.3. Thus, the price that the store pays for the sneakers is $39/1.3, or $30. At the end of the year sales associates get 20% off of this $30 price, therefore paying 80% of the price the shoe store pays. The cost to the employees is $30 × 0.80, or $24.

What is the slope of any line parallel to the y-axis in the (x, y) coordinate plane? F. −1 G. 0 H. 1 J. Undefined K. Cannot be determined from the given information

The correct answer is J. Any line parallel to the y-axis is a vertical line. Vertical lines have slopes that are undefined. Remember that the definition of slope is rise/run; vertical lines have no run and thus dividing rise by run is dividing by 0, making the quotient undefined.

What is the smallest possible value for the product of 2 integers that differ by 7? F. 8 G. 0 H. −6 J. −10 K. −12

The correct answer is K. The smallest possible value will occur when it is negative. A negative product will result only when one of the numbers is positive and one is negative. The possible pairs are then −1 and 6, −2 and 5, −3 and 4, −4 and 3, −5 and 2, and −6 and 1. Of these pairs, the smallest product is (−3)(4) = (−4)(3), or −12.

Melissa had 3 fewer apples than Marcia. Then, she gave 2 apples to Marcia. Now how many fewer apples does Melissa have than Marcia? F. 0 G. 2 H. 3 J. 5 K. 7

The correct answer is K. To solve this problem, it is useful to assign values to the number of apples that Melissa and Marcia both possess. If Marcia has 10 apples, Melissa has 10 − 3 = 7 apples. If Melissa gives 2 of her 7 apples to Marcia, Melissa is left with 7−2 = 5 apples. When Marcia receives 2 more apples, she has 10 + 2 = 12 apples. Since Marcia now has 12 apples and Melissa now has 5 apples, Melissa has 12−5 = 7 fewer apples than Marcia.