All practice questions

Question 481Question →

If $x$ , $y$ , and $z$ are non-zero real numbers such that $x y z^2 < 0$ , $x - y > 0$ , and $\frac{x}{z} < \frac{y}{z}$ , which of the following MUST be true?

\frac{x - z}{y} < 0

\frac{x - y}{z} > 0

x^2 - y^2 > 0

\frac{y - z}{x} < 0

\frac{x + z}{y} > 0

Show answer & explanation

Answer: $\frac{x - z}{y} < 0$

Answer

\frac{x - z}{y} < 0

From

z^2 > 0

, the inequality

x y z^2 < 0

implies

x y < 0

, so

x

and

y

have opposite signs. Since

x - y > 0 \implies x > y

,

x

must be positive (

x > 0

) and

y

must be negative (

y < 0

). Next, comparing

x > y

with

\frac{x}{z} < \frac{y}{z}

shows that dividing by

z

reversed the inequality sign, which proves

z

is negative (

z < 0

). Evaluating

\frac{x - z}{y}

: the numerator

x - z

is positive minus negative, which equals positive plus positive (strictly positive), while the denominator

y

is strictly negative. A positive value divided by a negative value is always negative, so

\frac{x - z}{y} < 0

MUST be true.

Step-by-Step Solution

1

Determine the relative signs of

x

and

y

using

x y z^2 < 0

.

x

and

y

must have opposite signs (

x y < 0

).

Since

z \neq 0

,

z^2 > 0

is always positive. Dividing

x y z^2 < 0

by

z^2

yields

x y < 0

.

2

Determine the individual signs of

x

and

y

using

x - y > 0

.

x > 0

(positive) and

y < 0

(negative).

x - y > 0 \implies x > y

. Since

x

and

y

have opposite signs and

x

is strictly greater than

y

,

x

must be positive and

y

must be negative.

3

Determine the sign of

z

using

\frac{x}{z} < \frac{y}{z}

.

z < 0

(negative).

We know

x > y

. When dividing both sides of

x > y

by

z

, the inequality direction reverses to

<

. An inequality sign flips if and only if the divisor

z

is negative.

4

Evaluate the expression

\frac{x - z}{y}

.

\frac{x - z}{y} < 0

Because

x > 0

and

z < 0

,

-z > 0

, so the numerator

x - z = x + (-z)

is the sum of two positive numbers, which is positive. The denominator

y

is negative. Dividing a positive number by a negative number yields a negative result.

Key Concept

Deducing variable signs from product conditions and inequality sign reversal rules

Estimated Time:2m 0s

Question 482Question →

A jeweler melts two gold alloys together. Alloy X is $80\%$ gold by weight, and Alloy Y is $50\%$ gold by weight. How many grams of Alloy X must be mixed with Alloy Y to create a $60$ -gram alloy that is $70\%$ gold by weight?

20

24

30

40

50

Show answer & explanation

Answer: $40$

Answer

The mass of Alloy X required is

40

grams.

By writing the gold balance equation

0.80x + 0.50(60 - x) = 0.70(60)

, we solve to find

0.30x = 12

, which gives

x = 40

grams of Alloy X.

Step-by-Step Solution

1

Define variables for the mass of each alloy.

Let

x

be the mass of Alloy X in grams. Since the total mixture is

60

grams, the mass of Alloy Y is

(60 - x)

grams.

The sum of the component masses equals the total mass of the resulting mixture.

2

Set up the equation for total gold content.

0.80x + 0.50(60 - x) = 0.70(60)

The amount of pure gold from Alloy X plus the amount of pure gold from Alloy Y must equal the amount of pure gold in the final

60

-gram mixture.

3

Solve for

x

.

0.80x + 30 - 0.50x = 42 \Rightarrow 0.30x = 12 \Rightarrow x = 40

Simplifying the algebraic equation yields the exact amount of Alloy X required.

Key Concept

Weighted Average in Mixture Problems

Estimated Time:1m 0s

Question 483Question →

A computer program randomly selects a single positive integer $x$ from the set $\{1, 2, 3, \dots, 300\}$ , where each integer has an equal probability of being chosen. What is the probability that $x$ is divisible by either 6 or 9, but not by 4?

Show answer & explanation

Answer: 0.14

Answer

The probability that

x

is divisible by 6 or 9, but not by 4, is

0.14

(or

\frac{7}{50}

).

To calculate the probability, determine the ratio of favorable outcomes to total outcomes (300). First, count integers divisible by 6 (50) or 9 (33). Subtracting the 16 multiples of 18 counted in both yields 67 integers. Next, identify how many of these 67 integers are divisible by 4: there are 25 multiples of 12 and 8 multiples of 36 (all 8 of which are already included in the multiples of 12), resulting in 25 integers divisible by 4. Subtracting 25 from 67 leaves 42 favorable integers. The probability is

42 / 300 = 0.14

.

Step-by-Step Solution

1

Determine the total sample space size.

The total number of equally likely outcomes is

|S| = 300

.

Single-event probability requires dividing the count of favorable outcomes by the total number of outcomes in the set.

2

Calculate the count of integers divisible by 6 or 9.

Count =

50 + 33 - 16 = 67

.

Apply the Principle of Inclusion-Exclusion: sum the multiples of 6 and 9, then subtract the multiples of

\text{lcm}(6, 9) = 18

to eliminate double counting.

3

Exclude integers from this set that are divisible by 4.

Count of multiples of 4 to exclude =

25 + 8 - 8 = 25

.

Integers divisible by both 6 and 4 are multiples of

\text{lcm}(6, 4) = 12

. Integers divisible by both 9 and 4 are multiples of

\text{lcm}(9, 4) = 36

. Using inclusion-exclusion,

25 + 8 - 8 = 25

integers must be excluded.

4

Find the number of favorable outcomes and compute the probability.

Favorable outcomes

|E| = 67 - 25 = 42

. Probability

P(E) = \frac{42}{300} = 0.14

.

Divide the favorable set size by the total sample space size.

Key Concept

Basic Single-Event Probability with Set Inclusion-Exclusion and Number Properties

Question 484Question →

An archival digitization laboratory uses two high-resolution scanner units, Unit $A$ and Unit $B$ , to process historic document collections. Working alone at its constant rate, Unit $A$ can digitize a standard collection in $24$ hours. Working alone at its constant rate, Unit $B$ can digitize the same collection in $40$ hours. Unit $A$ begins digitizing a standard collection alone. After $8$ hours of operation, Unit $B$ is turned on, and both units work together at their respective constant rates until the entire collection is digitized. How many total hours does it take to digitize the collection from start to finish?

Show answer & explanation

Answer: 18

Answer

The total time required to digitize the collection from start to finish is 18 hours.

Unit A completes

\frac{1}{3}

of the total job during its 8 solo hours. The remaining

\frac{2}{3}

of the job is completed by Units A and B working together at a combined rate of

\frac{1}{15}

per hour, taking 10 hours. Adding the 8 solo hours to the 10 combined hours yields a total time of 18 hours.

Step-by-Step Solution

1

Determine the individual work rates per hour for each unit.

Unit A's rate is

\frac{1}{24}

of the collection per hour, and Unit B's rate is

\frac{1}{40}

of the collection per hour.

Work rate is defined as the fraction of the job completed per unit of time (

R = \frac{1}{T}

).

2

Calculate the fraction of the work completed by Unit A alone during the first 8 hours.

Work completed

= 8 \times \frac{1}{24} = \frac{8}{24} = \frac{1}{3}

of the collection.

Work done equals rate multiplied by time (

W = R \times t

).

3

Calculate the remaining fraction of the collection to be digitized.

Remaining work

= 1 - \frac{1}{3} = \frac{2}{3}

of the collection.

The total job represents 1 whole unit.

4

Calculate the combined work rate when both units operate together.

Combined rate

= \frac{1}{24} + \frac{1}{40} = \frac{5}{120} + \frac{3}{120} = \frac{8}{120} = \frac{1}{15}

of the collection per hour.

When working simultaneously, individual rates add together.

5

Find the time required for both units working together to complete the remaining work.

Combined time

= \frac{2/3}{1/15} = \frac{2}{3} \times 15 = 10

hours.

Time equals remaining work divided by the combined rate (

t = \frac{W_{\text{remaining}}}{R_{\text{combined}}}

).

6

Calculate the total time required from start to finish.

Total time

= 8 \text{ hours} + 10 \text{ hours} = 18

hours.

Total time is the sum of the time Unit A worked alone and the time both units worked together.

Key Concept

Work Rate and Combined Work

Question 485Question →

If $x$ is a real number satisfying the equation $x^2 + \frac{18}{x^2 + 3} = 8$ , what is the value of $x^2$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x^2

is 6.

Substituting

u = x^2

converts the equation to

u + \frac{18}{u+3} = 8

. Multiplying through by

(u+3)

yields

u^2 + 3u + 18 = 8u + 24

, which simplifies to the quadratic equation

u^2 - 5u - 6 = 0

. Factoring gives

(u-6)(u+1) = 0

, so

u = 6

or

u = -1

. Because

x

must be a real number,

x^2

must be non-negative, eliminating

u = -1

. Thus,

x^2 = 6

.

Step-by-Step Solution

1

Substitute a temporary variable

u = x^2

into the given equation

u + \frac{18}{u + 3} = 8

Using substitution simplifies the fourth-degree rational equation into a standard quadratic structure.

2

Clear the denominator by multiplying the entire equation by

(u + 3)

u(u + 3) + 18 = 8(u + 3)

, which expands to

u^2 + 3u + 18 = 8u + 24

Eliminating fractions allows rearranging terms into polynomial form.

3

Rearrange terms into standard quadratic form

au^2 + bu + c = 0

u^2 - 5u - 6 = 0

Moving all terms to one side prepares the expression for factoring.

4

Factor the quadratic polynomial into linear binomial factors

(u - 6)(u + 1) = 0

, giving

u = 6

or

u = -1

The roots of the transformed equation represent candidate values for

x^2

.

5

Apply the constraint that

x

is a real number to test candidate values

Since

x^2 \geq 0

for all real

x

,

u = -1

is rejected, leaving

u = x^2 = 6

Squaring any real number yields a non-negative result.

Key Concept

Solving rational equations reducible to quadratics via algebraic substitution and rejecting non-real roots.

Estimated Time:2m 0s

Question 486Question →

Two fulfillment centers, Center X and Center Y, process customer orders at constant individual rates of $x$ orders per hour and $y$ orders per hour, respectively, where $x > y > 0$ . Working together simultaneously, Center X and Center Y complete a total batch of $N$ orders in $T$ hours. In a separate shift, Center X works alone for $h$ hours (where $0 < h < T$ ) to process a portion of the $N$ orders, after which Center Y works alone for $k$ hours to finish all remaining orders in the batch.

Which of the following algebraic statements MUST be true? Select all that apply.

Select all that apply

The ratio of the processing rates satisfies

\frac{x}{y} = \frac{k - T}{T - h}

.

The processing rate of Center Y is expressed as

y = \frac{N(T - h)}{T(k - h)}

.

The total sequential processing time satisfies

h + k = 2T

.

The ratio of the processing rates simplifies directly to

\frac{x}{y} = \frac{k}{h}

.

The processing rate of Center X is expressed as

x = \frac{N(k - T)}{T(k + h)}

.

Show answer & explanation

Answer: The ratio of the processing rates satisfies $\frac{x}{y} = \frac{k - T}{T - h}$ .; The processing rate of Center Y is expressed as $y = \frac{N(T - h)}{T(k - h)}$ .

Answer

The valid algebraic statements are the rate ratio

\frac{x}{y} = \frac{k - T}{T - h}

and the expression for Center Y's rate

y = \frac{N(T - h)}{T(k - h)}

.

Equating the total work from combined operation

N = (x + y)T

and sequential operation

N = xh + yk

yields

x(T - h) = y(k - T)

, which proves that the ratio

\frac{x}{y} = \frac{k - T}{T - h}

must be true. Furthermore, substituting

x = \frac{N}{T} - y

into

xh + yk = N

gives

\frac{Nh}{T} + y(k - h) = N

, which simplifies to

y = \frac{N(T - h)}{T(k - h)}

. Therefore, both of these algebraic formulations are correct.

Step-by-Step Solution

1

Formulate the total work equations for both operational scenarios.

From simultaneous operation:

N = (x + y)T = xT + yT

. From sequential operation:

N = xh + yk

.

Total work completed in both scenarios equals the batch size

N

.

2

Equate the two expressions for

N

to derive the relationship between rates and times.

xT + yT = xh + yk \implies xT - xh = yk - yT \implies x(T - h) = y(k - T)

.

Grouping like rate terms enables solving for the rate ratio

\frac{x}{y}

.

3

Divide by

y(T - h)

to find the rate ratio

\frac{x}{y}

.

\frac{x}{y} = \frac{k - T}{T - h}.

Since

0 < h < T

,

T - h > 0

, so the division is valid and confirms the rate ratio relation.

4

Express

x

in terms of

y, N, T

and substitute into the sequential work equation.

x = \frac{N}{T} - y \implies \left(\frac{N}{T} - y\right)h + yk = N \implies \frac{Nh}{T} + y(k - h) = N

.

Eliminating

x

isolates

y

in terms of given parameters

N, T, h, k

.

5

Solve for

y

.

y(k - h) = N - \frac{Nh}{T} = N\left(1 - \frac{h}{T}\right) = \frac{N(T - h)}{T} \implies y = \frac{N(T - h)}{T(k - h)}

.

This establishes the exact formula for

y

.

Key Concept

Multi-variable system modeling of rate, time, and work constraints.

Estimated Time:2m 30s

Question 487Question →

A container holds 60 cards, each marked with a distinct integer from 1 to 60, inclusive. If one card is selected at random, the probability that the integer on the selected card is divisible by 5, but divisible by neither 4 nor 6, can be expressed as a simplified fraction $\frac{a}{b}$ , where $a$ and $b$ are positive integers with no common factors greater than 1. What is the value of $a + b$ ?

Show answer & explanation

Answer: 17

Answer

The value of

a + b

is 17.

There are 60 equally likely outcomes. Out of 12 multiples of 5, exactly 4 integers (20, 30, 40, and 60) are divisible by 4 or 6. Thus, 12 - 4 = 8 integers satisfy the condition of being divisible by 5 but neither 4 nor 6. The single-event probability is 8/60 = 2/15. In simplest form, a = 2 and b = 15, yielding a + b = 17.

Step-by-Step Solution

1

Determine total number of possible outcomes.

Total outcomes

N = 60

.

Each of the 60 cards is equally likely to be selected.

2

Identify all multiples of 5 in the set

\{1, 2, \dots, 60\}

.

Multiples of 5:

\{5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60\}

, containing

\frac{60}{5} = 12

integers.

These represent all potential candidates before applying exclusion criteria.

3

Identify multiples of 5 that are also divisible by 4 or by 6.

Multiples of 5 divisible by 4:

\text{LCM}(5,4) = 20 \rightarrow \{20, 40, 60\}

(3 integers). Multiples of 5 divisible by 6:

\text{LCM}(5,6) = 30 \rightarrow \{30, 60\}

(2 integers). Overlap (divisible by both 4 and 6):

\text{LCM}(5,4,6) = 60 \rightarrow \{60\}

(1 integer).

An integer is divisible by both 5 and 4 if it is a multiple of 20, and by both 5 and 6 if it is a multiple of 30.

4

Apply the inclusion-exclusion principle to find the count of invalid multiples of 5.

Invalid multiples =

3 + 2 - 1 = 4

integers (

\{20, 30, 40, 60\}

). Favorable outcomes =

12 - 4 = 8

integers (

\{5, 10, 15, 25, 35, 45, 50, 55\}

).

The integer 60 is divisible by both 4 and 6 and must not be double-subtracted.

5

Calculate the probability, express it as a simplified fraction

\frac{a}{b}

, and compute

a + b

.

Probability

P = \frac{8}{60} = \frac{2}{15}

. Thus,

a = 2

,

b = 15

, and

a + b = 2 + 15 = 17

.

Dividing the numerator and denominator by 4 simplifies

\frac{8}{60}

to

\frac{2}{15}

, where

\text{gcd}(2, 15) = 1

.

Key Concept

Basic Single-Event Probability with Multiple Divisibility Constraints

Question 488Question →

A commercial bakery uses two automated dough mixing machines, Machine $A$ and Machine $B$ , to process large batches of dough. Working alone at its constant rate, Machine $A$ can process a full batch of dough in $12$ hours. Machine $B$ , working alone at its constant rate, can process a full batch of dough in $6$ hours. Machine $A$ starts processing a batch alone. After working for $3$ hours, Machine $B$ joins Machine $A$ , and both machines work together at their respective constant rates to finish the batch. What is the total number of hours required to process the entire batch of dough from start to finish?

Show answer & explanation

Answer: 6

Answer

The total number of hours required to process the entire batch of dough from start to finish is 6 hours.

Machine A processes 1/12 of the batch per hour. In the first 3 hours, Machine A finishes 3/12 = 1/4 of the batch, leaving 3/4 of the batch uncompleted. When Machine B joins, the combined processing rate is 1/12 + 1/6 = 1/4 batch per hour. Dividing the remaining 3/4 batch by the combined rate of 1/4 batch per hour yields 3 hours. Adding the initial 3 hours worked by Machine A alone gives a total elapsed time of 6 hours.

Step-by-Step Solution

1

Determine the individual work rates for Machine

A

and Machine

B

.

Machine

A

's rate is

\frac{1}{12}

batch/hour, and Machine

B

's rate is

\frac{1}{6}

batch/hour.

Work rate is the fraction of the job completed per hour (1 divided by total hours required alone).

2

Calculate the portion of the batch completed by Machine

A

operating alone during the first 3 hours.

Machine

A

completes

3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}

of the batch.

Work completed equals work rate multiplied by time worked.

3

Determine the remaining fraction of the batch to be completed.

Remaining work is

1 - \frac{1}{4} = \frac{3}{4}

of the batch.

The entire job corresponds to 1 whole batch.

4

Calculate the combined work rate of Machine

A

and Machine

B

.

Combined rate

= \frac{1}{12} + \frac{1}{6} = \frac{1}{12} + \frac{2}{12} = \frac{3}{12} = \frac{1}{4}

batch per hour.

When machines work together, their individual rates add up.

5

Calculate the time required for both machines working together to complete the remaining work.

Time together

= \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{3/4}{1/4} = 3

hours.

Time equals work divided by rate.

6

Calculate the total time required from start to finish.

Total time

= 3 \text{ hours (Machine A alone)} + 3 \text{ hours (both together)} = 6

hours.

The total time is the sum of time spent in each phase of the process.

Key Concept

Work Rate and Combined Work

Question 489Question →

The numbers of packages delivered by a courier on seven consecutive days are listed below:

5, 13, 2, 10, 15, 7, 4

What is the median number of packages delivered per day over this seven-day period?

5

7

8

10

13

Show answer & explanation

Answer: $7$

Answer

The median number of packages delivered per day is

7

.

To find the median of a dataset, the values must first be arranged in ascending order:

2, 4, 5, 7, 10, 13, 15

. Since there are

7

numbers, the median is the

4

th number in the ordered list, which is

7

.

Step-by-Step Solution

1

Arrange the data values in ascending order.

The ordered set is

2, 4, 5, 7, 10, 13, 15

.

Finding the median requires ordering the data sequentially first.

2

Identify the middle term of the ordered set containing

7

values.

The middle term is the

\frac{7+1}{2} = 4

th value, which is

7

.

For an odd number of data points

N

, the median is the

\frac{N+1}{2}

th element.

Key Concept

Median of a finite data set

Estimated Time:45s

Question 490Question →

An architectural firm initially allocates its annual project hours among three divisions—Residential, Commercial, and Urban Planning—in the ratio of $7 : 5 : 3$ , respectively. Mid-year, to meet new project demands, $300$ hours from Residential and $100$ hours from Commercial are transferred to Urban Planning. Following this transfer, the ratio of Commercial hours to Urban Planning hours becomes $1 : 1$ . What was the total number of project hours initially allocated across all three divisions?

Show answer & explanation

Answer: 3750

Answer

The total number of project hours initially allocated across all three divisions was 3,750 hours.

Representing the initial hours as

7x

,

5x

, and

3x

gives a total of

15x

hours. After the transfers, Commercial hours equal

5x - 100

and Urban Planning hours equal

3x + 300 + 100 = 3x + 400

. Setting these equal gives

5x - 100 = 3x + 400

, which simplifies to

2x = 500

, or

x = 250

. The initial total hours were

15 \times 250 = 3,750

.

Step-by-Step Solution

1

Define initial quantities using a multiplier variable

Let

x

represent the multiplier. Initial hours are Residential =

7x

, Commercial =

5x

, and Urban Planning =

3x

. Total initial hours =

15x

.

Ratios define the proportional relationship among the three divisions.

2

Determine updated hours after the transfers

Commercial hours become

5x - 100

. Urban Planning receives

300

hours from Residential and

100

hours from Commercial, becoming

3x + 300 + 100 = 3x + 400

.

Quantities added to Urban Planning must equal the sum of hours subtracted from the other two divisions.

3

Set up and solve the equation for the new ratio

Since Commercial and Urban Planning hours are now in a

1 : 1

ratio,

5x - 100 = 3x + 400 \implies 2x = 500 \implies x = 250

.

A

1 : 1

ratio means both quantities are equal.

4

Calculate the initial total project hours

Total initial hours =

15 \times 250 = 3,750

.

Substitute

x = 250

into the expression for total initial hours (

15x

).

Key Concept

Solving multi-part ratio word problems involving internal transfers by setting up algebraic equations with a single unknown multiplier.

Question 491Question →

How many integer values of $x$ satisfy the inequality $\frac{|2x - 7| - 3}{|x + 1| - 4} \leq 0$ ?

7

8

9

10

11

Show answer & explanation

Answer: 9

Answer

9 integer values satisfy the given inequality.

The quotient of two algebraic expressions is non-positive (

\leq 0

) when the numerator is zero (and denominator non-zero) or when the numerator and denominator have opposite signs. Testing these cases yields two disjoint intervals:

-5 < x \leq 2

and

3 < x \leq 5

. Within these intervals, the integers

-4, -3, -2, -1, 0, 1, 2, 4,

and

5

satisfy the inequality, yielding exactly 9 integer solutions.

Step-by-Step Solution

1

Determine where the numerator and denominator equal zero to set critical points and domain restrictions.

Numerator is zero at

|2x - 7| = 3 \implies x = 2

or

x = 5

. Denominator is zero at

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

or

x = 3

, which are excluded from the domain.

Division by zero is undefined, so

x \neq -5

and

x \neq 3

.

2

Analyze Case 1: Numerator

\geq 0

and Denominator

< 0

.

Numerator

\geq 0 \implies x \leq 2

or

x \geq 5

. Denominator

< 0 \implies -5 < x < 3

. Intersecting these gives

-5 < x \leq 2

.

A fraction is non-positive when the numerator and denominator have opposite signs.

3

Analyze Case 2: Numerator

\leq 0

and Denominator

> 0

.

Numerator

\leq 0 \implies 2 \leq x \leq 5

. Denominator

> 0 \implies x < -5

or

x > 3

. Intersecting these gives

3 < x \leq 5

.

A fraction is non-positive when the numerator and denominator have opposite signs.

4

Combine solution intervals and count integer solutions.

Combined solution set is

(-5, 2] \cup (3, 5]

. The integer values in this set are

-4, -3, -2, -1, 0, 1, 2, 4, 5

, giving a total count of 9.

Integrate all valid cases while excluding values that make the denominator zero.

Key Concept

Solving Rational Inequalities with Absolute Values

Estimated Time:2m 30s

Question 492Question →

An arithmetic sequence and a geometric sequence both have a first term equal to $2$ . The 3rd term of the arithmetic sequence is equal to the 2nd term of the geometric sequence, and the 7th term of the arithmetic sequence is equal to the 3rd term of the geometric sequence. If the common ratio of the geometric sequence is not equal to $1$ , what is the sum of the first $10$ terms of the arithmetic sequence?

20

30

54

65

130

Show answer & explanation

Answer: 65

Answer

The sum of the first 10 terms of the arithmetic sequence is 65.

By writing the equations for the given sequence terms (

2 + 2d = 2r

and

2 + 6d = 2r^2

), we substitute

d = r - 1

into the quadratic equation to find

r^2 - 3r + 2 = 0

. Since

r \neq 1

, we find

r = 2

, which gives a common difference

d = 1

. Applying the sum formula for an arithmetic sequence for

n = 10

yields

S_{10} = \frac{10}{2}(2(2) + 9(1)) = 5(13) = 65

.

Step-by-Step Solution

1

Express the 3rd and 7th terms of the arithmetic sequence and 2nd and 3rd terms of the geometric sequence.

Arithmetic terms:

a_3 = 2 + 2d

and

a_7 = 2 + 6d

. Geometric terms:

b_2 = 2r

and

b_3 = 2r^2

.

The

n

-th term of an arithmetic sequence is

a_n = a_1 + (n-1)d

and of a geometric sequence is

b_n = b_1 r^{n-1}

.

2

Set up the system of equations given by the problem.

2 + 2d = 2r \implies 1 + d = r \implies d = r - 1

, and

2 + 6d = 2r^2 \implies 1 + 3d = r^2

.

We are given

a_3 = b_2

and

a_7 = b_3

with

a_1 = b_1 = 2

.

3

Substitute

d = r - 1

into the second equation and solve for

r

.

1 + 3(r - 1) = r^2 \implies r^2 - 3r + 2 = 0 \implies (r - 1)(r - 2) = 0

. Since

r \neq 1

,

r = 2

.

Solving the quadratic equation yields two roots, but the problem excludes

r = 1

.

4

Determine the common difference

d

and compute the sum of the first 10 terms of the arithmetic sequence.

d = 2 - 1 = 1

.

S_{10} = \frac{10}{2}[2(2) + (10 - 1)(1)] = 5(4 + 9) = 65

.

The sum of the first

n

terms of an arithmetic sequence is given by

S_n = \frac{n}{2}[2a_1 + (n-1)d]

.

Key Concept

Combining Arithmetic and Geometric Sequence Formulas

Estimated Time:2m 0s

Question 493Question →

What is the sum of all distinct real solutions to the equation $(x^2 - 3x)^2 - 2(x^2 - 3x) - 8 = 0$ ?

2

3

6

0

8

Show answer & explanation

Answer: 6

Answer

The sum of all distinct real solutions is 6.

By defining

u = x^2 - 3x

, the original equation reduces to

u^2 - 2u - 8 = 0

, which factors as

(u - 4)(u + 2) = 0

, giving

u = 4

or

u = -2

. Substituting back for

x

gives two quadratic equations:

x^2 - 3x - 4 = 0

(yielding roots

x = 4

and

x = -1

) and

x^2 - 3x + 2 = 0

(yielding roots

x = 1

and

x = 2

). The four distinct real roots are

-1, 1, 2,

and

4

, and their sum is

-1 + 1 + 2 + 4 = 6

.

Step-by-Step Solution

1

Use algebraic substitution to simplify the disguised quadratic equation.

Let

u = x^2 - 3x

. The equation becomes

u^2 - 2u - 8 = 0

.

Recognizing the repeating quadratic block

x^2 - 3x

transforms the 4th-degree polynomial into a standard 2nd-degree quadratic in terms of

u

.

2

Factor the quadratic equation in terms of

u

to find its roots.

(u - 4)(u + 2) = 0 \implies u = 4 \text{ or } u = -2

.

Finding the values of

u

establishes the two quadratic equations in terms of

x

that need to be solved.

3

Substitute

x^2 - 3x

back for

u

and solve for

x

in Case 1 (

u = 4

).

x^2 - 3x = 4 \implies x^2 - 3x - 4 = 0 \implies (x - 4)(x + 1) = 0 \implies x = 4 \text{ or } x = -1

.

The discriminant

b^2 - 4ac = (-3)^2 - 4(1)(-4) = 25 > 0

, so there are two distinct real solutions:

x = 4

and

x = -1

.

4

Substitute

x^2 - 3x

back for

u

and solve for

x

in Case 2 (

u = -2

).

x^2 - 3x = -2 \implies x^2 - 3x + 2 = 0 \implies (x - 1)(x - 2) = 0 \implies x = 1 \text{ or } x = 2

.

The discriminant

b^2 - 4ac = (-3)^2 - 4(1)(2) = 1 > 0

, so there are two distinct real solutions:

x = 1

and

x = 2

.

5

Sum all distinct real solutions.

-1 + 1 + 2 + 4 = 6

.

All four solutions

\{-1, 1, 2, 4\}

are distinct and real.

Key Concept

Solving Disguised Quadratic Equations via Algebraic Substitution

Estimated Time:1m 30s

Question 494Question →

The price of a jacket was increased by $20\%$ . During a clearance sale, the new price was reduced by $10\%$ . What was the net percentage increase in the price of the jacket from its original value?

18\%

10\%

8\%

12\%

30\%

Show answer & explanation

Answer: $8\%$

Answer

An

8\%

net increase

Applying successive percentage changes requires multiplying the growth/decay factors. A

20\%

increase corresponds to a multiplier of

1.20

, and a

10\%

decrease corresponds to a multiplier of

0.90

. The net multiplier is

1.20 \times 0.90 = 1.08

, which represents a net increase of

8\%

over the original price.

Step-by-Step Solution

1

Assume a convenient original price to track percentage changes easily.

Let the original price of the jacket be

\$100

.

Choosing

\$100

as a base makes calculating percentages straightforward.

2

Calculate the price after the

20\%

increase.

\$\$100 + 0.20 \times \$100 = \$120

.

The price increases by

20\%

of the original price.

3

Calculate the price after the

10\%

clearance discount.

\$\$120 - 0.10 \times \$120 = \$108

.

The discount of

10\%

applies to the intermediate price of

\$120

, not the original

\$100

.

4

Determine the net percentage change from the original price.

\frac{\$108 - \$100}{\$100} \times 100\% = 8\%

.

Comparing the final price of

\$108

to the initial price of

\$100

gives a net increase of

8\%

.

Key Concept

Successive Percent Change

Estimated Time:45s

Question 495Question →

A student must choose $3$ books to read from a reading list of $6$ distinct books. How many different combinations of $3$ books can the student select?

15

18

20

60

120

Show answer & explanation

Answer: $20$

Answer

The student can select

20

different combinations of

3

books.

To find the number of ways to select

3

books from

6

distinct books when order does not matter, use the combination formula

\binom{n}{k} = \frac{n!}{k!(n-k)!}

. Evaluating

\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20

. Thus,

20

is the correct number of combinations.

Step-by-Step Solution

1

Identify the counting method

Since the order in which the books are selected does not matter, use the combinations formula

\binom{n}{k} = \frac{n!}{k!(n-k)!}

.

Selection of a group without regard to order requires combinations.

2

Substitute the given values into the formula

Here

n = 6

and

k = 3

, so

\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1}

.

Apply

n=6

available items and

k=3

chosen items.

3

Simplify the expression

\frac{120}{6} = 20.

Perform the basic arithmetic division.

Key Concept

Combinations and Group Selections

Estimated Time:45s

Question 496Question →

A financial analyst chooses two investment assets at random, one after another without replacement, from a portfolio consisting of $3$ technology stocks and $2$ utility stocks. The probability of achieving a target return depends on the composition of the selected assets:
- If both selected assets are technology stocks, the probability of achieving the target return is $0.80$ .
- If exactly one selected asset is a technology stock, the probability of achieving the target return is $0.50$ .
- If neither selected asset is a technology stock, the probability of achieving the target return is $0.10$ .

What is the probability, expressed as a decimal, that the analyst achieves the target return?

Show answer & explanation

Answer: 0.55

Answer

The probability that the analyst achieves the target return is 0.55.

To find the overall probability of achieving the target return, calculate the probability of each mutually exclusive composition of assets selected without replacement, and weight each by its conditional probability of success. The probability of selecting two technology stocks is (3/5) * (2/4) = 0.30. The probability of selecting one technology stock and one utility stock in either order is (3/5)*(2/4) + (2/5)*(3/4) = 0.60. The probability of selecting two utility stocks is (2/5) * (1/4) = 0.10. Multiplying each by its respective success rate (0.80, 0.50, and 0.10) yields 0.24 + 0.30 + 0.01 = 0.55.

Step-by-Step Solution

1

Determine the probabilities of the three mutually exclusive selection scenarios without replacement.

P(TT) = 0.30, P(1T) = 0.60, P(UU) = 0.10.

Since selections are made without replacement, the outcome of the first draw affects the total count and available assets for the second draw.

2

Calculate the joint probability of achieving the target return for each scenario.

P(TT and Target) = 0.24, P(1T and Target) = 0.30, P(UU and Target) = 0.01.

Multiply the probability of drawing each combination by the conditional probability of success for that specific combination.

3

Sum the joint probabilities across all possible scenarios.

Total Probability = 0.24 + 0.30 + 0.01 = 0.55.

The scenarios are mutually exclusive and exhaustive, so their sum gives the total overall probability of achieving the target return.

Key Concept

Probability of Dependent Events and Law of Total Probability

Estimated Time:2m 0s

Question 497Question →

If the quadratic equation $3x^2 - kx + 48 = 0$ has equal real roots, where $k > 0$ , and the roots of the quadratic equation $x^2 + kx - m = 0$ differ by 10, what is the value of $m$ ?

Show answer & explanation

Answer: -119

Answer

The value of

m

is

-119

.

Setting the discriminant of

3x^2 - kx + 48 = 0

equal to zero gives

k^2 - 4(3)(48) = 0

, yielding

k = 24

since

k > 0

. Substituting

k = 24

into the second equation gives

x^2 + 24x - m = 0

. According to Vieta's formulas, the sum of roots is

r_1 + r_2 = -24

and the product of roots is

r_1 r_2 = -m

. Using the algebraic identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

with the given root difference

|r_1 - r_2| = 10

yields

10^2 = (-24)^2 - 4(-m)

, which simplifies to

100 = 576 + 4m

. Subtracting 576 from both sides gives

4m = -476

, so

m = -119

.

Step-by-Step Solution

1

Find

k

using the discriminant condition for equal roots of

3x^2 - kx + 48 = 0

.

k = 24

A quadratic equation

ax^2 + bx + c = 0

has equal real roots when its discriminant

b^2 - 4ac = 0

. Here,

(-k)^2 - 4(3)(48) = 0

, so

k^2 = 576

. Since

k > 0

,

k = 24

.

2

Set up the algebraic relationship for the root difference of

x^2 + 24x - m = 0

.

(r_1 - r_2)^2 = 100

Given that the difference between the roots

r_1

and

r_2

is 10,

|r_1 - r_2| = 10

, which implies

(r_1 - r_2)^2 = 100

.

3

Apply Vieta's formulas to express

(r_1 - r_2)^2

in terms of

m

and solve.

m = -119

By Vieta's formulas for

x^2 + 24x - m = 0

, the sum of roots is

r_1 + r_2 = -24

and the product of roots is

r_1 r_2 = -m

. Using the identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, substitute the known values:

100 = (-24)^2 - 4(-m) = 576 + 4m

. Solving

576 + 4m = 100

gives

4m = -476

, so

m = -119

.

Key Concept

Discriminant analysis and root difference identity via Vieta's formulas

Question 498Question →

An investment fund initially holds shares of Technology, Healthcare, and Energy stocks in the ratio $5 : 3 : 2$ , respectively. After the fund sells $30$ shares of Technology stock and purchases $30$ shares of Energy stock, the ratio of Technology, Healthcare, and Energy shares becomes $4 : 3 : 3$ . What was the total number of shares in the investment fund initially?

Show answer & explanation

Answer: 300

Answer

The initial total number of shares in the investment fund was 300.

The initial shares can be represented as

5x

,

3x

, and

2x

for a total of

10x

shares. After selling

30

Technology shares and buying

30

Energy shares, the Healthcare shares remain unchanged at

3x

. Since Healthcare represents

3

parts in both the initial and final ratios (

5:3:2

and

4:3:3

), the value of one ratio part remains constant at

x

. Therefore, the decrease in Technology shares from

5x

to

4x

corresponds directly to

1x = 30

shares. Multiplying

x = 30

by the initial total of

10

parts gives

300

shares.

Step-by-Step Solution

1

Define initial quantities using a common ratio multiplier

x

Technology =

5x

, Healthcare =

3x

, Energy =

2x

. Total initial shares =

10x

.

Representing ratio terms algebraically preserves their proportional relationships.

2

Account for changes in share counts

New Technology shares =

5x - 30

, Healthcare shares =

3x

, new Energy shares =

2x + 30

.

30 shares of Technology were sold and 30 shares of Energy were added.

3

Set up an equation comparing the new ratio of Technology to Healthcare

\frac{5x - 30}{3x} = \frac{4}{3}

The new ratio of Technology to Healthcare is given as

4 : 3

.

4

Solve for multiplier

x

3(5x - 30) = 12x \implies 15x - 90 = 12x \implies 3x = 90 \implies x = 30

Cross-multiplying eliminates fractions to solve for the unknown multiplier.

5

Calculate the initial total number of shares

10x = 10 \times 30 = 300

The initial total is the sum of the initial parts (

5x + 3x + 2x = 10x

).

Key Concept

Ratio Modification and Algebraic Scaling

Estimated Time:1m 30s

Question 499Question →

An analytics team tracked the monthly recurring revenue (in thousands of dollars) for 10 distinct client accounts of a SaaS company. The arithmetic mean of the monthly revenues across these 10 accounts was $\$45$ thousand, and the median monthly revenue was $\$38$ thousand. If the revenue for each account was a distinct positive integer (in thousands of dollars), what is the maximum possible monthly revenue, in thousands of dollars, generated by the single highest-performing account?

Show answer & explanation

Answer: 241

Answer

The maximum possible monthly revenue generated by the single highest-performing account is 241 thousand dollars.

The total sum of the 10 distinct positive integer revenues is

10 \times 45 = 450

thousand dollars. To maximize the largest term (

x_{10}

), the sum of the remaining 9 terms (

x_1

through

x_9

) must be minimized. The smallest 4 positive integers are

1, 2, 3, 4

(sum =

10

). Given a median of 38, the two central terms must satisfy

x_5 + x_6 = 76

. Because all values are distinct integers and

x_5 < x_6

, the smallest possible integer value for

x_6

is 39 (making

x_5 = 37

). To minimize

x_7, x_8, x_9

, they should be the smallest distinct integers greater than 39, which are

40, 41, 42

. Summing these minimal values gives

10 + 76 + 40 + 41 + 42 = 209

. Subtracting this sum from the total sum yields

450 - 209 = 241

thousand dollars.

Step-by-Step Solution

1

Calculate the total revenue across all 10 client accounts.

Total sum =

10 \times 45 = 450

thousand dollars.

The sum of a data set is equal to the number of elements multiplied by the arithmetic mean.

2

Use the median to determine the sum of the 5th and 6th terms.

Since

n = 10

, the median is

\frac{x_5 + x_6}{2} = 38

, so

x_5 + x_6 = 76

.

In an ordered set of an even number of elements, the median is the average of the two central terms.

3

Formulate the minimization strategy to maximize the largest term (

x_{10}

).

To maximize

x_{10} = 450 - (x_1 + x_2 + \dots + x_9)

, minimize the sum of the first 9 terms.

For a fixed sum, maximizing one element requires minimizing all other elements in the set.

4

Minimize the first four terms (

x_1, x_2, x_3, x_4

).

Assign

x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4

, which sum to

10

.

The revenues are distinct positive integers, so the smallest possible values are the first 4 positive integers.

5

Minimize

x_6

to minimize the subsequent terms (

x_7, x_8, x_9

).

Since

x_5 < x_6

and

x_5 + x_6 = 76

, the smallest integer value for

x_6

is

39

(which sets

x_5 = 37

). Then set

x_7 = 40, x_8 = 41, x_9 = 42

.

Minimizing

x_6

allows

x_7, x_8, x_9

to take the smallest possible distinct integer values greater than

x_6

.

6

Calculate the maximum value for

x_{10}

.

x_{10} = 450 - (10 + 76 + 40 + 41 + 42) = 450 - 209 = 241

.

Subtracting the minimal sum of the first 9 elements from the total sum gives the theoretical maximum for the 10th term.

Key Concept

Extremal optimization of distinct statistics data sets given mean and median constraints

Question 500Question →

A manufacturer produces custom electric scooters at a manufacturing cost of $250 per unit. To establish the list price, the manufacturer marks up the cost price by 60%. During a promotional campaign, 70% of the manufactured scooters are sold at a 15% discount off the list price, and the remaining 30% are sold at a clearance discount of 35% off the list price. If the manufacturer incurs a fixed promotional overhead expense of$ 14,400, and the overall net profit across all scooters sold (after deducting the fixed overhead expense) is equal to 12% of the total manufacturing cost of all the scooters, how many scooters were manufactured and sold in total?

240

400

720

800

3,600

Show answer & explanation

Answer: 400

Answer

400 scooters

To find the total number of scooters

N

, first determine the list price:

250 \times 1.60 = \$400

. Next, calculate the selling prices for the two tiers:

400 \times 0.85 = \$340

for 70% of units, and

400 \times 0.65 = \$260

for 30% of units. The weighted average revenue per scooter is

0.70(340) + 0.30(260) = \$316

. The profit before fixed overhead per scooter is

316 - 250 = \$66

, yielding a total net profit of

66N - 14,400

. Setting this equal to the target net profit of 12% of total manufacturing costs (

0.12 \times 250N = 30N

) gives

66N - 14,400 = 30N

, which simplifies to

36N = 14,400

, or

N = 400

.

Step-by-Step Solution

1

Calculate the list price of a single scooter.

List price =

250 \times (1 + 0.60) = \$400

.

The list price is established by marking up the cost price ($250) by 60%.

2

Calculate the discounted selling prices for both sales categories.

Promotional price (70% of inventory) =

400 \times (1 - 0.15) = \$340

. Clearance price (30% of inventory) =

400 \times (1 - 0.35) = \$260

.

Discounts are applied directly to the list price.

3

Determine the weighted average selling price per unit and profit per unit before overhead.

Weighted average selling price =

0.70(340) + 0.30(260) = 238 + 78 = \$316

. Gross profit per unit =

316 - 250 = \$66

.

Combining the two sales segments gives the effective revenue generated per unit sold.

4

Set up and solve the net profit equation for total units

N

.

Total net profit =

66N - 14,400

. Target profit =

0.12 \times 250N = 30N

. Equating gives

66N - 14,400 = 30N \implies 36N = 14,400 \implies N = 400

.

Net profit after deducting fixed overhead must equal 12% of total manufacturing cost.

Key Concept

Weighted average selling price, markup vs. discount percentage bases, and multi-step net profit equation modeling.

Alternative Method

Work per unit on a percentage basis: List price is 160% of cost. Revenue from 70% of inventory is

0.70 \times 1.60 \times 0.85 = 0.952

of cost. Revenue from 30% of inventory is

0.30 \times 1.60 \times 0.65 = 0.312

of cost. Total revenue is

(0.952 + 0.312) = 1.264

of cost. Gross profit before overhead is

1.264 - 1.000 = 0.264

of cost. Net profit is

0.264 \times \text{Total Cost} - 14,400

. Setting this equal to

0.120 \times \text{Total Cost}

gives

0.144 \times \text{Total Cost} = 14,400

, so

\text{Total Cost} = \$100,000

. Since total cost is

250 \times N = 100,000

, we get

N = 400

.

Estimated Time:2m 0s

Answer

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