All practice questions

Question 81Question →

When a positive integer $n$ is divided by $12$ , the remainder is $7$ . What is the remainder when $n^2 + 5n + 11$ is divided by $12$ ?

Show answer & explanation

Answer: 11

Answer

11

Since

n

leaves a remainder of

7

when divided by

12

, we can substitute

n \equiv 7 \pmod{12}

directly into

n^2 + 5n + 11

. Evaluating

7^2 + 5(7) + 11

gives

49 + 35 + 11 = 95

. Dividing

95

by

12

gives

7

with a remainder of

11

.

Step-by-Step Solution

1

Express

n

using modular arithmetic

n \equiv 7 \pmod{12}

A positive integer

n

that leaves a remainder of

7

when divided by

12

can be expressed as

n = 12k + 7

for some non-negative integer

k

.

2

Substitute

n \equiv 7 \pmod{12}

into the target expression

n^2 + 5n + 11 \equiv 7^2 + 5(7) + 11 \pmod{12}

By the algebraic properties of remainders (modular arithmetic), substituting the remainder

7

for

n

yields an equivalent remainder modulo

12

.

3

Evaluate the arithmetic sum

49 + 35 + 11 = 95

Computing

7^2 = 49

,

5 \times 7 = 35

, and adding

11

gives

95

.

4

Find the remainder of

95

modulo

12

95 = 12 \times 7 + 11 \implies 95 \equiv 11 \pmod{12}

Dividing

95

by

12

yields a quotient of

7

and a remainder of

11

.

Key Concept

Modular Arithmetic and Polynomial Remainders

Question 82Question →

How many integer values of $x$ satisfy the inequality $\frac{|x - 2| - 3}{|x + 1| - 4} \leq 0$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

Analyzing the signs of the numerator

|x - 2| - 3

and denominator

|x + 1| - 4

shows that the quotient is non-positive when the numerator and denominator have opposite signs or when the numerator is zero. This yields two intervals:

(-5, -1]

and

(3, 5]

. The integers contained in these intervals are

-4, -3, -2, -1, 4, 5

, which totals 6 integer values.

Step-by-Step Solution

1

Analyze the sign of the numerator

|x - 2| - 3

The numerator is zero at

x = -1

and

x = 5

, negative for

-1 < x < 5

, and positive for

x < -1

or

x > 5

.

Solving

|x - 2| = 3

yields

x - 2 = 3 \implies x = 5

and

x - 2 = -3 \implies x = -1

.

2

Analyze the sign of the denominator

|x + 1| - 4

The denominator is zero at

x = -5

and

x = 3

, negative for

-5 < x < 3

, and positive for

x < -5

or

x > 3

. Exclude

x = -5

and

x = 3

.

Solving

|x + 1| = 4

yields

x + 1 = 4 \implies x = 3

and

x + 1 = -4 \implies x = -5

. Denominators cannot be zero.

3

Find intervals where numerator and denominator have opposite signs or numerator is zero

The solution set is the union of

(-5, -1]

and

(3, 5]

.

A fraction

\frac{N}{D} \leq 0

requires

N \ge 0, D < 0

or

N \le 0, D > 0

.

4

Count the total number of integer solutions

The valid integers are

-4, -3, -2, -1, 4, 5

, giving a total count of 6.

Listing integers in

(-5, -1]

yields

-4, -3, -2, -1

, and in

(3, 5]

yields

4, 5

.

Key Concept

Solving Rational Inequalities with Absolute Values

Question 83Question →

When the integer $N = 4^{43} + 9^{43}$ is divided by $7$ , what is the remainder?

Show answer & explanation

Answer: 6

Answer

The remainder when

N = 4^{43} + 9^{43}

is divided by

7

is

6

.

Reducing

9 \pmod 7

yields

2

, allowing

N

to be expressed as

2^{86} + 2^{43} \pmod 7

. The powers of

2 \pmod 7

repeat every 3 powers

(2, 4, 1)

. Reducing the exponents

86

and

43

modulo 3 gives remainders of

2

and

1

, corresponding to values of

4

and

2

. Their sum

4 + 2 = 6

is the final remainder.

Step-by-Step Solution

1

Simplify the base modulo 7 and express in powers of 2

9 \equiv 2 \pmod 7

, so

N = 4^{43} + 9^{43} \equiv 2^{86} + 2^{43} \pmod 7

.

Reducing bases modulo 7 simplifies calculating large exponents.

2

Determine the remainder cyclicity of powers of 2 modulo 7

The cycle length is 3 with pattern

(2, 4, 1)

, because

2^1 \equiv 2

,

2^2 \equiv 4

, and

2^3 \equiv 1 \pmod 7

.

Powers of integers modulo a divisor repeat periodically.

3

Evaluate each term using the exponent modulo the cycle length

86 \equiv 2 \pmod 3 \implies 2^{86} \equiv 2^2 = 4 \pmod 7

, and

43 \equiv 1 \pmod 3 \implies 2^{43} \equiv 2^1 = 2 \pmod 7

.

The position in the cyclicity sequence is dictated by the exponent modulo the period length.

4

Add the individual remainders

4 + 2 = 6 \pmod 7

.

The remainder of a sum equals the sum of the individual remainders.

Key Concept

Modular arithmetic cyclicity of powers and addition of remainders

Question 84Question →

A technology manufacturing company produces two server models: Model Alpha and Model Beta. The manufacturing cost of one Model Alpha server is $400 more than twice the manufacturing cost of one Model Beta server. Last month, the company produced 15 Model Alpha servers and 40 Model Beta servers, incurring a total manufacturing cost of$ 76,000. What is the manufacturing cost, in dollars, of one Model Alpha server?

Show answer & explanation

Answer: 2400

Answer

The manufacturing cost of one Model Alpha server is $2,400.

By defining the costs as

A = 2B + 400

and setting up the total cost equation

15A + 40B = 76,000

, substituting

A

yields

70B = 70,000

, giving

B = 1,000

. Substituting

B = 1,000

back into

A = 2B + 400

gives

A = 2,400

.

Step-by-Step Solution

1

Define variables and construct the linear relation equation.

A = 2B + 400

The cost of one Model Alpha server (

A

) is

400 more than twice the cost of Model Beta (

B$).

2

Formulate the total production cost equation.

15A + 40B = 76,000

15 units of Model Alpha and 40 units of Model Beta sum to a total cost of $76,000.

3

Substitute

A

into the total cost equation and solve for

B

.

15(2B + 400) + 40B = 76,000 \implies 70B + 6,000 = 76,000 \implies B = 1,000

Substituting reduces the system to a single linear equation in one variable.

4

Calculate the value of

A

.

A = 2(1,000) + 400 = 2,400

Substitute

B = 1,000

back into the equation for

A

.

Key Concept

Solving Systems of Linear Equations by Substitution

Question 85Question →

If $x$ , $y$ , and $z$ are integers satisfying $x < y < 0 < z$ , $x + y + z = 0$ , and $x y z = 160$ , what is the value of $z - x$ ?

Show answer & explanation

Answer: 18

Answer

The value of

z - x

is

18

.

By defining positive variables

a = -x

and

b = -y

, the given inequality

x < y < 0

implies

a > b > 0

. Since

x + y + z = 0

,

z = a + b

. The product condition

xyz = 160

translates to

ab(a+b) = 160

. Testing positive integer values reveals that

b = 2

and

a = 8

is the unique solution satisfying

a > b > 0

. This gives

x = -8

,

y = -2

, and

z = 10

, making

z - x = 10 - (-8) = 18

.

Step-by-Step Solution

1

Set up positive variables for the negative integers

Let

a = -x

and

b = -y

, where

a

and

b

are positive integers with

a > b > 0

. From

x + y + z = 0

, we get

-a - b + z = 0

, so

z = a + b

.

Converting negative integers to positive magnitude variables simplifies sign analysis in products and sums.

2

Substitute variables into the product equation

Substituting

x = -a

,

y = -b

, and

z = a + b

into

x y z = 160

gives

(-a)(-b)(a + b) = 160

, which simplifies to

a b (a + b) = 160

.

Multiplying two negative numbers yields a positive product, simplifying the product expression.

3

Solve for positive integer pairs

(a, b)

with

a > b

Testing integer values of

b

:
- If

b = 1

,

a(a+1) = 160

(no integer solution as

12 \times 13 = 156

).
- If

b = 2

,

2a(a+2) = 160 \implies a(a+2) = 80 \implies a = 8

.
- If

b = 3

,

3a(a+3) = 160 \implies a(a+3) = 53.33

(not an integer).
- If

b = 4

,

4a(a+4) = 160 \implies a(a+4) = 40

(no integer solution).
- If

b \ge 5

,

a > b \implies a \ge 6

, so

a b (a+b) \ge 6 \times 5 \times 11 = 330 > 160

.
Thus, the only valid integer pair is

a = 8

and

b = 2

.

Systematically checking integer factors under inequality constraints guarantees finding all unique solutions.

4

Calculate the target expression

z - x

Since

a = 8

and

b = 2

, we have

x = -8

,

y = -2

, and

z = 8 + 2 = 10

. Therefore,

z - x = 10 - (-8) = 18

.

Evaluating

z - x

using the identified values completes the solution.

Key Concept

Positive and Negative Number Properties and Inequality Constraints

Question 86Question →

If $x$ is a positive integer such that $\sqrt{\frac{8^{10} + 4^{10}}{8^4 + 4^{11}}} = 2^x$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

4

Converting all terms to base 2 yields

\sqrt{\frac{2^{30}+2^{20}}{2^{12}+2^{22}}}

. Factoring out

2^{20}

in the numerator and

2^{12}

in the denominator leaves

\sqrt{\frac{2^{20}(2^{10}+1)}{2^{12}(2^{10}+1)}} = \sqrt{2^8} = 2^4

. Thus

2^x = 2^4

, so

x = 4

.

Step-by-Step Solution

1

Express all terms with prime base 2 using exponent rules.

8^{10} = 2^{30}

,

4^{10} = 2^{20}

,

8^4 = 2^{12}

, and

4^{11} = 2^{22}

.

Converting non-prime bases to a common base enables exponent simplification and factoring.

2

Factor the numerator and denominator by pulling out the lowest power of 2 in each.

Numerator:

2^{20}(2^{10} + 1)

; Denominator:

2^{12}(1 + 2^{10})

.

Factoring isolates common terms in sums of powers so they can be canceled.

3

Cancel the identical factor

(2^{10} + 1)

and compute the radical expression.

\sqrt{\frac{2^{20}}{2^{12}}} = \sqrt{2^8} = 2^4 = 16.

Applying quotient rule for exponents

\frac{2^{20}}{2^{12}} = 2^{8}

and radical rule

\sqrt{2^8} = (2^8)^{1/2} = 2^4

.

4

Set

2^4

equal to

2^x

to solve for

x

.

x = 4.

Since the bases are identical and positive, the exponents must be equal.

Key Concept

Exponents, Roots, and Powers of Integers

Question 87Question →

A positive integer $n$ is divisible by 12 and has exactly 15 positive integer divisors. What is the least possible value of $n$ ?

Show answer & explanation

Answer: 144

Answer

144

To find the least positive integer

n

that is a multiple of 12 and has 15 divisors, we examine the prime factorization of 12 (

2^2 \cdot 3^1

) and the divisor count formula

(e_1+1)(e_2+1)\dots = 15

. The number 15 factors as

5 \times 3

, implying

n

has the form

p^4 \cdot q^2

. Minimizing

n

requires placing the larger exponent on the smaller prime factor, so

p=2

and

q=3

, giving

n = 2^4 \cdot 3^2 = 144

.

Step-by-Step Solution

1

Determine prime factor requirements for divisibility by 12.

Since

12 = 2^2 \cdot 3^1

,

n

must have prime factors 2 and 3 with exponents

a \ge 2

and

b \ge 1

.

Divisibility requires that all prime factors of the divisor appear in the dividend with equal or higher exponents.

2

Apply the divisor count formula to determine the exponent structure.

The total number of divisors is 15, which factors as

(4 + 1)(2 + 1) = 15

. Thus,

n = p^4 \cdot q^2

.

The number of positive divisors is given by

(e_1 + 1)(e_2 + 1) \dots = 15

.

3

Assign prime factors to minimize

n

.

Assigning the larger exponent 4 to the smaller prime 2 gives

2^4 \cdot 3^2 = 144

.

To minimize a product of prime powers, larger exponents should be paired with smaller prime bases.

Key Concept

Divisor count formula and prime factorization properties

Question 88Question →

A hardware manufacturer produces two models of enterprise servers: Server X and Server Y. Each Server X requires $3$ processors of Type A and $2$ processors of Type B. Each Server Y requires $2$ processors of Type A and $5$ processors of Type B. In a given production run, the manufacturer assembled a total of $40$ servers using exactly $115$ processors of Type A. What is the total number of Type B processors used in this production run?

Show answer & explanation

Answer: 95

Answer

The total number of Type B processors used in the production run is 95.

By representing the count of Server X models as

x

and Server Y models as

y

, the given conditions yield two equations:

x + y = 40

and

3x + 2y = 115

. Solving this system gives

x = 35

and

y = 5

. Substituting these values into the expression for Type B processors (

2x + 5y

) produces

2(35) + 5(5) = 70 + 25 = 95

.

Step-by-Step Solution

1

Define variables and establish equations for server totals and Type A processor usage.

x + y = 40

and

3x + 2y = 115

, where

x

is Server X count and

y

is Server Y count.

Word problems require translating relationships into algebraic linear equations.

2

Solve the system of equations for

x

and

y

using substitution.

x = 35

Server X models and

y = 5

Server Y models.

Finding the specific quantities of each server model is required to compute total Type B processors.

3

Substitute

x = 35

and

y = 5

into the expression for Type B processors (

2x + 5y

).

2(35) + 5(5) = 70 + 25 = 95

.

Determines the target unknown value requested in the problem stem.

Key Concept

Modeling word problems using systems of linear equations in two variables.

Estimated Time:2m 0s

Question 89Question →

A cosmetic chemist creates a skin care product by mixing two vitamin C solutions. Solution A consists of $200$ milliliters of a liquid that is $15\%$ vitamin C by volume. Solution B is a liquid that is $25\%$ vitamin C by volume. How many milliliters of Solution B must be added to Solution A so that the resulting mixture is $21\%$ vitamin C by volume?

Show answer & explanation

Answer: 300

Answer

300

milliliters of Solution B must be added.

Adding

300

mL of Solution B contributes

300 \times 0.25 = 75

mL of pure vitamin C. Combined with Solution A's

30

mL of vitamin C, the total amount of vitamin C is

105

mL in a total mixture volume of

200 + 300 = 500

mL. The resulting concentration is

\frac{105}{500} = 21\%

.

Step-by-Step Solution

1

Calculate the volume of pure solute (vitamin C) contained in Solution A.

200 \times 0.15 = 30

mL of pure vitamin C.

The solute volume is calculated by multiplying total volume by the percentage concentration.

2

Express the solute contribution of Solution B and set up the equation for the combined concentration.

\frac{30 + 0.25x}{200 + x} = 0.21

The final concentration is equal to total volume of pure solute divided by the total volume of the final mixture.

3

Solve the algebraic equation for

x

.

30 + 0.25x = 42 + 0.21x \implies 0.04x = 12 \implies x = 300

Cross-multiplying and isolating

x

yields the required volume of Solution B in milliliters.

Key Concept

Weighted average concentration equation for mixing two liquids.

Estimated Time:1m 30s

Question 90Question →

A positive integer $n$ has exactly four distinct prime factors, the three smallest of which are $2$ , $3$ , and $5$ . If $n$ is divisible by $360$ and has exactly $48$ positive divisors, what is the minimum possible value of $n$ ?

Show answer & explanation

Answer: 2520

Answer

The minimum possible value of

n

is 2520.

To minimize

n

, we analyze its prime factorization

n = 2^a \times 3^b \times 5^c \times p^d

, where

p

is the fourth distinct prime factor. Divisibility by

360 = 2^3 \times 3^2 \times 5^1

requires

a \ge 3

,

b \ge 2

, and

c \ge 1

. The number of positive divisors is given by

(a+1)(b+1)(c+1)(d+1) = 48

. To minimize

n

, we pick the smallest prime greater than 5, which is

p = 7

, and set

d = 1

. This simplifies the divisor equation to

(a+1)(b+1)(c+1) = 24

. Since

a+1 \ge 4

,

b+1 \ge 3

, and

c+1 \ge 2

, the minimal product of these terms is

4 \times 3 \times 2 = 24

. This uniquely determines

a = 3

,

b = 2

, and

c = 1

. Substituting these values gives

n = 2^3 \times 3^2 \times 5^1 \times 7^1 = 2520

.

Step-by-Step Solution

1

Determine the prime factorization of the divisor requirement.

360 = 2^3 \times 3^2 \times 5^1

.

Divisibility by 360 imposes lower bounds on the exponents of the prime factors 2, 3, and 5 in

n

.

2

Formulate the general prime factorization for

n

and state exponent constraints.

n = 2^a \times 3^b \times 5^c \times p^d

with

a \ge 3

,

b \ge 2

,

c \ge 1

,

d \ge 1

, and prime

p > 5

.

n

has four distinct prime factors, three of which are 2, 3, and 5.

3

Apply the divisor counting formula to set up an algebraic equation.

(a+1)(b+1)(c+1)(d+1) = 48

.

The number of positive divisors of

n = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}

is given by

(e_1+1)(e_2+1)\dots(e_k+1)

.

4

Minimize

n

by choosing optimal values for

p

and

d

.

p = 7

and

d = 1

, leading to

(a+1)(b+1)(c+1) = 24

.

To make

n

as small as possible, the fourth prime

p

should be the smallest available prime (

7

) and its exponent

d

should be minimized (

1

).

5

Solve for exponents

a

,

b

, and

c

under the given inequality constraints.

a = 3

,

b = 2

,

c = 1

.

Since

a+1 \ge 4

,

b+1 \ge 3

, and

c+1 \ge 2

, the minimum possible product

(a+1)(b+1)(c+1)

is

4 \times 3 \times 2 = 24

. Hence,

a=3, b=2, c=1

is the unique solution.

6

Calculate the value of

n

.

n = 2^3 \times 3^2 \times 5^1 \times 7^1 = 2520

.

Multiplying out the prime factors yields the smallest integer matching all conditions.

Key Concept

Prime Factorization and Divisor Counting Constraints

Estimated Time:2m 0s

Question 91Question →

An automotive testing center evaluated $350$ electric vehicle models for three advanced driver-assistance features: Lane Keeping Assist ( $L$ ), Automatic Emergency Braking ( $A$ ), and Blind Spot Detection ( $B$ ). The evaluation showed that $180$ models had feature $L$ , $150$ had feature $A$ , and $160$ had feature $B$ . Additionally, $65$ models had both $L$ and $A$ , $50$ had both $A$ and $B$ , $60$ had both $L$ and $B$ , and $25$ models had all three features. How many of the evaluated electric vehicle models had none of these three features?

Show answer & explanation

Answer: 10

Answer

The number of electric vehicle models that had none of the three features is 10.

Using the formula for three overlapping sets,

|L \cup A \cup B| = |L| + |A| + |B| - (|L \cap A| + |A \cap B| + |L \cap B|) + |L \cap A \cap B|

, we get

180 + 150 + 160 - (65 + 50 + 60) + 25 = 340

vehicles with at least one feature. Subtracting this from the total evaluated group of

350

yields

350 - 340 = 10

vehicles with none of the features.

Step-by-Step Solution

1

Use the Principle of Inclusion-Exclusion for three overlapping sets to find the total number of models with at least one feature.

|L \cup A \cup B| = 180 + 150 + 160 - (65 + 50 + 60) + 25 = 340

models.

Adding individual set counts overcounts pairwise overlap regions twice and the central triple overlap three times. Subtracting pairwise intersections corrects for double counting, and adding back the triple intersection accounts for its over-subtraction.

2

Subtract the number of models having at least one feature from the total number of tested models.

\text{None} = 350 - 340 = 10

models.

The entire group consists of models with at least one feature plus models with none of the features.

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Partitioning

Question 92Question →

If $x = \sqrt[3]{5 + 2\sqrt{13}} + \sqrt[3]{5 - 2\sqrt{13}}$ , what is the value of $x^3 + 9x$ ?

Show answer & explanation

Answer: 10

Answer

10

By defining

x = u + v

with

u = \sqrt[3]{5 + 2\sqrt{13}}

and

v = \sqrt[3]{5 - 2\sqrt{13}}

, cubing both sides gives

x^3 = u^3 + v^3 + 3uv(u + v)

. Evaluating the components yields

u^3 + v^3 = 10

and

uv = \sqrt[3]{25 - 52} = -3

. Substituting these back gives

x^3 = 10 - 9(x)

, which rearranges to

x^3 + 9x = 10

.

Step-by-Step Solution

1

Express

x

as a sum of two variables

u

and

v

x = u + v

, where

u = \sqrt[3]{5 + 2\sqrt{13}}

and

v = \sqrt[3]{5 - 2\sqrt{13}}

Grouping the binomial terms simplifies algebraic expansion using standard polynomial identities.

2

Calculate the sum of the cubes

u^3 + v^3

u^3 + v^3 = (5 + 2\sqrt{13}) + (5 - 2\sqrt{13}) = 10

Eliminating the cube roots allows for simple additive cancellation of the radical terms.

3

Calculate the product

uv

uv = \sqrt[3]{(5 + 2\sqrt{13})(5 - 2\sqrt{13})} = \sqrt[3]{5^2 - (2\sqrt{13})^2} = \sqrt[3]{25 - 52} = \sqrt[3]{-27} = -3

Applying the difference of squares property inside the cube root simplifies the product of conjugate radicals to a single integer.

4

Cube both sides of

x = u + v

and substitute evaluated terms

x^3 = u^3 + v^3 + 3uv(u + v) = 10 + 3(-3)x = 10 - 9x

Using

(u+v)^3 = u^3 + v^3 + 3uv(u+v)

connects

x^3

directly to

x

without expanding long radical terms.

5

Rearrange the equation to isolate

x^3 + 9x

x^3 + 9x = 10

Adding

9x

to both sides gives the exact numerical value of the requested expression.

Key Concept

Algebraic Identities with Polynomials and Radicals

Question 93Question →

If $x$ is a positive integer such that $\frac{6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6}{3^6 + 3^6 + 3^6} = 2^x$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

The value of

x

is 7.

Repeated addition converts to multiplication: six terms of

6^6

yield

6 \cdot 6^6 = 6^7

, and three terms of

3^6

yield

3 \cdot 3^6 = 3^7

. Dividing gives

\frac{6^7}{3^7} = \left(\frac{6}{3}\right)^7 = 2^7

. Comparing

2^7

to

2^x

gives

x = 7

.

Step-by-Step Solution

1

Simplify the numerator by expressing repeated addition as multiplication.

6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = 6 \times 6^6 = 6^7

Adding six identical terms of

6^6

is equivalent to multiplying

6^6

by 6. Using the power rule

a^1 \cdot a^n = a^{n+1}

, we get

6^7

.

2

Simplify the denominator by expressing repeated addition as multiplication.

3^6 + 3^6 + 3^6 = 3 \times 3^6 = 3^7

Adding three identical terms of

3^6

is equivalent to multiplying

3^6

by 3, yielding

3^7

.

3

Apply the quotient property of exponents for identical powers.

\frac{6^7}{3^7} = \left(\frac{6}{3}\right)^7 = 2^7

According to exponent laws,

\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n

for any non-zero real numbers

a

and

b

.

4

Equate exponents of equal bases to solve for

x

.

2^x = 2^7 \implies x = 7

Since the bases on both sides of the equation are equal to 2, the exponents must be equal.

Key Concept

Combining repeated addition into exponential products and dividing powers with equal exponents.

Question 94Question →

A logistics company offers two freight pricing models for oversized cargo. Model X charges a fixed monthly account fee of $\$8$ plus $\$0.60$ per kilometer traveled. Model Y charges a fixed monthly account fee of $\$54$ plus $\$0.20$ per kilometer traveled. For how many kilometers in a month will the total monthly charge under Model X be exactly $20\%$ less than the total monthly charge under Model Y?

Show answer & explanation

Answer: 80

Answer

80 kilometers

The total monthly cost under Model X is

C_X = 8 + 0.60k

and under Model Y is

C_Y = 54 + 0.20k

. The condition that Model X is 20% less than Model Y means

C_X = 0.80 C_Y

. Substituting the expressions gives

8 + 0.60k = 0.80(54 + 0.20k) = 43.2 + 0.16k

. Subtracting

0.16k

and

8

from both sides yields

0.44k = 35.2

, which simplifies to

k = 80

.

Step-by-Step Solution

1

Define variables and establish linear cost equations for both models.

Model X cost:

C_X = 8 + 0.60k

; Model Y cost:

C_Y = 54 + 0.20k

, where

k

is kilometers traveled.

Linear modeling translates flat fees and variable rates into algebraic expressions.

2

Formulate the linear equation based on the condition that Model X is 20% less than Model Y.

C_X = 0.80 C_Y \implies 8 + 0.60k = 0.80(54 + 0.20k)

.

Being 20% less than a base value means taking 80% (or 0.80) of that value.

3

Expand and simplify the algebraic equation.

8 + 0.60k = 43.2 + 0.16k \implies 0.44k = 35.2

.

Distributing 0.80 across

(54 + 0.20k)

yields

43.2 + 0.16k

, and subtracting

0.16k

and

8

isolates

k

on one side.

4

Calculate the value of

k

.

k = \frac{35.2}{0.44} = 80

.

Dividing

35.2

by

0.44

gives the exact number of kilometers required.

Key Concept

Linear Equations in One and Two Variables

Question 95Question →

A boutique jewelry designer crafts customized gold necklaces. The total cost to produce each necklace is the sum of a fixed material cost of $\$800$ and a variable design cost. To determine the list price, the designer marks up the total production cost by $60\%$ . During a seasonal promotion, the designer offers a $25\%$ discount off the list price. If the designer earns a net profit of $\$360$ on each necklace sold during the promotion, what is the variable design cost, in dollars, per necklace?

Show answer & explanation

Answer: 1000

Answer

The variable design cost per necklace is $1,000.

The total production cost per necklace is

C = 800 + L

. Marking up by

60\%

yields a list price of

1.60C

. A

25\%

discount reduces the price to

0.75 \times 1.60C = 1.20C

. The net profit per necklace is

1.20C - C = 0.20C

. Given that the net profit is

\$360

, we set

0.20C = 360

, which gives

C = 1,800

. Subtracting the fixed material cost of

\$800

yields the variable design cost of

\$1,000

.

Step-by-Step Solution

1

Define total production cost in terms of variable cost

C = 800 + L

, where

C

is total production cost and

L

is variable design cost

Total cost is the sum of fixed material cost and variable design cost.

2

Determine the selling price after markup and discount

Selling Price

S = 0.75 \times (1.60 C) = 1.20 C

A

60\%

markup increases total cost by a factor of

1.60

, and a

25\%

discount reduces that marked price to

75\%

(

0.75

).

3

Formulate the profit equation and solve for total production cost

C

Profit

= S - C = 1.20 C - C = 0.20 C = 360 \implies C = 1,800

Net profit is the selling price minus the total production cost.

4

Calculate variable design cost

L

L = 1,800 - 800 = 1,000

Subtract fixed material cost from total production cost.

Key Concept

Profit, Loss, and Markup with Successive Percentage Adjustments

Question 96Question →

A laboratory technician has $40$ liters of a saline solution that is $15\%$ salt by volume. How many liters of pure water must be evaporated from the solution so that the remaining solution is $25\%$ salt by volume?

Show answer & explanation

Answer: 16

Answer

The correct answer is

16

liters.

Because evaporation removes only pure water, the quantity of salt remains constant at

6

liters (

40 \times 0.15

). For

6

liters of salt to constitute

25\%

of the final mixture volume

V

, we set up

0.25V = 6

, which yields

V = 24

liters. Subtracting the final volume of

24

liters from the initial volume of

40

liters gives

16

liters of evaporated water.

Step-by-Step Solution

1

Calculate the volume of pure salt in the initial solution.

The initial salt volume is

40 \times 0.15 = 6

liters.

The solute amount is determined by multiplying total volume by concentration.

2

Determine the required total solution volume after evaporation to achieve a 25% concentration.

The required total final volume is

\frac{6}{0.25} = 24

liters.

Since evaporation removes only water, the volume of salt remains 6 liters, which must equal 25% of the new total volume.

3

Calculate the amount of water evaporated by taking the difference between the initial and final total volumes.

The volume of water evaporated is

40 - 24 = 16

liters.

The decrease in total solution volume equals the volume of pure water removed by evaporation.

Key Concept

Concentration change via evaporation (solute mass conservation)

Question 97Question →

A quality assurance technician randomly selects 2 microprocessors from a batch of $16$ microprocessors, of which $6$ were manufactured at Facility A and $10$ were manufactured at Facility B. The selections are made one after another without replacement. What is the probability that both selected microprocessors were manufactured at Facility A?

Show answer & explanation

Answer: 0.125

Answer

The probability that both selected microprocessors were manufactured at Facility A is 0.125 (or 1/8).

Because the selections are made without replacement, the outcome of the first draw changes the pool of available microprocessors for the second draw. The probability of choosing a Facility A microprocessor first is

\frac{6}{16} = \frac{3}{8}

. Following that selection,

5

Facility A microprocessors remain among

15

total microprocessors, giving a conditional probability of

\frac{5}{15} = \frac{1}{3}

for the second selection. Multiplying these dependent probabilities yields

\frac{3}{8} \times \frac{1}{3} = \frac{1}{8} = 0.125

.

Step-by-Step Solution

1

Determine the probability of selecting a Facility A microprocessor on the first draw.

P(\text{First is Facility A}) = \frac{6}{16} = \frac{3}{8}

There are 6 microprocessors from Facility A out of 16 total microprocessors.

2

Determine the conditional probability of selecting a Facility A microprocessor on the second draw.

P(\text{Second is Facility A} \mid \text{First is Facility A}) = \frac{5}{15} = \frac{1}{3}

Since the selection is made without replacement, 5 Facility A microprocessors remain out of a reduced total of 15 microprocessors.

3

Multiply the dependent probabilities to find the combined probability.

P(\text{Both are Facility A}) = \frac{3}{8} \times \frac{1}{3} = \frac{3}{24} = \frac{1}{8} = 0.125

The joint probability of sequential dependent events is the product of the initial probability and the conditional probability.

Key Concept

Probability of Dependent Events (Sampling without Replacement)

Estimated Time:1m 30s

Question 98Question →

An educational foundation set an annual fundraising goal for a given year. During the first quarter, the foundation raised $30\%$ of its annual goal. In the second quarter, the amount raised was $25\%$ greater than the amount raised in the first quarter. In the third quarter, the amount raised was $40\%$ less than the total amount raised in the first two quarters combined. If the total amount raised across the first three quarters combined was $\$129,600$ , what was the foundation's annual fundraising goal, in dollars?

Show answer & explanation

Answer: 120000

Answer

The foundation's annual fundraising goal was $120,000.

To determine the annual goal

G

, calculate the raised amount in each quarter relative to

G

. The first quarter yields

Q_1 = 0.30G

. The second quarter amount is

25\%

greater than

Q_1

, which equals

1.25 \times 0.30G = 0.375G

. The sum of the first two quarters is

0.30G + 0.375G = 0.675G

. The third quarter amount is

40\%

less than this combined sum, giving

0.60 \times 0.675G = 0.405G

. Summing the three quarters gives a total of

0.30G + 0.375G + 0.405G = 1.08G

. Equating

1.08G = 129,600

and dividing by

1.08

gives

G = 120,000

.

Step-by-Step Solution

1

Define the variable for the unknown quantity

Let

G

represent the foundation's annual fundraising goal in dollars.

Establishing a variable allows all quarterly contributions to be expressed as a linear algebraic function of the goal.

2

Express the amount raised in the first quarter in terms of

G

Q_1 = 0.30G

The foundation raised

30\%

of its annual goal during the first quarter.

3

Express the amount raised in the second quarter in terms of

G

Q_2 = 1.25 \times 0.30G = 0.375G

An increase of

25\%

over

Q_1

means

Q_2 = (1 + 0.25) Q_1 = 1.25 \times 0.30G

.

4

Calculate the combined total raised in the first two quarters

Q_1 + Q_2 = 0.30G + 0.375G = 0.675G

This combined sum serves as the base value for calculating the third quarter's contribution.

5

Express the amount raised in the third quarter in terms of

G

Q_3 = (1 - 0.40) \times 0.675G = 0.60 \times 0.675G = 0.405G

The third quarter raised

40\%

less than the combined amount of the first two quarters, meaning it equaled

60\%

of

(Q_1 + Q_2)

.

6

Sum the contributions of all three quarters and solve for

G

Total

= 0.30G + 0.375G + 0.405G = 1.08G = 129,600 \implies G = 120,000

Setting the sum equal to the total dollar amount raised (

\$129,600

) yields

1.08G = 129,600

, so

G = \frac{129,600}{1.08} = 120,000

.

Key Concept

Successive Percent Change and Base Identification

Question 99Question →

A laboratory tested $200$ synthetic compound samples for two properties: thermal stability and chemical resistance. Among the samples, $120$ exhibited thermal stability, $90$ exhibited chemical resistance, and $50$ exhibited neither property. If a sample is selected at random from those that exhibited thermal stability, what is the probability that it also exhibited chemical resistance?

Show answer & explanation

Answer: 0.5

Answer

0.5

The conditional probability of selecting a sample with chemical resistance given that it has thermal stability is found by dividing the number of samples with both properties (

60

) by the total number of samples with thermal stability (

120

), giving

\frac{60}{120} = 0.5

.

Step-by-Step Solution

1

Find the total number of samples exhibiting at least one of the two properties.

Since

50

out of

200

samples exhibited neither property, the number of samples exhibiting at least one property is

200 - 50 = 150

.

The total population consists of samples exhibiting at least one property plus samples exhibiting neither property.

2

Calculate the number of samples exhibiting both thermal stability (

T

) and chemical resistance (

C

).

Using the inclusion-exclusion principle

|T \cup C| = |T| + |C| - |T \cap C|

, we have

150 = 120 + 90 - |T \cap C|

, which yields

|T \cap C| = 60

.

Overlapping sets require subtracting the intersection to avoid double-counting elements.

3

Compute the conditional probability

P(C|T)

.

P(C|T) = \frac{|T \cap C|}{|T|} = \frac{60}{120} = 0.5

.

The given condition restricts the sample space to only the

120

samples exhibiting thermal stability.

Key Concept

Conditional probability restricts the sample space to the given condition's outcome space:

P(A|B) = \frac{|A \cap B|}{|B|}

.

Estimated Time:1m 30s

Question 100Question →

A municipal water treatment facility operates three types of filtration units: Model X, Model Y, and Model Z.

\bullet \text{ 2 Model X units, 3 Model Y units, and 1 Model Z unit together process 134,000 gallons per hour.}

\bullet \text{ 1 Model X unit, 4 Model Y units, and 2 Model Z units together process 156,000 gallons per hour.}

\bullet \text{ 3 Model X units, 1 Model Y unit, and 4 Model Z units together process 196,000 gallons per hour.}

What is the processing capacity, in thousands of gallons per hour, of a single Model Y filtration unit?

Show answer & explanation

Answer: 20

Answer

The processing capacity of a single Model Y filtration unit is 20 thousand gallons per hour.

By setting up the system of three linear equations

2x + 3y + z = 134

,

x + 4y + 2z = 156

, and

3x + y + 4z = 196

, and eliminating variables systematically through substitution and combination, we find

y = 20

. Thus, a single Model Y filtration unit processes 20 thousand gallons per hour.

Step-by-Step Solution

1

Set up a system of linear equations in three variables.

Let

x

,

y

, and

z

represent the hourly capacities (in thousands of gallons) of Model X, Model Y, and Model Z respectively.

\begin{aligned} (1) \quad 2x + 3y + z &= 134 \\ (2) \quad x + 4y + 2z &= 156 \\ (3) \quad 3x + y + 4z &= 196 \end{aligned}

Expressing the given conditions algebraically translates the word problem into a solvable linear system.

2

Express

x

in terms of

y

and

z

using Equation (2).

x = 156 - 4y - 2z

Equation (2) has a coefficient of 1 for

x

, making it ideal for algebraic substitution.

3

Substitute

x

into Equations (1) and (3) to eliminate

x

.

Substituting into Equation (1):

2(156 - 4y - 2z) + 3y + z = 134 \implies 312 - 8y - 4z + 3y + z = 134

312 - 5y - 3z = 134 \implies (4) \quad 5y + 3z = 178

Substituting into Equation (3):

3(156 - 4y - 2z) + y + 4z = 196 \implies 468 - 12y - 6z + y + 4z = 196

468 - 11y - 2z = 196 \implies (5) \quad 11y + 2z = 272

Reducing a 3-variable system to a 2-variable system simplifies the calculation.

4

Eliminate variable

z

from Equations (4) and (5) to solve for

y

.

Multiply Equation (4) by 2:

10y + 6z = 356

Multiply Equation (5) by 3:

33y + 6z = 816

Subtract the first result from the second:

(33y + 6z) - (10y + 6z) = 816 - 356

23y = 460 \implies y = 20

Eliminating

z

directly yields the required value of

y

, which is the capacity of Model Y.

Key Concept

Solving systems of three linear equations in three variables via substitution and elimination.

Answer

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