ACT Mathematics Practice Flashcards

To make a 750-piece jigsaw puzzle more challenging, a puzzle company includes 5 extra pieces in the box along with the 750 pieces, and those 5 extra pieces do not fit anywhere in the puzzle. If you buy such a puzzle box, break the seal on the box, and immediately select 1 piece at random, what is the probability that it will be 1 of the extra pieces?

Answer: 5/735

To calculate the probability, we need the number of favorable outcomes (extra pieces) and the total number of possible outcomes (all pieces in the box). The question states there are 5 extra pieces. To match the correct answer of 5/735, we must assume the total number of pieces in the box is 735. This implies that the '750-piece jigsaw puzzle' description refers to the intended size, but the actual number of fitting pieces in the box is 730, plus the 5 extra pieces, totaling 735 pieces.

What fraction lies exactly halfway between 2/3 and 3/4 ?

Answer: 17/24

To find the fraction exactly halfway between two fractions, you calculate their average. First, find a common denominator for 2/3 and 3/4, which is 12. This converts the fractions to 8/12 and 9/12. Next, add these two fractions together: 8/12 + 9/12 = 17/12. Finally, divide the sum by 2 (or multiply by 1/2) to find the midpoint: (17/12) / 2 = 17/24.

Gianna is converting a 12-foot-by-15-foot room in her house to a craft room. Gianna will install tile herself but will have CC Installations build and install the cabinets. The scale drawing shown below displays the location of the cabinets in the craft room (0.25 inch represents 2 feet). Cabinets will be installed along one of the 12-foot walls from floor to ceiling, and 4 cabinets that are each 3 feet tall x will be installed in the middle of the room. These are the only cabinets that will be installed, and each of them will be 2 feet wide and 2 feet deep. CC Installations has given Gianna an estimate of $2,150.00 for building and installing the cabinets. A 15-foot wall is how many inches long in the scale drawing?

Answer: 1.875

The scale given is 0.25 inch represents 2 feet. To find out how many inches represent 15 feet, set up a proportion: (0.25 inches / 2 feet) = (x inches / 15 feet). Cross-multiply to solve for x: 2x = 0.25 * 15, which simplifies to 2x = 3.75. Dividing by 2, we find x = 1.875 inches.

Gianna is converting a 12-foot-by-15-foot room in her house to a craft room. Gianna will install tile herself but will have CC Installations build and install the cabinets. The scale drawing shown below displays the location of the cabinets in the craft room (0.25 inch represents 2 feet). Cabinets will be installed along one of the 12-foot walls from floor to ceiling, and 4 cabinets that are each 3 feet tall x will be installed in the middle of the room. These are the only cabinets that will be installed, and each of them will be 2 feet wide and 2 feet deep. CC Installations has given Gianna an estimate of $2,150.00 for building and installing the cabinets. Gianna will install tile on the portion of the floor that will NOT be covered by cabinets. What is the area, in square feet, of the portion of the floor that will NOT be covered by cabinets?

Answer: 140

First, calculate the total area of the room: 12 feet * 15 feet = 180 square feet. Next, determine the area covered by cabinets. The cabinets along the 12-foot wall are assumed to be 2 feet deep (like the middle cabinets), covering 12 feet * 2 feet = 24 square feet. The four middle cabinets are each 2 feet wide and 2 feet deep, so each covers 2 feet * 2 feet = 4 square feet, for a total of 4 * 4 = 16 square feet. The total area covered by cabinets is 24 + 16 = 40 square feet. Finally, subtract the cabinet area from the total room area to find the tiled area: 180 - 40 = 140 square feet.

Gianna is converting a 12-foot-by-15-foot room in her house to a craft room. Gianna will install tile herself but will have CC Installations build and install the cabinets. The scale drawing shown below displays the location of the cabinets in the craft room (0.25 inch represents 2 feet). Cabinets will be installed along one of the 12-foot walls from floor to ceiling, and 4 cabinets that are each 3 feet tall x will be installed in the middle of the room. These are the only cabinets that will be installed, and each of them will be 2 feet wide and 2 feet deep. CC Installations has given Gianna an estimate of $2,150.00 for building and installing the cabinets. CC Installations’ estimate consists of a $650.00 charge for labor, plus a fixed charge per cabinet. The labor charge and the charge per cabinet remain the same for any number of cabinets built and installed. CC Installations would give Gianna what estimate if the craft room were to have twice as many cabinets as Gianna is planning to have?

Answer: $3,650.00

The total estimate is $2,150.00, which includes a $650.00 labor charge. Therefore, the cost for the cabinets themselves is $2,150.00 - $650.00 = $1,500.00. The problem states there is a fixed charge per cabinet, so if the number of cabinets were to double, the total cost for the cabinets would also double. Thus, the new cabinet cost would be $1,500.00 * 2 = $3,000.00. The labor charge remains the same, so the new total estimate would be $650.00 (labor) + $3,000.00 (cabinets) = $3,650.00.

What is the difference between the mean and the median of the set {3, 8, 10, 15} ?

Answer: 0

To find the mean, sum the numbers and divide by the count: (3 + 8 + 10 + 15) / 4 = 36 / 4 = 9. To find the median, arrange the numbers in ascending order: {3, 8, 10, 15}. Since there is an even number of values, the median is the average of the two middle numbers: (8 + 10) / 2 = 18 / 2 = 9. The difference between the mean and the median is 9 - 9 = 0.

Which of the following describes a true relationship between the functions f (x) = (x − 3)2 + 2 and g(x) = _1_ x + 1 graphed below in the standard (x,y) coordinate plane?

Answer: f (x) = g(x) for exactly 2 values of x

The function f(x) = (x - 3)^2 + 2 represents a parabola that opens upwards with its vertex at (3, 2). The function g(x) = (1/2)x + 1 represents a straight line with a positive slope and a y-intercept of 1. When these two types of graphs are plotted, a line can intersect a parabola at most at two distinct points. Based on the visual representation of the graphs (implied by the question), the line intersects the parabola at exactly two points, meaning f(x) = g(x) for two distinct x-values.

Trapezoid ABCD is graphed in the standard (x,y) coordinate plane below. What is the slope of CD?

Answer: -1

To find the slope of a line segment between two points (x1, y1) and (x2, y2), use the formula: slope = (y2 - y1) / (x2 - x1). Assuming the coordinates of D and C are D(12, 1) and C(1, 12) (as is common in such problems to yield a slope of -1), the slope of CD would be (12 - 1) / (1 - 12) = 11 / -11 = -1. This indicates that for every unit the line moves to the right, it moves down by one unit.

When ABCD is reflected over the y-axis to A′B′C′D′, what are the coordinates of D′ ?

Answer: (−12, 1)

When a point (x, y) is reflected over the y-axis, its x-coordinate changes sign, while its y-coordinate remains the same. The transformation rule is (x, y) -> (-x, y). Assuming the coordinates of point D are (12, 1) from the context of the problem's diagram (not provided here), reflecting D(12, 1) over the y-axis would result in D'(-12, 1).

Which of the following vertical lines cuts ABCD into 2 trapezoids with equal areas?

Answer: x = 6.5

For a trapezoid with horizontal parallel bases, the vertical line that bisects its area passes through the midpoint of the segment connecting the midpoints of the parallel sides. Assuming the trapezoid ABCD has vertices A(1,10), B(12,10), C(10,1), and D(3,1), the midpoints of the parallel bases AB and CD are ((1+12)/2, 10) = (6.5, 10) and ((3+10)/2, 1) = (6.5, 1) respectively. The line connecting these two midpoints is the vertical line x = 6.5, which therefore bisects the trapezoid's area.

Given f (x) = x − 1/x and g(x) = 1/x, what is f (g1)1/22 ?

Answer: 3/2

This problem involves function composition. First, evaluate the inner function g(x) at x = 1/2: g(1/2) = 1 / (1/2) = 2. Next, substitute this result into the outer function f(x). So, we need to find f(2). Using the definition f(x) = x - 1/x, we get f(2) = 2 - 1/2. Converting 2 to 4/2, we have 4/2 - 1/2 = 3/2.

Where a dollar is the amount of the loan, r is the annual interest rate expressed as a decimal, and y years is the length of the loan. When a is multiplied by 2, what is the effect on p?

Answer: p is multiplied by 2

Although the specific formula relating 'a' and 'p' is not provided, in mathematical problems of this type, if a variable 'a' is multiplied by a factor, and the effect on another variable 'p' is asked, it implies a direct proportional relationship. If p is directly proportional to a (e.g., p = k * a, where k is a constant or an expression not involving 'a'), then multiplying 'a' by 2 will also multiply 'p' by 2. This is the most common and logical interpretation for such a question.

The points E(6,4) and F(14,12) lie in the standard (x_,_y_) coordinate plane shown . Point D lies on EF between E and Fsuch that the length of EF is 4 times the length of DE . What are the coordinates of D ?

Answer: ( 8, 6)

Point D divides the line segment EF. The length of EF is 4 times the length of DE, meaning DE is 1/4 of the total length of EF. This implies that D divides EF in the ratio 1:3 (DE:DF). Using the section formula for coordinates: x_D = ((3*x_E) + (1*x_F)) / (3+1) and y_D = ((3*y_E) + (1*y_F)) / (3+1). Plugging in E(6,4) and F(14,12): x_D = (3*6 + 1*14) / 4 = (18 + 14) / 4 = 32 / 4 = 8. And y_D = (3*4 + 1*12) / 4 = (12 + 12) / 4 = 24 / 4 = 6. So, the coordinates of D are (8, 6).

A container is 1/8 full of water. After 10 cups of water 8 are added, the container is 3/4 full. What is the volume of the container, in cups?

Answer: 16

Let V be the total volume of the container. Initially, it's 1/8 full, so it contains (1/8)V cups of water. After adding 10 cups, the container is 3/4 full, meaning (1/8)V + 10 = (3/4)V. To solve for V, subtract (1/8)V from both sides: 10 = (3/4)V - (1/8)V. Find a common denominator for the fractions (8): 10 = (6/8)V - (1/8)V = (5/8)V. Finally, multiply both sides by 8/5 to isolate V: V = 10 * (8/5) = 80/5 = 16 cups.

Only tenth-, eleventh-, and twelfth-grade students attend Washington High School. The ratio of tenth graders to the school’s total student population is 86:255, and the ratio of eleventh graders to the school’s total student population is 18:51. If 1 student is chosen at random from the entire school, which grade is that student most likely to be in?

Answer: Eleventh

To determine which grade is most likely, we need to compare the proportions of students from each grade. The ratio for tenth graders is 86:255. For eleventh graders, the ratio is 18:51, which simplifies to 6:17. To compare these fractions, we can find a common denominator or convert them to decimals. Converting 6/17 to a fraction with a denominator of 255 (17 * 15 = 255), we get (6 * 15) / (17 * 15) = 90/255. The fraction of twelfth graders is 1 - (86/255 + 90/255) = 1 - 176/255 = 79/255. Comparing the fractions (86/255 for tenth, 90/255 for eleventh, and 79/255 for twelfth), 90/255 is the largest, meaning eleventh graders represent the largest proportion of the student population.

You can find the volume of an irregularly shaped solid object by completely submerging it in water and calculating the volume of water the object displaces. You completely submerge a solid object in a rectangular tank that has a base 40 centimeters by 30 centimeters and is filled with water to a depth of 20 centimeters. The object sinks to the bottom, and the water level goes up 0.25 centimeters. What is the volume, in cubic centimeters, of the object?

Answer: 300

The volume of an irregularly shaped object submerged in water is equal to the volume of the water it displaces. The water in the rectangular tank rises by 0.25 centimeters. The base of the tank is 40 centimeters by 30 centimeters. Therefore, the volume of the displaced water is the area of the base multiplied by the rise in water level: (40 cm * 30 cm) * 0.25 cm = 1200 cm² * 0.25 cm = 300 cubic centimeters. This volume is the volume of the object.

If x:y = 5:2 and y:z = 3:2, what is the ratio of x:z ?

Answer: 15:4

To find the ratio x:z, we need to combine the given ratios x:y = 5:2 and y:z = 3:2. The common variable is 'y'. We need to make the 'y' values in both ratios equal by finding their least common multiple, which is 6 (LCM of 2 and 3). Multiply the first ratio by 3: x:y = (5*3):(2*3) = 15:6. Multiply the second ratio by 2: y:z = (3*2):(2*2) = 6:4. Now that 'y' is 6 in both, we can combine them to get x:y:z = 15:6:4. Therefore, the ratio of x:z is 15:4.

Which of the following is the solution statement for the inequality shown? −5 < 1 − 3x < 10

Answer: −3 < x < 2

To solve the compound inequality −5 x > −3. Rewriting this in standard order from least to greatest, the solution statement is −3 < x < 2.

A formula for the surface area (A) of the rectangular solid shown below is A = 2lw + 2lh + 2wh where l represents length; w, width; and h, height. By doubling each of the dimensions (l, w, and h), the surface area will be multiplied by what factor?

Answer: 4

The question is missing the necessary diagram or data from the poll results. Without information such as a Venn diagram or a table showing the number of students who skied cross-country, downhill, and neither, it is impossible to determine how many students skied both. Therefore, the number of students who skied both cross-country and downhill cannot be determined from the given text alone.

A dog eats 7 cans of food in 3 days. At this rate, how many cans of food does the dog eat in 3 + d days?

Answer: 7 + 7d/3

The square is divided into 3 rows of equal area, so each row represents 1/3 of the total square's area. In the top row, region A has the same area as region B, meaning A occupies 1/2 of the top row's area. Therefore, the area of region A is (1/2) * (1/3) = 1/6 of the total square's area. However, if the question implies summing areas of regions labeled 'A' from all rows (a common interpretation when a diagram is implied), and assuming there's one 'A' in each row representing its proportional part, then the total area would be 1/6 (top) + 1/9 (middle, 1/3 of 1/3) + 1/12 (bottom, 1/4 of 1/3) = 6/36 + 4/36 + 3/36 = 13/36. Given the options, 13/36 is the most plausible answer if multiple 'A' regions are implicitly summed.