Tüm alıştırma soruları

Soru 521Soru →

Let $N = 2^4 \times 3^3 \times 5^2 \times 7$ . How many positive integer divisors of $N$ are even, divisible by 15, but not divisible by 9?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

Any divisor of

N = 2^4 \times 3^3 \times 5^2 \times 7^1

takes the form

2^a \times 3^b \times 5^c \times 7^d

. The condition that the divisor is even requires

a \ge 1

, which gives 4 choices (

a \in \{1, 2, 3, 4\}

). The condition that the divisor is divisible by 15 requires

b \ge 1

and

c \ge 1

. The condition that it is not divisible by 9 requires

b < 2

. Together,

b \ge 1

and

b < 2

mean

b = 1

(1 choice). The condition

c \ge 1

allows

c \in \{1, 2\}

(2 choices). The exponent

d

can be 0 or 1 (2 choices). Multiplying these choices gives

4 \times 1 \times 2 \times 2 = 16

.

Adım Adım Çözüm

1

Express the general prime factorization of a divisor

d = 2^a \times 3^b \times 5^c \times 7^d

with

0 \le a \le 4

,

0 \le b \le 3

,

0 \le c \le 2

,

0 \le d \le 1

Any divisor of

N

must consist only of the prime factors of

N

up to their respective maximum powers.

2

Apply the condition for even numbers

a \in \{1, 2, 3, 4\}

(4 choices)

An even integer must contain at least one factor of 2.

3

Apply the condition for divisibility by 15 and non-divisibility by 9

b = 1

(1 choice) and

c \in \{1, 2\}

(2 choices)

Divisibility by

15 = 3 \times 5

requires

b \ge 1

and

c \ge 1

. Non-divisibility by

9 = 3^2

requires

b < 2

. Thus

b

must be exactly 1.

4

Apply the condition for the exponent of 7

d \in \{0, 1\}

(2 choices)

There are no restrictions given for prime factor 7.

5

Calculate the product of choices

4 \times 1 \times 2 \times 2 = 16

By the fundamental counting principle, the total number of valid combinations of prime exponents is the product of the number of choices for each exponent.

Anahtar Kavram

Counting Divisors with Prime Factorization Restrictions

Soru 522Soru →

Set $S$ consists of a sequence of consecutive integers, ordered from least to greatest. If Set $S$ contains exactly $45$ integers and the sum of all the integers in Set $S$ is $405$ , what is the value of the smallest integer in Set $S$ ?

Cevabı ve açıklamayı göster

Cevap: -13

Cevap

The smallest integer in Set

S

is

-13

.

The average (arithmetic mean) of the set is calculated by dividing the total sum (

405

) by the number of terms (

45

), which equals

9

. In an evenly spaced set with an odd number of elements, the mean is equal to the median (middle term). Because there are

45

terms, exactly

22

terms lie below the median. Subtracting

22

from

9

yields

-13

as the smallest integer in the set.

Adım Adım Çözüm

1

Calculate the arithmetic mean and median of the set

Arithmetic mean = Median = 9

For an evenly spaced set, the arithmetic mean is equal to the median. Dividing the sum (

405

) by the number of terms (

45

) gives

9

.

2

Determine the number of terms preceding the median

22 terms precede the median

With

45

terms in total, the median is the

23\text{rd}

term, which leaves

\frac{45 - 1}{2} = 22

terms smaller than the median.

3

Calculate the smallest integer

Smallest integer =

-13

Subtracting

22

from the median gives

9 - 22 = -13

.

Anahtar Kavram

In any set of consecutive integers with

n

terms, the arithmetic mean equals the median. If

n

is odd, the smallest integer is given by

\text{Median} - \frac{n-1}{2}

.

Soru 523Soru →

If $x = \sqrt{3} + \sqrt{2}$ , what is the value of $x^4 + x^{-4}$ ?

98

100

102

144

14

Cevabı ve açıklamayı göster

Cevap: 98

Cevap

98

Rationalizing the denominator gives

x^{-1} = \sqrt{3} - \sqrt{2}

. Summing

x + x^{-1}

yields

2\sqrt{3}

. Squaring this sum gives

x^2 + 2 + x^{-2} = 12

, so

x^2 + x^{-2} = 10

. Squaring once more gives

x^4 + 2 + x^{-4} = 100

, which simplifies to

x^4 + x^{-4} = 98

.

Adım Adım Çözüm

1

Find the reciprocal expression

x^{-1}

x^{-1} = \frac{1}{\sqrt{3} + \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} = \sqrt{3} - \sqrt{2}

Rationalizing the denominator simplifies the expression for

x^{-1}

.

2

Calculate the sum

x + x^{-1}

x + x^{-1} = (\sqrt{3} + \sqrt{2}) + (\sqrt{3} - \sqrt{2}) = 2\sqrt{3}

Adding the original expression and its reciprocal eliminates the

\sqrt{2}

term.

3

Square

x + x^{-1}

to find

x^2 + x^{-2}

(x + x^{-1})^2 = x^2 + 2(x)(x^{-1}) + x^{-2} = x^2 + 2 + x^{-2} = (2\sqrt{3})^2 = 12 \implies x^2 + x^{-2} = 10

Using the algebraic identity

(a+b)^2 = a^2 + 2ab + b^2

allows finding

x^2 + x^{-2}

.

4

Square

x^2 + x^{-2}

to find

x^4 + x^{-4}

(x^2 + x^{-2})^2 = x^4 + 2(x^2)(x^{-2}) + x^{-4} = x^4 + 2 + x^{-4} = 10^2 = 100 \implies x^4 + x^{-4} = 98

Squaring the second-power sum and subtracting the cross-term 2 yields the required value.

Anahtar Kavram

Rationalizing radical denominators and binomial squaring of reciprocals

Tahmini Süre:1m 30s

Soru 524Soru →

Set $A$ consists of 5 consecutive even integers, and Set $B$ consists of 5 consecutive even integers. The smallest element of Set $B$ is 6 greater than the largest element of Set $A$ . If $S$ represents the standard deviation of the combined set of 10 integers formed by merging Set $A$ and Set $B$ , what is the value of $S^2$ ?

Cevabı ve açıklamayı göster

Cevap: 57

Cevap

The value of

S^2

is 57.

Each subset of 5 consecutive even integers has an internal variance of 8. Because the mean of Set A is

x

and the mean of Set B is

x+14

, the combined mean is

x+7

. Each group mean is 7 units away from the overall mean. By the parallel axis theorem for variance, the combined variance is the within-group variance plus the squared distance from the combined mean:

8 + 7^2 = 57

.

Adım Adım Çözüm

1

Define Set A algebraically and calculate its mean and variance.

Set

A = \{x-4, x-2, x, x+2, x+4\}

. Mean

\mu_A = x

, and variance

\sigma_A^2 = \frac{(-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2}{5} = \frac{40}{5} = 8

.

Symmetrically defining 5 consecutive even integers simplifies variance calculation.

2

Define Set B relative to Set A.

The largest element of Set

A

is

x+4

. The smallest element of Set

B

is

(x+4) + 6 = x+10

. Thus, Set

B = \{x+10, x+12, x+14, x+16, x+18\}

with mean

\mu_B = x+14

and variance

\sigma_B^2 = 8

.

Translating the word problem into exact algebraic representations for Set B.

3

Calculate the mean of the combined 10-element set.

The combined mean

\mu = \frac{5(x) + 5(x+14)}{10} = x+7

.

The combined mean is essential to evaluate overall variation.

4

Calculate the variance

S^2

of the combined set.

Using the combined variance formula

S^2 = \frac{1}{2}[\sigma_A^2 + (\mu_A - \mu)^2] + \frac{1}{2}[\sigma_B^2 + (\mu_B - \mu)^2] = \frac{1}{2}[8 + (-7)^2] + \frac{1}{2}[8 + 7^2] = 8 + 49 = 57

.

Combines internal set variances with the variance between group means.

Anahtar Kavram

Combined Variance and Standard Deviation of Sets

Soru 525Soru →

An automated security gate opens successfully on any given approach with a probability of $0.8$ , independently of other attempts. If a driver approaches the gate $3$ times, the probability that the gate opens successfully at least once is $0.992$ .

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is true because the probability of failing all

3

independent attempts is

(1 - 0.8)^3 = 0.008

, making the probability of at least one successful opening

1 - 0.008 = 0.992

.

The statement accurately reflects the complementary probability principle

P(\text{at least one}) = 1 - P(\text{none})

. For

3

independent trials with single-trial failure probability

0.2

, the probability of all trials failing is

0.2^3 = 0.008

. Subtracting this from

1

yields

0.992

.

Adım Adım Çözüm

1

Find the probability of failure on a single trial

P(\text{failure}) = 1 - 0.8 = 0.2

The failure of a trial is the complement of its success.

2

Calculate the probability of zero successes across 3 independent trials

P(\text{0 successes}) = (0.2)^3 = 0.008

For independent events, the joint probability is the product of individual probabilities.

3

Apply the complementary probability rule to find the probability of at least one success

P(\text{at least 1 success}) = 1 - 0.008 = 0.992

The event 'at least one success' is the exact complement of 'zero successes'.

Anahtar Kavram

Complementary Probability and At-Least-One Scenarios

Soru 526Soru →

A data set consists of $8$ positive integers with an arithmetic mean of $15$ and a unique mode of $12$ . What is the maximum possible value of an integer in this data set?

Cevabı ve açıklamayı göster

Cevap: 81

Cevap

81

The total sum of the

8

integers is

8 \times 15 = 120

. To maximize the single largest integer, the sum of the remaining

7

integers must be minimized. Since

12

is the unique mode,

12

must appear at least twice, and no other value can appear more than once. The smallest positive integers that can occupy the first

5

positions without creating another mode are

1, 2, 3, 4,

and

5

. Placing

12

in the

6\text{th}

and

7\text{th}

positions minimizes the sum of the first

7

elements to

1 + 2 + 3 + 4 + 5 + 12 + 12 = 39

. Thus, the maximum possible value for the largest integer is

120 - 39 = 81

.

Adım Adım Çözüm

1

Calculate the total sum of the 8 positive integers.

Total Sum =

8 \times 15 = 120

.

The sum of a set of numbers equals the arithmetic mean multiplied by the number of elements.

2

Minimize the sum of the first 7 integers to maximize the 8th integer.

Minimum sum of the first 7 integers =

1 + 2 + 3 + 4 + 5 + 12 + 12 = 39

.

To maximize the largest integer, the remaining 7 integers must be as small as possible. Since 12 is the unique mode, 12 must appear at least twice, and no other integer can appear more than once. Placing two 12s at the highest available positions among the 7 terms (

a_6 = 12

and

a_7 = 12

) and selecting the smallest distinct positive integers (

1, 2, 3, 4, 5

) for the first 5 terms minimizes their total sum.

3

Subtract the minimum sum of the 7 smallest terms from the total sum.

Maximum integer =

120 - 39 = 81

.

Subtracting the smallest possible sum of 7 terms from the fixed sum of 120 yields the maximum possible value for the 8th term.

Anahtar Kavram

Maximizing an element in a data set under mean and mode constraints

Soru 527Soru →

If real numbers $x$ and $y$ satisfy the system of linear equations:

7x + 4y = 113

3x + 6y = 87

what is the value of $x + y$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The value of

x + y

is 20.

Adding the two equations yields

10x + 10y = 200

. Dividing the entire equation by 10 directly gives

x + y = 20

. Alternatively, solving the system yields

x = 11

and

y = 9

, whose sum is

11 + 9 = 20

.

Adım Adım Çözüm

1

Add the two equations together to combine like terms.

(7x + 4y) + (3x + 6y) = 113 + 87 \implies 10x + 10y = 200

Recognizing that adding the equations creates symmetric coefficients of 10 for both

x

and

y

allows a direct solution for the sum

(x + y)

.

2

Divide the combined equation by 10.

\frac{10(x + y)}{10} = \frac{200}{10} \implies x + y = 20

Isolating

(x + y)

directly avoids the extra computation of solving for

x

and

y

individually.

Anahtar Kavram

Solving Systems of Linear Equations by Combination

Tahmini Süre:1m 30s

Soru 528Soru →

Let $x$ and $y$ be non-zero integers such that $-6 \le x \le 6$ and $-6 \le y \le 6$ . If $x^3 y < 0$ and $\frac{x}{y^2} < -\frac{1}{4}$ , what is the minimum possible value of $x - y$ ?

Cevabı ve açıklamayı göster

Cevap: -10

Cevap

The minimum possible value of

x - y

is

-10

.

Analyzing the signs shows x must be negative and y must be positive. Multiplying x/y² < -1/4 by 4y² gives y² < -4x. Testing x = -6 gives y² < 24, so the largest positive integer y is 4. The expression x - y reaches its minimum value of -6 - 4 = -10.

Adım Adım Çözüm

1

Determine the signs of variables x and y

x < 0 and y > 0

Since y² > 0 for any non-zero integer y, x/y² < -1/4 requires x < 0. Furthermore, x³y < 0 requires x³ and y to have opposite signs; since x < 0 implies x³ < 0, y must be positive.

2

Transform the inequality without altering its sign direction

y² < -4x

Multiplying x/y² < -1/4 by 4y² > 0 yields 4x < -y², which rearranges to y² < -4x.

3

Evaluate candidate integer pairs to minimize x - y

The minimum value of x - y is -10 when x = -6 and y = 4

To minimize x - y, select the most negative integer x and the largest allowable positive integer y. Setting x = -6 gives y² < 24, making max integer y = 4. Therefore, x - y = -6 - 4 = -10.

Anahtar Kavram

Deducing signs from products and quotients, and manipulating inequalities involving non-zero variables.

Soru 529Soru →

A medical distributor purchased a lot of $60$ identical diagnostic kits for a total cost of $\$4,800$ . The distributor marked up the cost price of each kit by $50\%$ to set its regular retail price. After selling $40$ kits at the regular retail price, the distributor sold the remaining $20$ kits at a clearance discount of $30\%$ off the regular retail price. What was the distributor's total net profit, in dollars, from the sale of all $60$ diagnostic kits?

Cevabı ve açıklamayı göster

Cevap: 1680

Cevap

The distributor's total net profit from the sale of all 60 diagnostic kits is $1,680.

To calculate net profit, first determine the unit cost:

\frac{\$4,800}{60} = \$80

. The marked-up regular price is

\$80 \times 1.50 = \$120

. Selling

40

kits at

\$120

yields

\$4,800

. The remaining

20

kits are sold at a

30\%

discount off

\$120

, giving a clearance price of

\$120 \times 0.70 = \$84

per kit and revenue of

20 \times \$84 = \$1,680

. Total revenue is

\$4,800 + \$1,680 = \$6,480

. Subtracting the total cost of

\$4,800

yields a net profit of

\$1,680

.

Adım Adım Çözüm

1

Determine the cost price per unit.

Cost per kit is

\frac{\$4,800}{60} = \$80

.

Dividing total initial cost by total units gives the unit cost price.

2

Calculate the regular retail price per unit.

Regular retail price is

\$80 \times (1 + 0.50) = \$120

.

The

50\%

markup is applied directly to the cost price of

\$80

.

3

Calculate revenue from the full-price sales segment.

Revenue from

40

kits is

40 \times \$120 = \$4,800

.

Multiply quantity sold at full price by regular retail price.

4

Determine clearance price and revenue for remaining units.

Clearance price is

\$120 \times (1 - 0.30) = \$84

. Revenue from

20

kits is

20 \times \$84 = \$1,680

.

The clearance discount of

30\%

is taken off the regular retail price, not cost.

5

Calculate total revenue and net profit.

Total revenue is

\$4,800 + \$1,680 = \$6,480

. Net profit is

\$6,480 - \$4,800 = \$1,680

.

Net profit equals total revenue minus total cost.

Anahtar Kavram

Profit, Loss, and Markup

Soru 530Soru →

A freight boat travels downstream along a river from Dock A to Dock B in $4$ hours. On the return trip upstream from Dock B to Dock A, engine trouble causes the boat's speed in still water to decrease by $25\%$ for the entire return trip. If the speed of the river current is a constant $3$ miles per hour and the upstream return trip takes $8$ hours, what is the distance, in miles, between Dock A and Dock B?

Cevabı ve açıklamayı göster

Cevap: 84

Cevap

The distance between Dock A and Dock B is 84 miles.

The correct distance of 84 miles is determined by accounting for both the constant river current of 3 mph and the 25% reduction in still-water boat speed on the return trip. Setting the downstream distance formula equal to the upstream distance formula yields a still-water initial speed of 18 mph, giving a one-way distance of 84 miles.

Adım Adım Çözüm

1

Define variables for the boat's speed in still water and write expressions for effective speeds.

Let

v

be the initial speed of the boat in still water (in mph). The downstream speed is

(v + 3)

mph. Due to a

25\%

reduction on the return leg, the boat's upstream speed in still water is

0.75v

mph, making the effective upstream speed

(0.75v - 3)

mph.

Effective speed downstream equals still-water speed plus current speed, while effective speed upstream equals still-water speed minus current speed.

2

Set up distance equations for both directions using

\text{Distance} = \text{Rate} \times \text{Time}

.

Downstream distance:

D = 4(v + 3)

. Upstream distance:

D = 8(0.75v - 3)

.

The distance between Dock A and Dock B is the same in both directions.

3

Equate the two distance expressions and solve for

v

.

4(v + 3) = 8(0.75v - 3) \implies v + 3 = 1.5v - 6 \implies 0.5v = 9 \implies v = 18

mph.

Equating the two representations of distance allows for solving the single unknown variable

v

.

4

Calculate the total distance

D

.

D = 4(18 + 3) = 4(21) = 84

miles.

Substituting

v = 18

back into the downstream distance formula yields the exact distance.

Anahtar Kavram

Rate, Time, and Distance with Currents and Variable Speeds

Tahmini Süre:2m 30s

Soru 531Soru →

A commercial real estate company manages three office buildings: Building A, Building B, and Building C. Building A contains $20,000$ square feet of rentable space leased at an average rate of $\$45$ per square foot per year. Building B contains $30,000$ square feet of rentable space leased at an average rate of $\$50$ per square foot per year. Building C contains $50,000$ square feet of rentable space. If the overall weighted average rental rate for all three buildings combined is $\$52$ per square foot per year, what is the average annual rental rate per square foot, in dollars, for Building C?

Cevabı ve açıklamayı göster

Cevap: 56

Cevap

The average annual rental rate for Building C is

56

dollars per square foot.

To find the average annual rental rate for Building C, determine the total revenue generated by all three buildings combined. With a total square footage of

20,000 + 30,000 + 50,000 = 100,000

sq ft and an overall weighted average rate of

\$52

/sq ft, the total revenue is

100,000 \times 52 = 5,200,000

dollars. Subtracting the revenue generated by Building A (

20,000 \times 45 = 900,000

dollars) and Building B (

30,000 \times 50 = 1,500,000

dollars) leaves

5,200,000 - 2,400,000 = 2,800,000

dollars that must be contributed by Building C. Dividing this by Building C's

50,000

square feet yields an average rate of

56

dollars per square foot.

Adım Adım Çözüm

1

Determine total combined rentable area and total combined revenue

Total area

= 20,000 + 30,000 + 50,000 = 100,000

sq ft. Total revenue

= 100,000 \times 52 = 5,200,000

dollars.

Weighted average rate multiplied by total square footage gives total annual rental revenue across all properties.

2

Calculate known annual revenues from Building A and Building B

Building A revenue

= 20,000 \times 45 = 900,000

dollars. Building B revenue

= 30,000 \times 50 = 1,500,000

dollars. Combined A and B revenue

= 2,400,000

dollars.

Multiplying individual component weights by their respective rates yields individual group totals.

3

Subtract known revenues from total revenue and divide by Building C's square footage

Building C revenue

= 5,200,000 - 2,400,000 = 2,800,000

dollars. Building C rate

= \frac{2,800,000}{50,000} = 56

dollars.

Isolating the remaining revenue requirement and dividing by Building C's area yields its average rental rate per square foot.

Anahtar Kavram

Weighted Average = Total Weighted Sum / Total Weight Sum

Soru 532Soru →

A testing kit contains $8$ electronic sensors, of which exactly $3$ are uncalibrated and $5$ are fully calibrated. If $3$ sensors are chosen at random without replacement, what is the probability that at least $1$ of the chosen sensors is uncalibrated?

\frac{5}{28}

\frac{15}{28}

\frac{23}{28}

\frac{55}{56}

\frac{387}{512}

Cevabı ve açıklamayı göster

Cevap: $\frac{23}{28}$

Cevap

The correct answer is

\frac{23}{28}

.

To find the probability that at least one sensor is uncalibrated, calculate 1 minus the probability that no selected sensors are uncalibrated. Out of 8 sensors, there are

\binom{8}{3} = 56

ways to pick 3. The number of ways to pick 3 calibrated sensors from the 5 available is

\binom{5}{3} = 10

. Thus,

P(\text{no uncalibrated}) = \frac{10}{56} = \frac{5}{28}

. Subtracting this from 1 gives

1 - \frac{5}{28} = \frac{23}{28}

.

Adım Adım Çözüm

1

Calculate the total number of ways to choose 3 sensors out of 8.

\binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56

total outcomes.

Selection is made without replacement and order does not matter.

2

Calculate the complementary event: selecting 0 uncalibrated sensors (all 3 selected sensors are calibrated).

\binom{5}{3} = \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10

outcomes with all calibrated sensors.

There are 5 calibrated sensors available, and 3 must be selected from them.

3

Find the probability of the complementary event (no uncalibrated sensors selected).

P(\text{no uncalibrated}) = \frac{10}{56} = \frac{5}{28}

.

Divide the favorable outcomes for selecting all calibrated sensors by total possible outcomes.

4

Apply the complementary probability principle to find

P(\text{at least one uncalibrated})

.

P(\text{at least one uncalibrated}) = 1 - P(\text{no uncalibrated}) = 1 - \frac{5}{28} = \frac{23}{28}

.

The sum of probabilities of complementary events is 1.

Anahtar Kavram

Complementary Probability: P(at least one X) = 1 - P(no X)

Soru 533Soru →

For decades, urban planners assumed that constructing additional highways was the most effective method to alleviate traffic congestion in growing metropolitan areas. However, recent urban transportation studies demonstrate that expanding road capacity often triggers a phenomenon known as "induced demand," wherein newly generated road space rapidly attracts additional drivers who previously utilized public transit or traveled during off-peak hours. Consequently, traffic congestion typically returns to pre-expansion levels within a few years. To address urban mobility more sustainably, modern transit analysts advocate for congestion pricing models and expanded high-capacity public transport networks rather than continued roadway expansion.

Which of the following best states the primary purpose of the passage?

To explain why expanding road capacity fails to resolve traffic congestion long-term and present alternative approaches.

To define the concept of induced demand and detail how driver travel times fluctuate during peak hours.

To denounce traditional urban planners for intentionally creating traffic congestion in growing cities.

To argue that traditional highway expansion remains superior to public transit networks despite recent studies.

To prove that implementing congestion pricing completely eliminates all traffic delays across metropolitan areas.

Cevabı ve açıklamayı göster

Cevap: To explain why expanding road capacity fails to resolve traffic congestion long-term and present alternative approaches.

Cevap

The primary purpose of the passage is to explain why expanding road capacity fails to resolve traffic congestion long-term and present alternative approaches.

The author introduces a traditional practice, shows why it fails using the concept of induced demand, and concludes by presenting modern alternatives such as congestion pricing and expanded transit networks. The option stating that the purpose is to explain why expanding road capacity fails and present alternative approaches encompasses this full scope accurately.

Adım Adım Çözüm

1

Analyze passage structure and locate main claim

The passage opens with a traditional view (highway expansion alleviates congestion), introduces a structural pivot ('However'), presents evidence of failure ('induced demand'), and concludes with recommended modern alternatives.

Tracking structural transitions reveals the author's primary concern.

2

Synthesize total scope

The passage evaluates traditional road expansion, explains why it fails, and proposes alternative solutions.

The correct main idea statement must cover both the limitation of the old method and the proposal of new methods.

3

Evaluate option scope and eliminate traps

The statement emphasizing the failure of expanding road capacity and the presentation of alternative approaches correctly captures the main intent.

Other options focus on narrow supporting details, exaggerate tone, or misstate the author's position.

Anahtar Kavram

Identifying Primary Purpose in Reading Comprehension

Soru 534Soru →

A car rental agency charges a fixed daily rate plus a constant fee per mile driven. A customer who rented a car for 3 days and drove 150 miles paid a total of $180. Another customer who rented the same model of car for 5 days and drove 200 miles paid a total of$ 270. What is the fixed daily rate, in dollars, charged by the rental agency?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The fixed daily rate charged by the rental agency is 30 dollars.

Formulating the equations

3d + 150m = 180

and

5d + 200m = 270

and solving for the daily rate

d

yields

d = 30

.

Adım Adım Çözüm

1

Define variables and formulate the system of linear equations.

Let

d

be the fixed daily rate (in dollars) and

m

be the cost per mile driven (in dollars). The scenario translates to:
Equation 1:

3d + 150m = 180

Equation 2:

5d + 200m = 270

Total charge is the linear combination of daily fixed costs and per-mile variable costs.

2

Simplify Equation 1 to express

d

in terms of

m

.

Dividing Equation 1 by 3 yields

d + 50m = 60

, so

d = 60 - 50m

.

Simplifying equations reduces computation error when using substitution.

3

Substitute the expression for

d

into Equation 2 to solve for

m

.

5(60 - 50m) + 200m = 270 \Rightarrow 300 - 250m + 200m = 270 \Rightarrow -50m = -30 \Rightarrow m = 0.60

Substitution eliminates variable

d

, leaving a linear equation in one variable.

4

Calculate the fixed daily rate

d

.

d = 60 - 50(0.60) = 60 - 30 = 30

Substituting

m = 0.60

gives the exact daily fixed rate.

Anahtar Kavram

Setting up and solving a system of two linear equations in two variables using elimination or substitution.

Soru 535Soru →

What is the smallest positive integer $n$ such that $180 \times n$ is a perfect square and $450 \times n$ is a perfect cube?

Cevabı ve açıklamayı göster

Cevap: 1620

Cevap

The smallest positive integer

n

is 1620.

By writing 180 as

2^2 \times 3^2 \times 5^1

and 450 as

2^1 \times 3^2 \times 5^2

, we can analyze the exponents needed for

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

. To make

180n

a perfect square,

a

and

b

must be even and

c

must be odd. To make

450n

a perfect cube,

1+a

,

2+b

, and

2+c

must be multiples of 3. Finding the minimal non-negative integers that satisfy both conditions yields

a = 2

,

b = 4

, and

c = 1

. Thus,

n = 2^2 \times 3^4 \times 5^1 = 1620

.

Adım Adım Çözüm

1

Express 180 and 450 in terms of their prime factorizations.

180 = 2^2 \times 3^2 \times 5^1

and

450 = 2^1 \times 3^2 \times 5^2

Prime factorization exposes the exponent requirements for perfect powers.

2

Determine the constraints on exponents of

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

.

For

180n

to be a square,

a

must be even,

b

must be even, and

c

must be odd. For

450n

to be a cube,

1+a

,

2+b

, and

2+c

must be multiples of 3.

A number is a perfect square if all prime exponents are even, and a perfect cube if all prime exponents are multiples of 3.

3

Find the minimal values for

a, b, c

.

a = 2, b = 4, c = 1

a=2

is even and makes

1+2=3

;

b=4

is even and makes

2+4=6

;

c=1

is odd and makes

2+1=3

.

4

Compute

n

.

n = 2^2 \times 3^4 \times 5^1 = 1620

Multiplying the prime powers together yields the smallest integer

n

.

Anahtar Kavram

Prime Factorization and Exponent Rules for Perfect Powers

Soru 536Soru →

A commercial machinery supplier purchased a batch of industrial generators. The supplier marked up the wholesale cost of each generator by $40\%$ to set the original retail price. During an end-of-quarter promotion, the supplier offered a $15\%$ discount off the original retail price. If a customer purchased a generator at the promotional price for $\$4,760$ , what was the supplier's profit, in dollars, on that sale?

Cevabı ve açıklamayı göster

Cevap: 760

Cevap

The supplier's profit on the sale was $760.

Let

C

represent the wholesale cost of a generator. A

40\%

markup yields an original retail price of

1.40C

. Applying a

15\%

discount to this retail price produces a promotional selling price of

0.85 \times 1.40C = 1.19C

. Setting

1.19C = 4,760

gives

C = 4,000

. The profit is the promotional selling price minus wholesale cost:

\$4,760 - \$4,000 = \$760

.

Adım Adım Çözüm

1

Define the relationship between wholesale cost and original retail price.

Original Retail Price =

1.40 \times C

A

40\%

markup on wholesale cost

C

increases the price to

140\%

of

C

.

2

Apply the discount to calculate the promotional selling price.

Selling Price =

0.85 \times (1.40 \times C) = 1.19 \times C

A

15\%

discount off the retail price leaves

85\%

of the retail price.

3

Solve for the wholesale cost

C

.

C = \frac{4,760}{1.19} = 4,000

Given that the final selling price is

\$4,760

, set

1.19C = 4,760

and divide.

4

Calculate the profit made on the sale.

Profit =

\$4,760 - \$4,000 = \$760

Profit equals total revenue (selling price) minus total cost.

Anahtar Kavram

Successive markup and discount calculations with distinct base values.

Soru 537Soru →

An integer $N$ has exactly three distinct prime factors $p$ , $q$ , and $r$ , such that $N = p^a \cdot q^b \cdot r^c$ , where $a$ , $b$ , and $c$ are positive integers with $a < b < c$ . If $N$ has $24$ positive factors and $N^2$ has $105$ positive factors, what is the value of $a + b + c$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

a + b + c

is

6

.

For an integer with prime factorization

p^a q^b r^c

, the number of positive divisors is

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

. For

N^2 = p^{2a} q^{2b} r^{2c}

, the number of positive divisors is

(2a+1)(2b+1)(2c+1)

. Factoring

105

into three odd terms greater than

1

yields

3 \times 5 \times 7

. Matching these terms in increasing order gives

2a+1=3 \implies a=1

,

2b+1=5 \implies b=2

, and

2c+1=7 \implies c=3

. Checking

(1+1)(2+1)(3+1) = 24

confirms these values. Summing

a+b+c

yields

1+2+3=6

.

Adım Adım Çözüm

1

Write the formula for the number of positive factors of

N

and

N^2

The number of positive factors of

N = p^a q^b r^c

is

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

. Since

N^2 = p^{2a} q^{2b} r^{2c}

, the number of positive factors of

N^2

is

(2a+1)(2b+1)(2c+1) = 105

.

The number of divisors of a prime-factored integer is found by adding 1 to each exponent in its prime factorization and multiplying the results.

2

Factor

105

into three odd factors greater than

1

The prime factorization of

105

is

3 \times 5 \times 7

. The only way to express

105

as a product of three integers greater than

1

is

3 \times 5 \times 7

.

Since

a, b, c

are positive integers, each term

2a+1, 2b+1, 2c+1

must be an odd integer greater than

1

.

3

Assign the factors using the inequality condition

a < b < c

Since

a < b < c

, it follows that

2a+1 < 2b+1 < 2c+1

. Therefore:

2a+1 = 3 \implies a = 1

;

2b+1 = 5 \implies b = 2

;

2c+1 = 7 \implies c = 3

.

Matching the ordered values of

2a+1, 2b+1, 2c+1

with the sorted factors

3, 5, 7

uniquely determines

a, b,

and

c

.

4

Verify with the factor count for

N

and calculate the sum

(1+1)(2+1)(3+1) = 2 \times 3 \times 4 = 24

, which matches the given condition. The sum

a + b + c = 1 + 2 + 3 = 6

.

Verification confirms the solution satisfies all constraints.

Anahtar Kavram

Prime Factorization and Divisor Counting Formula

Tahmini Süre:1m 30s

Soru 538Soru →

At a semiconductor manufacturing facility, silicon wafers are produced across three consecutive shifts: Shift X, Shift Y, and Shift Z. The ratio of the number of wafers produced in Shift X to Shift Y is $3 : 2$ , and the ratio of the number of wafers produced in Shift Y to Shift Z is $4 : 5$ . The defect rate of wafers produced in Shift X is $2.0\%$ , and the defect rate of wafers produced in Shift Y is $3.5\%$ . If the combined defect rate for all wafers produced across the three shifts is $3.0\%$ , what is the defect rate, expressed as a percentage, of the wafers produced in Shift Z?

Cevabı ve açıklamayı göster

Cevap: 3.8

Cevap

The defect rate of the wafers produced in Shift Z is 3.8%.

To find the defect rate of Shift Z, first determine the combined ratio of wafer production for Shift X, Shift Y, and Shift Z. Given Shift X : Shift Y = 3 : 2 and Shift Y : Shift Z = 4 : 5, scale Shift X : Shift Y to 6 : 4 so that Shift Y has the same ratio value in both expressions. The unified ratio is Shift X : Shift Y : Shift Z = 6 : 4 : 5, representing 15 total parts. Applying the weighted average formula gives (6 * 2.0% + 4 * 3.5% + 5 * r_Z) / 15 = 3.0%. Multiplying both sides by 15 yields 12.0 + 14.0 + 5 * r_Z = 45.0, which simplifies to 5 * r_Z = 19.0. Dividing by 5 gives r_Z = 3.8%.

Adım Adım Çözüm

1

Find the combined ratio of production volumes across Shift X, Shift Y, and Shift Z.

Shift X : Shift Y = 3 : 2 = 6 : 4, and Shift Y : Shift Z = 4 : 5. Therefore, Shift X : Shift Y : Shift Z = 6 : 4 : 5.

Combining the separate ratios into a single compound ratio establishes the weighting factor for each shift.

2

Calculate the total parts and weight of each shift.

Total parts = 6 + 4 + 5 = 15 parts.

The weight of each shift in the weighted average corresponds to its share of the total 15 parts.

3

Formulate the weighted average equation for the overall defect rate.

(6 * 2.0 + 4 * 3.5 + 5 * r_Z) / 15 = 3.0

The overall combined defect rate is the weighted sum of individual defect rates divided by the total number of parts.

4

Solve for the unknown defect rate r_Z of Shift Z.

12.0 + 14.0 + 5 * r_Z = 45.0 => 26.0 + 5 * r_Z = 45.0 => 5 * r_Z = 19.0 => r_Z = 3.8

Algebraic simplification yields the exact defect rate for Shift Z.

Anahtar Kavram

Weighted Average of Combined Sets

Tahmini Süre:2m 0s

Soru 539Soru →

Consider the following passage:

For decades, agroecologists attributed the soil fertility degradation observed under continuous monoculture systems primarily to the mechanical depletion of macro-nutrients such as nitrogen and phosphorus. Consequently, soil management protocols prioritized synthetic fertilizer supplementation to offset these elemental deficits. However, recent longitudinal studies on bio-rhizosphere dynamics have challenged this paradigm. Researchers observed that long-term synthetic fertilization often accelerates, rather than mitigates, systemic productivity declines by disrupting indigenous mycorrhizal fungi communities essential for micronutrient transport and soil aggregate stability.

While proponents of high-input chemical agriculture contend that synthetic inputs remain indispensable for maintaining short-term yields under global food security pressures, emerging ecological models suggest a different mechanism. These models indicate that chemical saturation induces a microbial starvation state: plants provided with readily accessible inorganic nitrogen reduce their exudation of carbon compounds, effectively starving the beneficial fungal networks that protect root systems from pathogenic invasion. Thus, the fundamental driver of long-term soil exhaustion is not merely chemical depletion, but a biologically mediated breakdown of symbiotic rhizosphere relations. Therefore, sustainable yield stabilization requires shifting management strategies from exclusive nutrient replacement toward rhizosphere microbiome preservation.

True or False: The primary purpose of the passage is to contend that long-term soil productivity losses stem fundamentally from microbial disruption rather than simple elemental exhaustion, and to advocate for a shift toward preserving rhizosphere ecosystems.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True. The primary purpose of the passage is to argue that soil degradation under continuous monoculture is primarily driven by microbial disruption rather than chemical depletion, and to advocate for rhizosphere-focused soil management.

The statement correctly encapsulates the author's central argument and rhetorical goal. The passage refutes the conventional belief that soil degradation is purely chemical and argues that synthetic inputs cause biological starvation of fungal networks, concluding that soil management must prioritize microbiome preservation.

Adım Adım Çözüm

1

Analyze the structural pivots and overall progression of the passage.

The passage moves from a traditional paradigm (chemical depletion) to new longitudinal evidence (microbial breakdown caused by synthetic saturation) and ends with a normative recommendation (rhizosphere preservation).

Determining primary purpose requires identifying where the passage places its ultimate rhetorical focus rather than focusing exclusively on background details or opposing views.

2

Evaluate whether the statement captures the complete passage scope or merely an isolated detail.

The statement captures both the main thesis (microbial disruption over chemical depletion) and the author's ultimate conclusion (shifting strategies to rhizosphere microbiome preservation).

A correct primary purpose statement must synthesize the central claim and main conclusion of the text.

Anahtar Kavram

Identifying Primary Purpose and Main Idea

Soru 540Soru →

Passage:
For decades, plant ecologists assumed that carbon allocation within ectomycorrhizal fungal networks operated strictly under a mutualistic trade-off model, wherein host trees exchanged photosynthetically derived sugars for mineral nutrients at fixed stoichiometric ratios. However, recent isotopic tracing studies have revealed a more dynamic system of reciprocal reward mechanisms, where trees adjust carbon fluxes based on real-time nutrient delivery rates from specific fungal partners. While some researchers interpret these findings as evidence of market-like decision-making in plant-microbe symbioses, this framework oversimplifies the complex physiological constraints involved. Fungal hyphae are not autonomous agents conducting transactions; rather, their nutrient release is often driven by localized soil moisture gradients and passive enzymatic diffusion. Thus, framing carbon transfer as strategic economic exchange obscures the passive physical processes that fundamentally govern nutrient transport across fungal membranes. A comprehensive model must integrate physiological bioenergetics with network-level nutrient dynamics, recognizing that observed allocation patterns emerge from biochemical feedbacks rather than active resource trading.

Statement: The primary purpose of the passage is to challenge a market-based economic framework applied to plant-fungal nutrient exchange and advocate for a model integrating physiological constraints.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True. The statement correctly describes the primary purpose of the passage.

The statement accurately identifies the primary purpose because the author explicitly argues against using market concepts to describe nutrient exchanges in mycorrhizal networks and advocates instead for a model grounded in physiological dynamics.

Adım Adım Çözüm

1

Analyze the structural organization and main focus of the passage.

The passage transitions from introducing empirical findings on plant-fungal networks to criticizing the 'market-like decision-making' framework used by some researchers.

Tracking structural transitions reveals the author's core thesis and communicative goal.

2

Identify the author's critique of the prevailing interpretation.

The author argues that treating fungal exchange as economic trading 'oversimplifies' physiological limits and 'obscures' passive physical mechanisms.

Understanding the author's stance distinguishes their argument from secondary viewpoints mentioned in the text.

3

Examine the author's ultimate conclusion and recommendation.

The final sentence asserts that a valid model must integrate physiological bioenergetics and biochemical feedbacks rather than active resource trading.

The concluding recommendation confirms the primary purpose of the passage.

4

Compare the passage's primary purpose to the provided evaluation statement.

The statement asserts that the primary purpose is to challenge the market-based framework and advocate for a physiological approach, which perfectly mirrors the synthesized main idea.

Direct alignment between the statement and the passage analysis confirms a True evaluation.

Anahtar Kavram

Identifying Primary Purpose in Reading Comprehension

Cevap

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Sık Yapılan Hatalar

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Anahtar Kavram