ACT study set math

Word Problems

1) Label your variables 2) ID what you're solving for 3) Translate words into algebra, one step at a time

A television that normally sells for $260.00 is reduced in price by 15%. What is the reduced price of the television? A. $221.00 B. $245.00 C. $258.00 D. $275.00 E. $299.00

A. $221.00

In the standard (x,y)coordinate plane, what is the slope of the line containing the points (−2,4) and (2,4)? A. 0 B. 1/3 C. 2 D. 3 E. 4

A. 0

The price and demand for a company's product are related by the equation p(q+4)=100, where p is is the price per unit, in dollars, and q is the number of units the company is able to sell. How many units can the company sell if their price per unit is $10.00? A. 36 B. 40 C. 44 D. 360 E. 386

A. 36

This spring, Malik purchased 3 trays of pepper plants and 5 trays of tomato plants for $26. If each tray of tomato plants costs twice as much as each tray of pepper plates, how much did he pay for all the tomato plates? A. $9.75 B. $13.00 C. $16.25 D. $20.00 E. $23.00

D. $20.00

To keep up with rising expenses, a motel manager needs to raise the $40.00 room rate by 22%. What will be the new rate? A. $40.22 B. $42.20 C. $48.00 D. $48.80 E. $62.00

D. $48.80

What is the solution for y of the equation 3y + 2/3 = 5y A. -1/2 B. -1/3 C. 1/6 D. 1/3 E.4/3

D. 1/3 Isolate y on one side of the equation by doing the same thing to both sides. Subtract 3y from both sides and then divide both sides by 2: 3y − 3y + 2/3 = 5y − 3y. So, 2/3 = 2y, and y = 1/3.

In the standard (x, y) coordinate plane, the coordinates of the endpoints of AJAJare (10,14) and (22,6). What is the x-coordinate of the midpoint of AJAJ ? A. 10 B. 12 C. 14 D. 16 E. 32

D. 16

An ice cream store determines the cost of its sundaes by using the formula C= 0.50s + 0.35n + 0.25t, where C is the total cost in dollars, s is the number of scoops of ice cream, n is the number of scoops of nuts, and t is the number of liquid toppings. A Nutty Sundae costs $3.55. It has 3 scoops of nuts and 2 different liquid toppings. How many scoops of ice cream are in this sundae? A. 1 B. 2 C. 3 D. 4 E. 5

D. 4 Plug the given values into the formula and solve for s, the number of scoops of ice cream: 3.55 = 0.50s + 0.35(3) + 0.25(2). This simplifies to 3.55 = 0.50s + 1.05 + 0.50, so 2.00 = 0.50s. Multiply both sides by 2 to get s on the right side, which gives 4.00 = s.

For real numbers a, b, and c, the ratio of a to b is 3:4 and the ratio of b to c is 3:5. What is the ratio of a to c ? A. 1:1 B. 3:5 C. 4:5 D. 5:4 E. 9:20

D. 9:20

If 4x = 3x - 1 then x= ? A. y+1 B. 4/3y - 1 C. 4/3y + 1 D. 4y-1/3 E. 4y+1/3

E. 4y+1/3 Solve the equation 4y=3x−14y=3x-1 for x in terms of y. First, add 1 to both sides to obtain 4y+1=3x4y+1=3x. Then divide both sides by 3 to get 4y+1/3 = x.

Senja has 105 solid-colored disks that are either red, blue, or green. She lines them up on the floor and finds that there are 30 more red disks than green and 21 more green disks than blue. How many red disks does she have? A. 11 B. 18 C. 32 D. 52 E. 62

E. 62 Convert the words into equations and solve for the number of red disks. Let r equal the number of red disks, b the number of blue disks, and gg the number of green disks. There are 30 more red disks than green, so r = g+ 30r = g + 30. There are 21 more green disks than blue, so g = b + 21g = b + 21. There are a total of 105 disks, so r + b + g=105. The question asks for the number of red disks, so state all the quantities in terms of r. Rewrite the first equation as g = r−30g = r-30 and the second equation as b = g−21b = g-21. Since you know that g = r−30g = r-30, substitute r−30r-30 for g into the second equation to obtain b=(r−30)−21=r−51b=(r-30)-21=r-51. Substitute r−51 for b and r−30r for g into the equation r + b + g = 105r + b + g=105 to yield r+(r−51)+(r−30)=105r+(r-51)+(r-30)=105. This simplifies to 3r−81 = 1053r-81 = 105, which further simplifies to 3r = 1863r = 186. Divide both sides by 3 to get r=62.

Rates

Rate x Time = Distance Use whichever two parts you have to find the part you're missing.

Percent Change

Percent change = amount/original x 100%

Common Multiplier

a number multiplied by both the numerator and denominator to get an equivalent ratio

Ratio

one number compared to another number

Percent

part/whole = x 100%

Justin's goal this past summer was to save an average of $6.00 per week for 10 weeks. He saved a total of $56.00 for the 10 weeks. On average, how much more should Justin have saved each week to reach his goal? A. $0.11 B. $0.40 C. $0.50 D. $0.56 E. $0.93

B. $0.40 If Justin had saved $6 per week for 10 weeks, he would have saved a total of $6 × 10 = $60. Since he only saved $56, he was $60 − $56 = $4 short. So he should have saved $4/ 10 weeks = $0.40

What is the slope of the line in the standard (x, y) coordinate plane that has equation 2x + 3y = 4 ? A. -3/2 B. -2/3 C. 2/3 D. 3/2 E. 3

B. -2/3 Convert the equation to slope-intercept format, y = mx + b. The coefficient of x will be the slope. Subtract 2x from both sides of the given equation, and then divide both sides by 3 to solve for y.

This year, 75% of the graduating class of Harriet Tubman High School had taken at least 8 math courses. Of the remaining class members, 60% had taken 6 or 7 math courses. What percent of the graduating class had taken fewer than 6 math courses? A. 0% B. 10% C. 15% D. 30% E. 45%

B. 10%

Due to inflation, a car that formerly sold for $15,000 now sells for 10% more. Which of the following calculations gives the current cost, in dollars, of the car? A. 15,000 + 10 B. 15,000 + 15,000(0.01) C. 15,000 + 15,000(0.10) D. 15,000 + 15,000(10) E. 15,000(0.10)

B. 15,000 + 15,000(0.01)

Slope - intercept form of a line

y= mx+b m=slope b=y intercept

A line in the standard (x,y) coordinate plane is parallel to the x-axis and 5 units above it. Which of the following is an equation of this line? A. y=5 B. x=5 C. y=5x D. y= x+5 E. x = y+5

A. y =5 Parallel lines in the standard (x,y) coordinate plane have the same slope but different y-intercepts. The x-axis is defined by the equation y = 0; since slope is the change in y over the change in x, the slope of the line y = 0 is 0. Thus, a line parallel to the x-axis but 5 units above it would have a y-intercept of 5 and be defined by the equation y = 5.

Hannah has a part-time job making holiday wreaths. This week, she made x medium wreaths, twice as many large wreaths as medium wreaths, and 1/2 as many small wreaths as medium wreaths. In terms of x, how many wreaths did Hannah make this week? A. 3x B. 3.5x C.4x D. 4.5x E. 6.5x

B. 3.5x Convert the wording into an algebraic expression, using x as the number of medium wreaths. The number of large wreaths was 2x, and the number of small wreaths was 1/2x. Thus, the total number of wreaths was x + 2x + 1/2x = 3 1/2x

Apel's Apple Farm has 2 rectangular orchards that are each 500 ft long and 800 ft wide. Each of these 2 orchards is divided into 20-foot-by-20-foot squares, and every square contains exactly 1 apple tree. Each tree is staked to be vertical. Each stake is 2 ft from the base of the tree along the level ground. A taut wire connects the stake at ground level to the tree at a spot on the tree 5 ft from the ground. Last year, the farm's average yield per producing tree was 200 apples. This year, 43% of the trees in the first orchard are producing fruit. How many trees in the first orchard are producing fruit? A. 215 B. 285 C. 430 D. 500 E. 570

B. 430

Each point on a line in the standard (x,y)coordinate plane is equidistant from (7,3)and (-3,3). Which of the following is an equation of that line? A. x=0 B. x=2 C. x=3 D. x=4 E. x=10

B. x=2

The number of bricks, B, needed to build a wall of length L feet and uniform height H feet can be found by the equation B = 7LHB = 7LH. A wall of uniform height that is 16 feet long is constructed using 280 bricks. What is the height, in feet, of the wall? A. 0.25 B. 1.75 C. 2.5 D. 17.5 E. 40

C. 2.5 Substitute the values for L and B into the equation B = 7LH and solve for H: 280 = 7(16)H = 112H So, H= 280/ 112 = 2.5

What is the slope of the line given by the equation 3x−8y=−16? A. -8 B. -3 C. 3/8 D. 8/3 E. 3

C. 3/8

Two parallel lines are drawn in the standard (x,y) coordinate plane. One line has slope 1/4. The other line passes through the points (2,4) and (9,y). What is the value of y ? A. 1 3/4 B. 4 C. 5 3/4 D. 11 E. 18

C. 5 3/4

A baseball team played its first 20 games and won 12 of them. Then, the team went on a losing streak and lost its next 4 games. How many consecutive additional victories does the baseball team need in order to bring its winning percentage back to at least what it was just before this 4-game losing streak? A. 2 B. 4 C. 6 D. 8 E. 12

C. 6

At Rex's Service Station, 15 oil changes are performed each hour. At Star's Service Station, 12 oil changes are performed each hour. Which of the following expressions gives the number of oil changes that are performed in 8 hours at these 2 stations? A. 12(15)/8 B. 8(12)(15) C. 8(15) + 12 D. 8 (15 - 12) E. 8(15 + 22)

E. 8 (15 + 22) Convert the wording into an algebraic expression. Use the rate formula, Quantity = Rate × Time, and be certain to include the units in your calculations. At Rex's Service Station, in 8 hours there are 8(15)8(15) oil changes performed; at Star's Service Station, there are 8(12)8(12) performed. The total is thus 8(15)+8(12), which can also be stated as 8(15+12).

In a certain high school, exactly 4 out of every 15 students are in the high school band. If 300 students go to the high school, how many are in the band? A. 20 B. 24 C. 60 D. 75 E. 80

E. 80

For the equation y = kx + g representing a line in the standard (x, y) coordinate plane, g represents the: A. vertex B. focus C. slope D. x-intercept E. y-intercept

E. y - intercept The slope-intercept format for a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept. In the given equation, g corresponds to b, so g is the y-intercept. Note that the y-intercept is g when x equals 0.

Whenever the equation 5x + 3 = 2y is true, x = ? F. 2/5y - 3 G. 2/5y - 3/5 H. 2y - 3 J. 2y + 3 K. 5/2y + 3/2

G. 2/5y - 3/5

What is the slope of the line given by the equation -2x + 5y = 20? F. 2 G. 5 H. 2/5 J. 5/2 K. -2

H. 2/5

Percent Word Problems

Identify what the question is asking Write down the information in the problem Convert percents to fractions or decimals

-Multiplication; Division -Addition; Subtraction

Isolating Variables

Solving Equations

Isolating Variables; Solving for one variable in terms of another; Word Problems

In the standard (x,y) coordinate plane, what is the midpoint of the line segment with endpoints (2,3) and (8,9)? F. 11 G. (5/2, 17/2) H. (3,3) J. (5,6) K. (6,6)

J. (5,6)

In the standard (x,y) coordinate plane, the slope of the line through the points (2,2b) and (b,0) is 2. Which of the following is the value of b? F. 2/5 G. 1/2 H. 4/5 J. 1 K. 2

J. 1

Solving in "terms of"

Solving for a variable with a different variable in its solution; a variable will be part of the final expression. E.g. x in terms of y: x = 2y - 3

Proportions

two ratios set equal to each other if you know three parts of any proportion, you cross-multiply to find the fourth