All practice questions

Question 781Question →

A museum cataloged $250$ historical artifacts. Each artifact was subjected to at least one of three preservation procedures: micro-scanning, chemical analysis, or radiocarbon dating. Exactly $140$ artifacts underwent micro-scanning, $120$ underwent chemical analysis, and $100$ underwent radiocarbon dating. If $40$ artifacts underwent both micro-scanning and chemical analysis, $35$ underwent both chemical analysis and radiocarbon dating, and $45$ underwent both micro-scanning and radiocarbon dating, how many artifacts underwent all three procedures?

Show answer & explanation

Answer: 10

Answer

10 artifacts underwent all three preservation procedures.

Using the Inclusion-Exclusion Principle for three sets, the total union is equal to the sum of individual set sizes (

140 + 120 + 100 = 360

), minus the sum of two-set overlaps (

40 + 35 + 45 = 120

), plus the three-set overlap. Setting up

250 = 360 - 120 + x

yields

250 = 240 + x

, so

x = 10

.

Step-by-Step Solution

1

Set up the Inclusion-Exclusion formula for three overlapping sets.

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

The total size of the union equals the sum of the individual set sizes minus the pairwise intersections plus the triple intersection.

2

Substitute the given values into the formula.

250 = 140 + 120 + 100 - (40 + 35 + 45) + |A \cap B \cap C|

Since every artifact underwent at least one procedure, the total number of artifacts equals the union of all three sets.

3

Compute the sum of individual sets and pairwise intersections to isolate the unknown.

250 = 360 - 120 + |A \cap B \cap C| \Rightarrow 250 = 240 + |A \cap B \cap C|

Simplifying the arithmetic terms allows direct calculation of the triple intersection.

4

Subtract

240

from

250

to find the final value.

|A \cap B \cap C| = 10

Solving the linear equation yields the number of artifacts undergoing all three procedures.

Key Concept

Principle of Inclusion-Exclusion for Three Sets

Question 782Question →

An agricultural processing plant produces a livestock feed blend consisting of corn, soybeans, and oats in an initial weight ratio of $5 : 3 : 2$ , respectively. To increase the protein content of a $1,200\text{ kg}$ batch of this blend, a technician removes a certain quantity of oats and replaces it with an equal weight of soybeans. If the resulting weight ratio of corn to soybeans to oats becomes $10 : 9 : 1$ , how many kilograms of oats were replaced by soybeans?

120

180

240

540

720

Show answer & explanation

Answer: 180

Answer

180 kg of oats were replaced by soybeans.

The correct answer demonstrates that since corn is unchanged (

600\text{ kg}

), it corresponds to

10

parts in the new

10 : 9 : 1

ratio. This yields

60\text{ kg}

per part in the new ratio. Oats decrease from

240\text{ kg}

(

2

initial parts of

120\text{ kg}

) to

60\text{ kg}

(

1

final part of

60\text{ kg}

), indicating that

180\text{ kg}

of oats were replaced by an equal weight of soybeans.

Step-by-Step Solution

1

Determine the initial weight of each component in the 1,200 kg batch.

The initial ratio is

5 : 3 : 2

, giving a total of

5 + 3 + 2 = 10

parts. Each part corresponds to

\frac{1,200\text{ kg}}{10} = 120\text{ kg}

. Therefore: Corn

= 5 \times 120 = 600\text{ kg}

, Soybeans

= 3 \times 120 = 360\text{ kg}

, Oats

= 2 \times 120 = 240\text{ kg}

.

Finding the absolute initial quantities provides baseline values before the substitution occurs.

2

Analyze the invariant quantity after the substitution.

Since oats are replaced by an equal weight of soybeans, the total weight of the batch remains

1,200\text{ kg}

, and the weight of corn remains unchanged at

600\text{ kg}

.

Identifying unchanged quantities allows direct scaling of the new ratio.

3

Calculate the value of one ratio unit in the new ratio.

The new ratio of corn to soybeans to oats is

10 : 9 : 1

. The

600\text{ kg}

of corn represents

10

parts of the new ratio. Thus,

1\text{ part} = \frac{600\text{ kg}}{10} = 60\text{ kg}

. Alternatively, total parts

= 10 + 9 + 1 = 20

parts, so

1\text{ part} = \frac{1,200\text{ kg}}{20} = 60\text{ kg}

.

Determining the scale of the new ratio allows finding the new weight of each ingredient.

4

Compute the weight of oats replaced.

The new oats weight is

1 \times 60\text{ kg} = 60\text{ kg}

. The replaced quantity is Initial Oats

-

Final Oats

= 240\text{ kg} - 60\text{ kg} = 180\text{ kg}

.

The difference between initial and final oat weights equals the replaced quantity.

Key Concept

Solving multi-part ratio changes using invariant quantities and constant total mass scaling.

Estimated Time:2m 0s

Question 783Question →

A box contains 90 tokens, numbered sequentially from $1$ to $90$ , inclusive. If one token is drawn at random from the box, what is the probability that the number on the token is a multiple of $4$ or $6$ , but not a multiple of $8$ ?

\frac{13}{45}

\frac{1}{3}

\frac{19}{90}

\frac{2}{9}

\frac{7}{30}

Show answer & explanation

Answer: $\frac{19}{90}$

Answer

The probability that the selected number is a multiple of 4 or 6, but not a multiple of 8, is

\frac{19}{90}

.

To find the probability, determine the number of favorable outcomes out of 90 total outcomes. The number of multiples of 4 is 22, and the number of multiples of 6 is 15. The numbers that are multiples of both 4 and 6 are multiples of 12, of which there are 7. By inclusion-exclusion, the number of integers that are multiples of 4 or 6 is

22 + 15 - 7 = 30

. Since all multiples of 8 are automatically multiples of 4, all 11 multiples of 8 in the range are included in these 30 numbers. Excluding the multiples of 8 leaves

30 - 11 = 19

favorable integers. Thus, the probability is

\frac{19}{90}

.

Step-by-Step Solution

1

Find the count of multiples of 4 and multiples of 6 in the range from 1 to 90.

Multiples of 4:

\lfloor \frac{90}{4} \rfloor = 22

. Multiples of 6:

\lfloor \frac{90}{6} \rfloor = 15

.

Identify the size of each individual set of multiples.

2

Calculate the number of elements in the union of multiples of 4 or 6 using the Inclusion-Exclusion Principle.

Multiples of both 4 and 6 are multiples of

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

. Count of multiples of 12:

\lfloor \frac{90}{12} \rfloor = 7

. Union size:

22 + 15 - 7 = 30

.

Avoid double-counting numbers that are divisible by both 4 and 6.

3

Identify and subtract the multiples of 8 within this union.

Multiples of 8 in the range:

\lfloor \frac{90}{8} \rfloor = 11

. Since every multiple of 8 is also a multiple of 4, all 11 multiples of 8 are contained within the union. Favorable outcomes:

30 - 11 = 19

.

Filter out numbers that satisfy the exclusion condition (multiples of 8).

4

Compute the single-event probability.

P = \frac{19}{90}

.

Divide the number of favorable outcomes (19) by the total sample space (90).

Key Concept

Basic Single-Event Probability with Principle of Inclusion-Exclusion and Set Restriction

Estimated Time:2m 0s

Question 784Question →

A storage tank is completely filled with $150$ liters of a solution containing Compounds X, Y, and Water in a ratio of $2:3:5$ by volume. First, $30$ liters of the mixture are drained and replaced with $30$ liters of a liquid mixture that is $20\%$ Compound Y by volume. Next, $50$ liters of the resulting mixture are drained and replaced with $50$ liters of another liquid mixture that is $70\%$ Compound Y by volume. What is the percentage of Compound Y, by volume, in the final mixture?

Show answer & explanation

Answer: 42

Answer

The final mixture contains 42% of Compound Y by volume.

To find the final concentration, track the exact volume of Compound Y through both stages. Initially, the tank holds

150 \times \frac{3}{10} = 45

liters of Y. In the first replacement, draining

30

liters (

20\%

of the tank) leaves

80\%

of Y, which is

36

liters; adding

30

liters of

20\%

Y mixture adds

6

liters, bringing Y to

42

liters. In the second replacement, draining

50

liters (

\frac{1}{3}

of the tank) leaves

\frac{2}{3}

of Y, which is

28

liters; adding

50

liters of

70\%

Y mixture adds

35

liters, yielding

63

liters of Y. The final concentration is

\frac{63}{150} \times 100\% = 42\%

.

Step-by-Step Solution

1

Calculate initial volume of Compound Y in the tank

Initial Y volume = 45 liters

The ratio X:Y:Water is 2:3:5, giving a total of 2 + 3 + 5 = 10 parts. The fraction of Y is 3/10. For a 150-liter tank, Y = 150 * (3/10) = 45 liters.

2

Calculate Compound Y after the first draining and replacement cycle

Compound Y volume after first replacement = 42 liters

Draining 30 liters removes 30 / 150 = 1/5 of the mixture. Y remaining = 45 * (1 - 1/5) = 36 liters. Replacing with 30 liters of 20% Y adds 30 * 0.20 = 6 liters of Y. Total Y = 36 + 6 = 42 liters.

3

Calculate Compound Y after the second draining and replacement cycle

Compound Y volume after second replacement = 63 liters

Draining 50 liters removes 50 / 150 = 1/3 of the current mixture. Y remaining = 42 * (1 - 1/3) = 28 liters. Replacing with 50 liters of 70% Y adds 50 * 0.70 = 35 liters of Y. Total Y = 28 + 35 = 63 liters.

4

Compute final percentage of Compound Y

42%

The final volume of the tank remains 150 liters. Percentage of Y = (63 / 150) * 100% = 42%.

Key Concept

Multi-Stage Mixture Removal and Replacement

Question 785Question →

A textile manufacturing plant uses two automated weaving looms, Loom $P$ and Loom $Q$ , to produce standardized fabric orders. Working alone at its constant rate, Loom $P$ can complete a full fabric order in $8$ hours, while Loom $Q$ working alone at its constant rate can complete the same order in $12$ hours. Loom $P$ begins working on a full order alone. After $3$ hours, Loom $Q$ joins Loom $P$ , and both looms work together at their respective constant rates until the order is completed. How many total hours does it take, from the moment Loom $P$ begins, to complete the entire fabric order?

3

6

7.5

10

15.5

Show answer & explanation

Answer: $6$

Answer

6

hours

In the first

3

hours, Loom

P

completes

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

of the job, leaving

\frac{5}{8}

of the job remaining. When Loom

Q

joins, their combined rate is

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

of the job per hour. The time needed to complete the remaining

\frac{5}{8}

of the job is

\frac{5/8}{5/24} = 3

hours. Adding the initial

3

hours of solo work gives a total of

6

hours.

Step-by-Step Solution

1

Determine the individual hourly work rates of Loom

P

and Loom

Q

.

Rate of Loom

P = \frac{1}{8}

order/hour; Rate of Loom

Q = \frac{1}{12}

order/hour.

Work rate is the reciprocal of the total time needed to complete one full unit of work.

2

Calculate the fraction of the order completed by Loom

P

during its

3

hours of solo operation.

Work completed by Loom

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

of the order.

Work done equals rate multiplied by time spent working.

3

Find the remaining fraction of the order to be completed.

Remaining work =

1 - \frac{3}{8} = \frac{5}{8}

of the order.

The total job represents

1

whole unit of work.

4

Calculate the combined work rate when both looms operate together.

Combined rate =

\frac{1}{8} + \frac{1}{12} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24}

order/hour.

When entities work together, their individual rates add up.

5

Determine the time required for both looms together to finish the remaining work.

Time together =

\frac{5/8}{5/24} = \frac{5}{8} \times \frac{24}{5} = 3

hours.

Time equals remaining work divided by the combined work rate.

6

Calculate the total time from start to finish.

Total time =

3 \text{ hours (solo)} + 3 \text{ hours (combined)} = 6 \text{ hours}

.

The total elapsed time is the sum of the solo operating time and the combined operating time.

Key Concept

Work Rate and Combined Work

Estimated Time:2m 0s

Question 786Question →

In a geometric sequence of positive numbers, the first term is $3$ and the common ratio is $2$ . What is the value of the 5th term of this sequence?

48

96

1,296

24

16

Show answer & explanation

Answer: $48$

Answer

The 5th term of the sequence is

48

.

The

n

-th term of a geometric sequence is given by

a_n = a_1 \cdot r^{n-1}

. Substituting

a_1 = 3

,

r = 2

, and

n = 5

gives

a_5 = 3 \cdot 2^4 = 3 \cdot 16 = 48

.

Step-by-Step Solution

1

Identify the formula for the

n

-th term of a geometric sequence.

a_n = a_1 \cdot r^{n-1}

In any geometric sequence, each term is obtained by multiplying the previous term by the common ratio

r

.

2

Substitute the given values into the formula.

a_5 = 3 \cdot 2^{5-1} = 3 \cdot 2^4

The first term

a_1 = 3

, common ratio

r = 2

, and term number

n = 5

.

3

Calculate the exponent and final value.

a_5 = 3 \cdot 16 = 48

Evaluate

2^4 = 16

first according to order of operations, then multiply by 3.

Key Concept

Geometric Sequence

n

-th Term Formula

Question 787Question →

An automated freight transport vessel completes a round trip along a $120$ -mile coastal canal route from Port Alpha to Port Beta and back. On the return trip from Port Beta to Port Alpha, due to opposing currents, the vessel's constant speed was $20$ miles per hour slower than its constant speed on the trip from Port Alpha to Port Beta. If the total travel time for the entire round trip was $5$ hours, what was the average speed of the vessel, in miles per hour, for the entire round trip?

40

48

50

60

30

Show answer & explanation

Answer: 48

Answer

The average speed of the vessel for the entire round trip is 48 miles per hour.

The total distance traveled for the round trip is

120 + 120 = 240

miles. Setting up the time equation

\frac{120}{v} + \frac{120}{v-20} = 5

gives the quadratic equation

v^2 - 68v + 480 = 0

, which factors into

(v-60)(v-8) = 0

. Since speed must be greater than

20

mph for a valid return trip, the outgoing speed is

60

mph and the return speed is

40

mph. Dividing the total distance of

240

miles by the total time of

5

hours yields an overall average speed of

48

miles per hour.

Step-by-Step Solution

1

Define variables and establish time equations for each leg of the journey.

Let

v

be the vessel's speed from Port Alpha to Port Beta in miles per hour. The speed from Port Beta to Port Alpha is

v - 20

mph. Time outgoing is

\frac{120}{v}

hours and time returning is

\frac{120}{v - 20}

hours.

Travel time equals distance divided by rate.

2

Set up and simplify the quadratic equation for total travel time.

\frac{120}{v} + \frac{120}{v - 20} = 5 \implies \frac{24}{v} + \frac{24}{v - 20} = 1 \implies 24(v - 20) + 24v = v(v - 20) \implies v^2 - 68v + 480 = 0

.

The sum of the travel times for both legs must equal the total given trip duration of 5 hours.

3

Solve the quadratic equation to determine the valid speed

v

.

Factoring

(v - 60)(v - 8) = 0

gives roots

v = 60

or

v = 8

. Since the return speed

v - 20

must be positive,

v = 60

mph.

A speed of

v = 8

mph would produce a negative return speed, which is physically invalid.

4

Calculate the average speed for the entire round trip.

Total distance

= 120 + 120 = 240

miles. Total time

= 5

hours. Average speed

= \frac{240}{5} = 48

miles per hour.

Average speed over a multi-leg trip is defined as total distance divided by total time.

Key Concept

Equation Modeling and Average Speed over Multi-Leg Journeys

Estimated Time:2m 0s

Question 788Question →

A delivery van travels from Warehouse A to Warehouse B at a constant speed of $40$ miles per hour and returns along the exact same route from Warehouse B to Warehouse A at a constant speed of $60$ miles per hour. If the distance between Warehouse A and Warehouse B is $120$ miles, what is the average speed of the delivery van for the entire round trip, in miles per hour?

Show answer & explanation

Answer: 48

Answer

The average speed for the entire round trip is

48

miles per hour.

To find the average speed for the round trip, divide the total distance (

240

miles) by the total time taken (

5

hours). The time taken for the first leg is

\frac{120}{40} = 3

hours, and for the return leg is

\frac{120}{60} = 2

hours. Thus, the average speed is

\frac{240}{5} = 48

miles per hour.

Step-by-Step Solution

1

Determine the total distance traveled during the round trip.

The distance from Warehouse A to B is

120

miles, making the total round-trip distance

120 + 120 = 240

miles.

Average speed requires the total distance for all legs of the journey.

2

Calculate time taken for each leg and find total time.

Time taken at

40

mph is

\frac{120}{40} = 3

hours. Time taken at

60

mph is

\frac{120}{60} = 2

hours. Total time =

3 + 2 = 5

hours.

Time equals distance divided by rate (

t = \frac{d}{r}

).

3

Divide total distance by total time to obtain average speed.

Average speed =

\frac{240 \text{ miles}}{5 \text{ hours}} = 48

miles per hour.

The defining formula for average speed is

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

.

Key Concept

Average speed for a multi-leg journey is always total distance divided by total time, not the arithmetic mean of the speeds.

Question 789Question →

If $|3 - 2x| = 7$ , what is the product of all possible real values of $x$ ?

-10

-3

3

5

10

Show answer & explanation

Answer: $-10$

Answer

The product of all possible real values of

x

is

-10

.

Solving

|3 - 2x| = 7

yields two equations:

3 - 2x = 7

, which gives

x = -2

, and

3 - 2x = -7

, which gives

x = 5

. Multiplying these two values together gives

(-2) \times 5 = -10

.

Step-by-Step Solution

1

Set up two linear equations corresponding to the positive and negative cases of the absolute value expression.

3 - 2x = 7

or

3 - 2x = -7

By definition,

|a| = b

(where

b \ge 0

) implies

a = b

or

a = -b

.

2

Solve the first linear equation for

x

.

-2x = 4 \implies x = -2

Subtract 3 from both sides and divide by

-2

.

3

Solve the second linear equation for

x

.

-2x = -10 \implies x = 5

Subtract 3 from both sides and divide by

-2

.

4

Calculate the product of the two solutions.

(-2) \times 5 = -10

The question asks for the product of all possible real values of

x

.

Key Concept

Absolute Value Equations

Question 790Question →

If $y = 3^k$ , which of the following expressions is equivalent to $9^{k+1}$ for all real values of $k$ ?

9y^2

9y

3y^2

y^2 + 1

y^3

Show answer & explanation

Answer: $9y^2$

Answer

The expression

9y^2

is equivalent to

9^{k+1}

.

Applying the product rule of exponents gives

9^{k+1} = 9^k \cdot 9^1

. Expressing 9 as

3^2

gives

9^k = (3^2)^k = (3^k)^2

. Substituting

y = 3^k

yields

9 \cdot y^2 = 9y^2

.

Step-by-Step Solution

1

Rewrite

9^{k+1}

using the product rule of exponents (

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

).

9^{k+1} = 9^k \cdot 9^1 = 9 \cdot 9^k

Separating the exponent addition into a product allows simplification of the constant factor.

2

Express the base 9 in terms of base 3.

9^k = (3^2)^k = (3^k)^2

Since

y = 3^k

, converting base 9 to

3^2

allows substitution of

y

into the expression.

3

Substitute

y = 3^k

into the expression.

9 \cdot (3^k)^2 = 9y^2

Replacing

3^k

with

y

yields the simplified algebraic expression in terms of

y

.

Key Concept

Properties of exponents including power of a power

(a^m)^n = a^{mn}

and product of powers

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

.

Estimated Time:45s

Question 791Question →

If $x$ and $y$ are positive real numbers such that $x^2 - y^2 = 105$ and $x + y = 15$ , what is the value of $(x - 2y)^2$ ?

Show answer & explanation

Answer: 9

Answer

The value of

(x - 2y)^2

is 9.

Factoring

x^2 - y^2

into

(x - y)(x + y)

gives

(x - y)(15) = 105

, which simplifies to

x - y = 7

. Solving the system of equations

x + y = 15

and

x - y = 7

gives

x = 11

and

y = 4

. Substituting these values into the target expression

(x - 2y)^2

yields

(11 - 2(4))^2 = 3^2 = 9

.

Step-by-Step Solution

1

Apply the difference of squares factoring identity to

x^2 - y^2

(x - y)(x + y) = 105

The difference of two squares

x^2 - y^2

factors into

(x - y)(x + y)

.

2

Calculate the value of

x - y

x - y = 7

Since

x + y = 15

, dividing 105 by 15 gives

x - y = 7

.

3

Solve the system of equations for

x

and

y

x = 11

and

y = 4

Adding

(x + y) + (x - y) = 15 + 7

yields

2x = 22 \implies x = 11

. Subtracting

(x + y) - (x - y) = 15 - 7

yields

2y = 8 \implies y = 4

.

4

Evaluate the expression

(x - 2y)^2

9

Substitute

x = 11

and

y = 4

into

(x - 2y)^2

to obtain

(11 - 2(4))^2 = (11 - 8)^2 = 3^2 = 9

.

Key Concept

Difference of Squares Factoring and Systems of Equations

Question 792Question →

A private equity firm invested a total of $\$1,000,000$ divided between two portfolio companies, Company A and Company B. During the first year, the value of Company A increased by $x\%$ , while the value of Company B decreased by $x\%$ , where $x > 0$ . At the end of the first year, the value of Company A was $\$720,000$ , and the combined value of both companies was $\$1,040,000$ . If the value of Company B decreased by an additional $x\%$ during the second year relative to its value at the end of the first year, what was the value, in dollars, of Company B at the end of the second year?

Show answer & explanation

Answer: 256000

Answer

The value of Company B at the end of the second year was 256,000 dollars.

By representing the initial investments as algebraic expressions involving the multiplier

k = \frac{x}{100}

and setting up the total initial capital equation

A_0 + B_0 = 1,000,000

, we obtain a quadratic equation in

k

. Solving

(5k-1)^2 = 0

yields

k = 0.2

(

x = 20\%

). Decreasing Company B's Year 1 value of

\$320,000

by

20\%

gives

\$256,000

.

Step-by-Step Solution

1

Find the value of Company B at the end of Year 1

Year 1 value of Company B =

1,040,000 - 720,000 = 320,000

dollars

The total combined value of both companies at the end of Year 1 is given as $1,040,000.

2

Formulate algebraic expressions for initial values using rate

k = \frac{x}{100}

A_0 = \frac{720,000}{1+k}

and

B_0 = \frac{320,000}{1-k}

Company A increased by

x\%

so

A_1 = A_0(1+k)

; Company B decreased by

x\%

so

B_1 = B_0(1-k)

.

3

Set up and simplify the quadratic equation for the combined initial investment

\frac{720,000}{1+k} + \frac{320,000}{1-k} = 1,000,000 \implies 25k^2 - 10k + 1 = 0

Dividing by

80,000

yields

\frac{9}{1+k} + \frac{4}{1-k} = 12.5

. Multiplying through by

2(1-k^2)

leads to

2[9(1-k) + 4(1+k)] = 25(1-k^2)

.

4

Solve for

k

and determine percentage

x

(5k - 1)^2 = 0 \implies k = 0.2

, so

x = 20\%

Factoring the perfect square quadratic expression gives a unique solution for

k

.

5

Compute Company B's value at the end of Year 2

320,000 \times (1 - 0.2) = 256,000

dollars

Company B's value decreases by an additional

20\%

of its Year 1 value (

320,000

).

Key Concept

Algebraic Modeling of Rational/Quadratic Equations from Multi-Step Percent Change Scenarios

Estimated Time:2m 30s

Question 793Question →

For a constant $a > 0$ , the quadratic equation $x^2 - ax + (2a + 1) = 0$ has two real roots, $r$ and $s$ . If $r^2 + s^2 = 43$ , what is the value of $a$ ?

Show answer & explanation

Answer: 9

Answer

The value of

a

is 9.

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

x^2 - ax + (2a + 1) = 0

is

r + s = a

and the product of the roots is

rs = 2a + 1

. Using the identity

r^2 + s^2 = (r + s)^2 - 2rs

, substitute the Vieta expressions to obtain

r^2 + s^2 = a^2 - 2(2a + 1) = a^2 - 4a - 2

. Setting this equal to 43 yields

a^2 - 4a - 2 = 43

, which simplifies to

a^2 - 4a - 45 = 0

. Factoring the quadratic gives

(a - 9)(a + 5) = 0

, giving solutions

a = 9

or

a = -5

. Because the problem specifies that

a > 0

,

a

must be 9. Checking the discriminant

\Delta = (-9)^2 - 4(1)(19) = 5 > 0

confirms that two real roots exist.

Step-by-Step Solution

1

Express the sum and product of the roots in terms of

a

using Vieta's formulas.

r + s = a

and

rs = 2a + 1

For a standard quadratic equation

x^2 + bx + c = 0

, the sum of the roots is

-b

and the product of the roots is

c

.

2

Relate

r^2 + s^2

to

(r + s)

and

rs

.

r^2 + s^2 = (r + s)^2 - 2rs = a^2 - 2(2a + 1) = a^2 - 4a - 2

Expanding

(r + s)^2 = r^2 + 2rs + s^2

allows expressing

r^2 + s^2

in terms of known quantities.

3

Substitute

r^2 + s^2 = 43

into the equation and solve for

a

.

a^2 - 4a - 2 = 43 \implies a^2 - 4a - 45 = 0 \implies (a - 9)(a + 5) = 0

Rearranging terms forms a new quadratic equation in terms of

a

.

4

Apply the positivity constraint

a > 0

and verify that the roots are real.

Since

a > 0

,

a = 9

. The discriminant of the original equation is

\Delta = (-9)^2 - 4(1)(19) = 5 > 0

, confirming real roots exist.

The question specifies that

a

is positive and that

r

and

s

are real numbers.

Key Concept

Vieta's Formulas and Quadratic Modeling

Question 794Question →

At an intellectual property law firm, a senior partner reviewed $180$ patent applications across three technical domains: Artificial Intelligence, Biotechnology, and Clean Energy. The audit revealed that $25$ applications belonged to none of these three domains. Furthermore, $90$ applications involved Artificial Intelligence, $85$ involved Biotechnology, and $80$ involved Clean Energy. If the number of applications that belonged to exactly two of these domains was three times the number of applications that belonged to all three domains, how many patent applications belonged to exactly one domain?

Show answer & explanation

Answer: 75

Answer

The total number of patent applications that belonged to exactly one domain is

75

.

The total number of applications in at least one domain is

180 - 25 = 155

. Let

E_1

be the number of applications in exactly one domain,

E_2

in exactly two domains, and

x

in all three domains. The total union is

E_1 + E_2 + x = 155

, while the sum of individual set sizes is

|A| + |B| + |C| = E_1 + 2E_2 + 3x = 90 + 85 + 80 = 255

. Subtracting the union equation from the sum equation gives

E_2 + 2x = 100

. Given

E_2 = 3x

, substituting gives

5x = 100 \implies x = 20

, which implies

E_2 = 60

. Finally, subtracting

E_2

and

x

from the total union yields

E_1 = 155 - 60 - 20 = 75

.

Step-by-Step Solution

1

Calculate the total number of applications in at least one domain

|A \cup B \cup C| = 180 - 25 = 155

Subtracting the applications belonging to none of the domains from the overall total yields the total number of unique applications covered by the three domains combined.

2

Formulate regional Venn diagram equations

E_1 + E_2 + x = 155

and

E_1 + 2E_2 + 3x = 255

Summing individual set counts counts elements in exactly two domains twice and elements in all three domains three times.

3

Deduce the relationship between

E_2

and

x

E_2 + 2x = 100

Subtracting the equation for total union from the sum of individual sets isolates the overcounted regions.

4

Use the given proportion

E_2 = 3x

to solve for

x

and

E_2

x = 20

and

E_2 = 60

Substituting

E_2 = 3x

into

E_2 + 2x = 100

gives

5x = 100

, so

x = 20

and

E_2 = 60

.

5

Find the number of applications belonging to exactly one domain (

E_1

)

E_1 = 75

Subtracting the count of applications in exactly two domains (

60

) and all three domains (

20

) from the total union (

155

) gives

155 - 60 - 20 = 75

.

Key Concept

Three-Set Venn Diagram Region Partitioning

Question 795Question →

A delivery truck completes an outbound trip along Route X of length $d$ miles at a constant average speed of $v$ miles per hour. On the return trip along the exact same route, heavy traffic reduces the truck's average speed by $20$ miles per hour, which increases the travel time for the return trip by $2$ hours compared to the outbound trip. Later, the truck completes a trip along Route Y, which has the same distance $d$ as Route X, at an average speed of $(v + 60)$ miles per hour in exactly $2$ hours. What was the truck's average speed $v$ , in miles per hour, on the outbound trip along Route X?

40

50

60

80

150

Show answer & explanation

Answer: $60$

Answer

The truck's average outbound speed

v

on Route X was

60

miles per hour.

By setting up time equations for the two legs of Route X, the distance

d

is expressed as

d = \frac{v(v-20)}{10}

. Combining this with the relation for Route Y (

d = 2(v+60)

) yields the quadratic equation

v^2 - 40v - 1200 = 0

. Factoring gives

(v - 60)(v + 20) = 0

, yielding the valid positive speed

v = 60

miles per hour.

Step-by-Step Solution

1

Express the distance

d

in terms of

v

using the time difference on Route X.

d = \frac{v(v - 20)}{10}

The outbound time is

t_1 = \frac{d}{v}

and the return time is

t_2 = \frac{d}{v - 20}

. Given

t_2 - t_1 = 2

, we have

\frac{d}{v - 20} - \frac{d}{v} = 2 \implies \frac{20d}{v(v - 20)} = 2 \implies d = \frac{v(v - 20)}{10}

.

2

Express the distance

d

in terms of

v

using the conditions for Route Y.

d = 2(v + 60)

Route Y has distance

d

, speed

(v + 60)

mph, and takes

2

hours, so

d = \text{rate} \times \text{time} = 2(v + 60)

.

3

Equate the two expressions for

d

to form a quadratic equation in terms of

v

.

v^2 - 40v - 1200 = 0

Setting

\frac{v(v - 20)}{10} = 2(v + 60)

yields

v^2 - 20v = 20v + 1200

, which simplifies to

v^2 - 40v - 1200 = 0

.

4

Solve the quadratic equation for

v

and verify physical feasibility.

v = 60

Factoring

(v - 60)(v + 20) = 0

gives solutions

v = 60

or

v = -20

. Since speed must be positive and greater than

20

mph (to avoid a zero or negative return speed),

v = 60

mph.

Key Concept

Algebraic Modeling of Multi-Leg Motion using Systems of Rational and Quadratic Equations

Question 796Question →

A box contains 12 cards numbered consecutively from 1 through 12. Two cards are selected at random from the box without replacement. Given that the sum of the numbers on the two selected cards is even, what is the probability that at least one of the selected cards has a prime number on it?

\frac{19}{30}

\frac{19}{66}

\frac{2}{3}

\frac{7}{15}

\frac{15}{22}

Show answer & explanation

Answer: $\frac{19}{30}$

Answer

The conditional probability that at least one selected card has a prime number on it, given that their sum is even, is

\frac{19}{30}

.

The option offering 19/30 is correct. Given that the sum of the two cards is even, both cards must be even or both cards must be odd. Choosing 2 even cards from 6 available gives 15 pairs, and choosing 2 odd cards from 6 available gives 15 pairs, making 30 possible pairs in total for the restricted sample space. Among the even cards, 2 is prime while 4, 6, 8, 10, and 12 are non-prime (5 numbers). Among the odd cards, 3, 5, 7, and 11 are prime (4 numbers) while 1 and 9 are non-prime (2 numbers). The pairs containing no primes consist of 2 non-prime evens (10 pairs) and 2 non-prime odds (1 pair), giving 11 non-prime pairs. Subtracting from 30 yields 19 pairs with at least one prime. Thus, the conditional probability is 19/30.

Step-by-Step Solution

1

Determine the restricted sample space (Condition B: Sum of two cards is even)

Total outcomes in Condition B = 30

The sum of two integers is even if both are even or both are odd. Among numbers 1 to 12, there are 6 even numbers ({2, 4, 6, 8, 10, 12}) and 6 odd numbers ({1, 3, 5, 7, 9, 11}). The number of ways to pick 2 even cards is

\binom{6}{2} = 15

, and 2 odd cards is

\binom{6}{2} = 15

. Total pairs with an even sum =

15 + 15 = 30

.

2

Categorize the numbers 1 through 12 by parity and primality

Prime evens = {2} (1 number); Non-prime evens = {4, 6, 8, 10, 12} (5 numbers); Prime odds = {3, 5, 7, 11} (4 numbers); Non-prime odds = {1, 9} (2 numbers)

Note that 1 is not a prime number, and 2 is the only even prime number.

3

Count the number of pairs in the restricted sample space with NO prime numbers

11 non-prime pairs

Pairs of two evens with no primes come from non-prime evens:

\binom{5}{2} = 10

pairs. Pairs of two odds with no primes come from non-prime odds:

\binom{2}{2} = 1

pair. Total non-prime pairs =

10 + 1 = 11

.

4

Calculate favorable outcomes (Event A ∩ B) and the conditional probability

P(A|B) = 19/30

Favorable pairs with at least one prime =

30 - 11 = 19

. Therefore,

P(\text{At least one prime} \mid \text{Even sum}) = \frac{19}{30}

.

Key Concept

Conditional Probability with Restricted Sample Space

Estimated Time:2m 0s

Question 797Question →

In a software development department, the numbers of Frontend developers, Backend developers, and DevOps engineers were initially in the ratio of $4 : 5 : 3$ , respectively. After the department hired $6$ additional DevOps engineers, with no changes to the number of Frontend or Backend developers, the ratio of Backend developers to DevOps engineers became $1 : 1$ . What is the total number of developers and engineers in the department after these new hires?

30

36

42

48

78

Show answer & explanation

Answer: 42

Answer

The total number of developers and engineers in the department after the new hires is 42.

By setting the initial counts of Frontend developers, Backend developers, and DevOps engineers as

4x

,

5x

, and

3x

, adding 6 to DevOps yields

3x + 6

. Equating Backend and DevOps counts gives

5x = 3x + 6

, so

x = 3

. Substituting

x = 3

into the post-hire counts gives 12 Frontend, 15 Backend, and 15 DevOps, summing to a total of 42 employees.

Step-by-Step Solution

1

Define initial quantities using a common ratio multiplier.

Let the initial numbers of Frontend developers, Backend developers, and DevOps engineers be

4x

,

5x

, and

3x

, respectively.

Ratios allow representing unknown totals in terms of a single multiplier

x

.

2

Express the updated counts after hiring new DevOps engineers.

Frontend =

4x

, Backend =

5x

, and DevOps =

3x + 6

.

Only the DevOps group increases by 6 while other group sizes remain unchanged.

3

Set up an equation based on the new Backend-to-DevOps ratio of

1 : 1

.

5x = 3x + 6 \implies 2x = 6 \implies x = 3

.

A

1 : 1

ratio means the number of Backend developers equals the new number of DevOps engineers.

4

Calculate the final total number of employees.

Frontend =

4(3) = 12

, Backend =

5(3) = 15

, DevOps =

3(3) + 6 = 15

. Total =

12 + 15 + 15 = 42

.

Summing all three roles after adding the 6 new hires gives the final count.

Key Concept

Ratio scaling and setting up algebraic equations from modified multi-part ratios.

Estimated Time:1m 30s

Question 798Question →

A wholesale distributor purchased a shipment of 50 units of Model A laptops and 30 units of Model B laptops for a total cost of $4,000. To set the list price for each model, the distributor marked up the unit cost of Model A by 40% and the unit cost of Model B by 50%. During a clearance promotion, Model A was sold at a 10% discount off its list price, and Model B was sold at a 20% discount off its list price. If the distributor earned a total profit of$ 995 on the entire shipment of 80 laptops, what was the original purchase cost of one Model A laptop?

$25

$50

$60.50

$65

$75

Show answer & explanation

Answer: $65

Answer

The original purchase cost of one Model A laptop was $65.

The original purchase cost of one Model A laptop is found by setting up a system of two linear equations: one for total cost (

50x + 30y = 4,000

, or

5x + 3y = 400

) and one for total profit (

50(0.26x) + 30(0.20y) = 995

, or

13x + 6y = 995

). Solving this system yields

x = 65

, which accurately reflects the unit cost of Model A.

Step-by-Step Solution

1

Define variables and set up the total cost equation.

Let

x

be the unit cost of Model A and

y

be the unit cost of Model B. The equation is

50x + 30y = 4,000

, which simplifies to

5x + 3y = 400

.

The distributor purchased 50 units of Model A and 30 units of Model B for a total cost of $4,000.

2

Calculate the selling price and profit per unit for each model.

For Model A:

\text{List Price} = 1.40x

,

\text{Selling Price} = 1.40x \times 0.90 = 1.26x

, so

\text{Profit per unit} = 1.26x - x = 0.26x

.
For Model B:

\text{List Price} = 1.50y

,

\text{Selling Price} = 1.50y \times 0.80 = 1.20y

, so

\text{Profit per unit} = 1.20y - y = 0.20y

.

Selling prices are derived by applying the discount to the list price, and unit profit is selling price minus cost price.

3

Set up the total profit equation.

Total profit is

50(0.26x) + 30(0.20y) = 995

, which simplifies to

13x + 6y = 995

.

Summing the total profit from all 50 units of Model A and 30 units of Model B equals the given total profit of $995.

4

Solve the system of linear equations for

x

.

Multiply

5x + 3y = 400

by 2 to get

10x + 6y = 800

. Subtract this from

13x + 6y = 995

to obtain

(13x - 10x) + (6y - 6y) = 995 - 800 \implies 3x = 195 \implies x = 65

.

Eliminating

y

isolates the variable

x

, representing the unit purchase cost of Model A.

Key Concept

Profit, Loss, and Markup with Successive Percentages

Question 799Question →

A diner offers a three-course meal deal where a customer selects $1$ appetizer from a choice of $4$ , $1$ main dish from a choice of $5$ , and $1$ dessert from a choice of $3$ . How many different three-course meal combinations can a customer order?

12

20

35

60

120

Show answer & explanation

Answer: $60$

Answer

The total number of unique three-course meal combinations is

60

.

According to the Fundamental Counting Principle, when a process involves a series of independent decision stages, the total number of possible outcomes is the product of the number of choices available at each stage. Multiplying

4

appetizers by

5

main dishes and

3

desserts yields

4 \times 5 \times 3 = 60

distinct meal combinations.

Step-by-Step Solution

1

Identify the number of independent choices available at each decision stage

Appetizer choices =

4

, Main dish choices =

5

, Dessert choices =

3

Each course selection represents an independent decision stage.

2

Apply the Fundamental Counting Principle by multiplying the number of options for all stages

4 \times 5 \times 3 = 60

The total number of multi-stage outcomes is the product of the number of options at each individual stage.

Key Concept

Fundamental Counting Principle

Question 800Question →

A wholesaler purchases a desk for $\$250$ and marks up the cost price by $20\%$ to establish the retail price. If the desk is sold at the retail price, what is the profit, in dollars, earned on the sale?

Show answer & explanation

Answer: 50

Answer

The profit earned on the sale of the desk is

\$50

.

The profit is calculated by finding

20\%

of the cost price of

\$250

, which is

0.20 \times 250 = 50

.

Step-by-Step Solution

1

Identify the cost price and markup rate from the problem statement.

Cost price

= \$250

; Markup rate

= 20\%

.

Markup is calculated directly as a percentage of the original cost price.

2

Multiply the cost price by the markup percentage to find the profit.

\text{Profit} = 0.20 \times 250 = 50

.

Taking

20\%

of the

\$250

purchase price gives the exact dollar markup/profit.

Key Concept

Dollar profit is calculated as the markup percentage multiplied by the cost price.

Answer

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