# ACT Math : Equations / Inequalities

## Example Questions

Jack has 14 coins consisting of nickels and dimes that total $0.90. How many nickels does Jack have? Possible Answers: 8 10 12 6 4 Correct answer: 10 Explanation: In order to solve this question we must first set up two equations. We know the number of nickels and the number of dimes equals 14 (n + d = 14). We also know the value of nickels and dimes. For the second equation we simply multiply the number of nickels we have by their value, added to the number of dimes we have by their value to get the total (0.05n + 0.10d = 0.90). Solve the first equation for n giving us n = 14 – d. We can then substitute 14 – d into the second equation wherever there is an “n”. Giving us 0.05 (14 – d) + 0.10d = 0.90. When we solve the equation we find the number of dimes is d = 4; therefore the remaining 10 coins must be nickels. ### Example Question #31 : Linear / Rational / Variable Equations If a = 1/3b and b = 4c, then in terms of c, a – b + c = ? Possible Answers: c 5/3c –5/3c –11/3c Correct answer: –5/3c Explanation: To begin we must find how a and c relate to each other. Using the second equation we know that we can plug in 4c everywhere there is a b in the first equation, giving us a = 4/3c. Now we can plug into the last equation. We plug in 4/3c for a, 4c for b, and leave c as it is. We must find a common denominator (4/3c – 12/3c + 3/3c) and add the numerators to find that our equation equals –5/3c. ### Example Question #11 : How To Find The Solution To An Equation If x= 8, then x2(4/(3 – x))(2/(4 – x)) – (4/x2) = ? Possible Answers: 22 35 16 15 0 Correct answer: 15 Explanation: There is really no need to alter this equation using algebra. Simply find that x = 2 and plug in. We see that 4(4)(1) – (1)=15. Remember to use correct order of operations here (parentheses, exponents, multiplication, division, addition, subtraction). ### Example Question #71 : Algebra Find the intersection of the following two equations: 3x + 4y = 6 15x - 4y = 3 Possible Answers: (0.5, 1.125) (1, 0.5) (18, 0) (0.2, 0) (3, 4) Correct answer: (0.5, 1.125) Explanation: The point of intersection for two lines is the same as the values of x and y that mutually solve each equation. Although you could solve for one variable and replace it in the other equation, use elementary row operations to add the two equations since you have a 4y and -4y: 3x + 4y = 6 15x - 4y = 3 18x = 9; x = 0.5 You can now plug x into the first equation: 3 * 0.5 + 4y = 6; 1.5 +4y = 6; 4y = 4.5; y = 1.125 Therefore, our point of intersection is (0.5, 1.125) ### Example Question #32 : Linear / Rational / Variable Equations Two cars start 25 mile apart and drive away from each other in opposite directions at speeds of 50 and 70 miles per hour. In approximately how many minutes will they be 400 miles apart? Possible Answers: None of the other answers 3.33 200 3.125 187.5 Correct answer: 187.5 Explanation: The cars have a distance from each other of 25 + 120t miles, where t is the number of hours, 25 is their initial distance and 120 is 50 + 70, or their combined speeds. Solve this equation for 400: 25 + 120t = 400; 120t = 375; t = 3.125 However, the question asked for minutes, so we must multiply this by 60: 3.125 * 60 = 187.5 minutes. ### Example Question #33 : Linear / Rational / Variable Equations A given university has an average professor pay of$40,000 a year and an average administrator pay of $45,000 per year. If the ratio of professors to administrators is 4 to 3, and the total pay for professors and administrators in a year is$40,415,000, how many professors does the college have?

548

475

411

500

375

548

Explanation:

Set up a system of linear equations based on our data:

40,000P + 45,000A = 40,415,000

P/A = 4/3

To make things easiest, solve the second equation for A in terms of P:

A = (3/4) P

Replace this value into the first equation:

40,000P + 45,000 * (3/4)P = 40,415,000

Simplify:

40,000P + 33,750P = 40,415,000

73,750P = 40,415,000

P = 548 (The number of professors)

### Example Question #81 : Algebra

Abby works at a car dealership and receives a commission "c" which is a percent of the profit the dealership makes from the sale, which is the difference between the price "p" of the car and the value "v" of the car. How much, in dollars, does the dealership earn per transaction?

(p – v)(0.01c)

pv(0.01c)

(p – v)(1 – c)

(p – v)(1 – 0.01c)

p(v – 0.01c)

(p – v)(1 – 0.01c)

Explanation:

To show that c is of the profit of the transaction, we must represent the profit as the difference between the price and the value of the car, or "(p – v)"

To show that Abby's commission in dollars is a percentage of the profit, we use 0.01 * c to convert the commission she earns to a percent.

To shift the earnings from Abby to the dealership (which is what the question requires of us), we must take 1 – 0.01c since this will accommodate for the remaining percentage. For example, it shifts 75% (0.75) to 25% (1 – 0.75 or 0.25).

Putting this all together, we get a final expression of:

(p – v)(1 – 0.01c) = dealership earnings

Check answer with arbitrary values: letting p = 300, v = 200, and c = 20, we get a value of 80 which makes sense as the $100 profit must be distributed evenly between Abby ($20) and the dealership ($80). ### Example Question #21 : How To Find The Solution To An Equation A theme park charges$10 for adults and $5 for kids. How many kids tickets were sold if a total of 548 tickets were sold for a total of$3750?

346

157

269

431

248

346

Explanation:

Let c = number of kids tickets sold. Then (548 – c) adult tickets were sold. The revenue from kids tickets is $5c, and the total revenue from adult tickets is$10(548 – c). Then,

5c + 10(548 – c) = 3750

5c + 5480 – 10c = 3750

5c = 1730

c = 346.

We can check to make sure that this number is correct:

$5 * 346 tickets +$10 * (548 – 346) tickets = \$3750 total revenue

### Example Question #44 : Equations / Inequalities

Two palm trees grow next to each other in Luke's backyard. One of the trees gets sick, so Luke cuts off the top 3 feet. The other tree, however, is healthy and grows 2 feet. How tall are the two trees if the healthy tree is now 4 feet taller than the sick tree, and together they are 28 feet tall?

11 and 17 feet

cannot be determined

12 and 16 feet

14 and 14 feet

8 and 20 feet

12 and 16 feet

Explanation:

Let s stand for the sick tree and h for the healthy tree. The beginning information about cutting the sick tree and the healthy tree growing is actually not needed to solve this problem! We know that the healthy tree is 4 feet taller than the sick tree, so h = s + 4.

We also know that the two trees are 28 feet tall together, so s + h = 28. Now we can solve for the two tree heights.

Plug h = s + 4 into the second equation: (s + 4) + s = 28. Simplify and solve for h: 2s = 24 so s = 12. Then solve for h: h = s + 4 = 12 + 4 = 16.

### Example Question #41 : Linear / Rational / Variable Equations

Solve for z

3(z + 4)3 – 7 = 17

4

2

–8

–2

8

–2

Explanation:

1. Add 7 to both sides

3(z + 4)3 – 7 + 7= 17 + 7

3(z + 4)3 = 24

2. Divide both sides by 3

(z + 4)3 = 8

3. Take the cube root of both sides

z + 4 = 2

4. Subtract 4 from both sides

z = –2