Tüm alıştırma soruları

Soru 121Soru →

For all positive real numbers $x$ and $y$ , if $x^2 y^3 = 108$ and $x^3 y^2 = 72$ , then $x + y = 5$ .

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

The statement is True.

Multiplying the two equations yields

(xy)^5 = 7776 = 6^5

, so

xy = 6

. Dividing the second equation by the first yields

\frac{x}{y} = \frac{2}{3}

, meaning

x = \frac{2}{3}y

. Substituting this into

xy = 6

gives

y = 3

and

x = 2

, making

x + y = 5

.

Adım Adım Çözüm

1

Multiply the two system equations to find the product

xy

.

(x^2 y^3)(x^3 y^2) = x^5 y^5 = (xy)^5 = 108 \times 72 = 7776 = 6^5

, so

xy = 6

.

Multiplying exponential expressions with common bases allows adding their exponents to form a unified power

(xy)^5

.

2

Divide the second equation by the first equation to find the relationship between

x

and

y

.

\frac{x^3 y^2}{x^2 y^3} = \frac{x}{y} = \frac{72}{108} = \frac{2}{3}, which simplifies to

x = \frac{2}{3}y

.

Dividing exponential expressions subtracts their exponents, yielding the simple ratio of the variables.

3

Solve for individual values of

x

and

y

and evaluate

x + y

.

Substituting

x = \frac{2}{3}y

into

xy = 6

gives

\frac{2}{3}y^2 = 6 \implies y = 3

and

x = 2

. Therefore,

x + y = 5

.

Solving the system confirms that

x = 2

and

y = 3

are the unique positive real solutions.

Anahtar Kavram

System of exponential equations solved by combining power products and ratios.

Tahmini Süre:1m 30s

Soru 122Soru →

A home decor retailer imports hand-woven area rugs at a cost price of $500$ dollars each. The retailer marks up the cost price by $60\%$ to establish the regular list price. During a seasonal clearance sale, each rug is sold at a discount of $25\%$ off its regular list price. If the retailer incurs an additional non-refundable freight charge of $40$ dollars per rug, which is included in the total cost of each rug, what is the net profit percentage earned by the retailer relative to the total cost of each rug?

11\frac{1}{9}\%

10\%

20\%

25\%

7.5\%

Cevabı ve açıklamayı göster

Cevap: $11\frac{1}{9}\%$

Cevap

The net profit percentage earned on the total cost of each rug is 11 1/9%.

The total cost of importing each rug includes both the initial cost price (

500) and the freight fee (

40), yielding a total cost base of

540. A 60% markup on the

500 cost price gives a list price of

500 * 1.60 =

800. Applying a 25% discount to the

800 list price yields a final selling price of

800 * 0.75 =

600. The net profit is

600 -

540 =

60. Expressed as a percentage of the total cost base of

540, the net profit percentage is (

60 / $540) * 100% = 1/9 * 100% = 11 1/9%.

Adım Adım Çözüm

1

Calculate the total cost per rug

Total Cost =

500 +

40 =

540

Total cost must include both the initial purchase cost price and the additional freight charge.

2

Determine the regular list price after markup

List Price =

500 \times (1 + 0.60) =

500 \times 1.60 =

800

The markup of 60% is applied to the initial cost price of $500.

3

Calculate the discounted selling price

Selling Price =

800 \times (1 - 0.25) =

800 \times 0.75 =

600

The 25% discount is applied to the regular list price of $800.

4

Determine net profit per rug

Net Profit = Selling Price - Total Cost =

600 -

540 =

60

Net profit is the difference between final revenue (selling price) and total expenses (total cost).

5

Calculate the net profit percentage based on total cost

\text{Net Profit \%} = \left(\frac{60}{540}\right) \times 100\% = \frac{1}{9} \times 100\% = 11\frac{1}{9}\%

Percentage profit must be evaluated against the total cost base of $540.

Anahtar Kavram

Profit, Loss, and Markup Base Calculations

Tahmini Süre:1m 30s

Soru 123Soru →

In a survey of $200$ software developers regarding their proficiency in three programming languages—Python, Java, and C++:

- $110$ developers are proficient in Python.
- $95$ developers are proficient in Java.
- $85$ developers are proficient in C++.
- $20$ developers are proficient in all three languages.
- $30$ developers are proficient in none of the three languages.

If the total number of developers proficient in exactly two of these languages is twice the number of developers proficient in Python and Java only, how many developers are proficient in C++ only?

15

25

35

40

45

Cevabı ve açıklamayı göster

Cevap: $25$

Cevap

The number of developers proficient in C++ only is 25.

The correct answer is 25. Subtracting the 30 developers proficient in none of the languages leaves 170 developers who are proficient in at least one language. Using the 3-set overlap equations, the number of developers proficient in exactly two languages is calculated to be 80. Since the number proficient in Python and Java only is half of this (40), the sum of the remaining two double-overlap regions (Java & C++ only, and Python & C++ only) must also equal 40. Subtracting this combined double-overlap count of 40 and the triple-overlap count of 20 from the total C++ count of 85 leaves 25 developers proficient in C++ only.

Adım Adım Çözüm

1

Calculate the total number of developers proficient in at least one language.

\text{Total proficient} = 200 - 30 = 170

Developers proficient in none of the three languages must be excluded from the universe of set elements.

2

Apply the 3-set Venn Diagram formula relating total unique elements, individual set sums, exactly-two regions, and all-three regions.

\text{Total} = E_1 + E_2 + E_3

, where

E_1

is exactly one,

E_2

is exactly two, and

E_3

is exactly three. Also,

|P| + |J| + |C| = E_1 + 2E_2 + 3E_3

.

Sum of individual sets:

110 + 95 + 85 = 290

.
So,

290 = E_1 + 2E_2 + 3(20) \implies E_1 + 2E_2 = 230

.

From total elements:

170 = E_1 + E_2 + 20 \implies E_1 + E_2 = 150

.

Subtracting the two equations gives

E_2 = 230 - 150 = 80

.

Each element in

E_2

is counted twice in the sum of individual sets, and each element in

E_3

is counted three times.

3

Determine the breakdown of the double-overlap regions.

Given that

E_2 = 2 \times (\text{Python and Java only})

, we have

\text{Python and Java only} = \frac{80}{2} = 40

.

The remaining double-overlap regions involving C++ are

(\text{Java and C++ only}) + (\text{Python and C++ only}) = E_2 - 40 = 80 - 40 = 40

.

The total number of developers proficient in exactly two languages consists of three disjoint groups: Python/Java only, Java/C++ only, and Python/C++ only.

4

Calculate the number of developers proficient in C++ only.

|C| = (\text{C++ only}) + (\text{Java and C++ only}) + (\text{Python and C++ only}) + (\text{All 3}) 85 = (\text{C++ only}) + 40 + 20 85 = (\text{C++ only}) + 60 \text{C++ only} = 25

Subtract all overlapping subsets containing C++ from the total proficiency count of set C.

Anahtar Kavram

Three-Set Overlapping Venn Diagrams and Subset Decompositions

Tahmini Süre:2m 30s

Soru 124Soru →

At an investment firm with $200$ portfolio managers, $110$ managers invest in Equities, $95$ invest in Fixed Income, and $80$ invest in Commodities. Furthermore, $45$ managers invest in both Equities and Fixed Income, $35$ invest in both Equities and Commodities, and $30$ invest in both Fixed Income and Commodities. If $15$ managers do not invest in any of these three asset classes, how many portfolio managers invest in all three asset classes?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The correct answer is 10 portfolio managers.

Subtracting the 15 managers who do not invest in any asset class from the total 200 yields 185 managers in at least one asset class. Applying the 3-set inclusion-exclusion formula

185 = (110 + 95 + 80) - (45 + 35 + 30) + x

simplifies to

185 = 175 + x

, which gives

x = 10

for the number of managers investing in all three asset classes.

Adım Adım Çözüm

1

Determine the total number of portfolio managers investing in at least one asset class.

185 managers

Subtracting the 15 managers who invest in none of the three asset classes from the total of 200 gives the size of the union of the three sets:

200 - 15 = 185

.

2

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

|E \cup F \cup C| = |E| + |F| + |C| - (|E \cap F| + |E \cap C| + |F \cap C|) + |E \cap F \cap C|

To avoid overcounting elements present in multiple set intersections, pairwise intersections are subtracted and the triple intersection is added back.

3

Substitute known values into the equation and solve for the unknown triple intersection

x

.

x = 10

185 = 110 + 95 + 80 - (45 + 35 + 30) + x \implies 185 = 285 - 110 + x \implies 185 = 175 + x \implies x = 10

.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle

Soru 125Soru →

If $4^x + 4^x + 4^x + 4^x = 2^{10}$ , what is the value of $x$ ?

\frac{5}{4}

2

\frac{5}{2}

4

5

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The value of

x

is

4

.

Combining the four identical terms on the left side gives

4 \cdot 4^x = 4^{1+x}

. Converting

4

to base

2

yields

(2^2)^{x+1} = 2^{2x+2}

. Setting this equal to the right side

2^{10}

gives the linear equation

2x + 2 = 10

, which solves to

x = 4

.

Adım Adım Çözüm

1

Combine the repeated addition on the left side of the equation.

4^x + 4^x + 4^x + 4^x = 4 \cdot 4^x = 4^{x+1}

Adding four identical terms is equivalent to multiplying the term by 4.

2

Convert all bases to 2 so that both sides can be compared.

4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x+2}

Since

4 = 2^2

, applying the power of a power rule

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

expresses the left side with base 2.

3

Equate the exponents and solve for

x

.

2x + 2 = 10 \implies 2x = 8 \implies x = 4

If

b^u = b^v

for a positive base

b \neq 1

, then

u = v

.

Anahtar Kavram

Combining like exponential terms by factoring and applying base conversion rules

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

and

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

.

Soru 126Soru →

Passage:
While early marine biologists posited that hydrothermal vent ecosystems relied exclusively on chemical compounds emitted directly from Earth's crust to sustain life, recent deep-sea explorations have revealed a more complex ecological dynamic. Researchers discovered that photosynthetic energy from surface waters occasionally reaches these abyssal zones via falling organic debris, known as marine snow. This external organic input significantly supplements the primary chemosynthetic production of vent bacteria, indicating that hydrothermal ecosystems are not entirely isolated from surface-driven biological processes.

Statement: The primary purpose of the passage is to argue that hydrothermal vent ecosystems depend entirely on chemical emissions from Earth's crust without any influence from surface waters.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

False

The statement accurately describes the outdated hypothesis mentioned at the start of the passage, but fails to capture the passage's primary purpose, which is to demonstrate that vent ecosystems receive supplementary surface inputs.

Adım Adım Çözüm

1

Identify the author's thesis and structural pivot in the passage.

The passage begins with an old belief ('relied exclusively') and pivots ('recent deep-sea explorations have revealed') to show that surface organic matter ('marine snow') also feeds these ecosystems.

Structural transition words indicate that the author is updating, rather than affirming, the traditional exclusive dependence view.

2

Compare the statement to the primary purpose of the passage.

The statement asserts that the passage defends exclusive crust dependency, which directly contradicts the author's main point that surface inputs supplement vent life.

The statement misrepresents the discredited initial premise as the main objective of the text.

Anahtar Kavram

Identifying Primary Purpose and Main Idea

Soru 127Soru →

An antique dealer purchased a vintage writing desk for $\$600$ . To prepare the desk for sale, the dealer spent $\$40$ on restoration. The dealer then marked up the initial purchase price of $\$600$ by $50\%$ to establish the list price. During a seasonal promotion, the desk was sold at a $20\%$ discount off the list price. What was the dealer's profit as a percentage of the total expenditure (purchase price plus restoration cost)?

Cevabı ve açıklamayı göster

Cevap: 12.5

Cevap

The dealer's profit as a percentage of the total expenditure is

12.5\%

.

To find the net profit percentage on total expenditure, first compute total cost (

\$600 + \$40 = \$640

). Next, compute list price by applying a

50\%

markup to the initial purchase price (

\$600 \times 1.50 = \$900

). Then compute the selling price after a

20\%

discount on list price (

\$900 \times 0.80 = \$720

). Net profit is selling price minus total cost (

\$720 - \$640 = \$80

). Finally, express net profit as a percentage of total cost:

(\$80 / \$640) \times 100\% = 12.5\%

.

Adım Adım Çözüm

1

Calculate the total cost (expenditure)

Total cost =

\$600 + \$40 = \$640

The total investment includes both the purchase price of the desk and the mandatory restoration fee incurred before sale.

2

Determine the list price

List price =

\$600 \times 1.50 = \$900

The markup of

50\%

is applied specifically to the initial purchase price of

\$600

.

3

Determine the selling price

Selling price =

\$900 \times 0.80 = \$720

The

20\%

discount reduces the list price of

\$900

.

4

Calculate the net dollar profit

Net profit =

\$720 - \$640 = \$80

Profit is calculated as total revenue (selling price) minus total cost.

5

Calculate the net profit percentage

Net profit percentage =

(\$80 / \$640) \times 100\% = 12.5\%

The percentage return must be evaluated using total expenditure (

\$640

) as the base.

Anahtar Kavram

Calculating successive percentage markups and discounts with secondary cost components

Tahmini Süre:1m 45s

Soru 128Soru →

A logistics company maintains a fleet consisting of three types of delivery vehicles: Vans, Medium Trucks, and Heavy Trucks. The ratio of the number of Vans to Medium Trucks is $3 : 2$ , and the ratio of the number of Medium Trucks to Heavy Trucks is $4 : 1$ . If the average fuel efficiency is $24$ miles per gallon for the Vans and $15$ miles per gallon for the Medium Trucks, and the average fuel efficiency for the entire combined fleet is $19$ miles per gallon, what is the average fuel efficiency, in miles per gallon, of the Heavy Trucks?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The average fuel efficiency of the Heavy Trucks is 5 miles per gallon.

Combining the given vehicle ratios gives a unified proportion of 6 Vans to 4 Medium Trucks to 1 Heavy Truck (total 11 units). The total weighted efficiency sum is

6(24) + 4(15) + 1(x) = 144 + 60 + x = 204 + x

. Setting this equal to the combined fleet total of

11 \times 19 = 209

yields

x = 5

miles per gallon.

Adım Adım Çözüm

1

Unify the two relative ratios into a single compound ratio for all three vehicle groups.

Vans : Medium Trucks : Heavy Trucks = 6 : 4 : 1, giving a total of 11 ratio units.

Both individual ratios share Medium Trucks as a common element (

2

units in the first ratio and

4

units in the second). Multiplying the first ratio by

2

aligns the Medium Truck count.

2

Set up the weighted mean equation using group weights and individual group averages.

Weighted Sum = 6(24) + 4(15) + 1(x) = 204 + x, where total elements = 11.

The sum of (group quantity × group mean) divided by total quantity equals the combined average.

3

Equate to total fleet mean and solve for the unknown average fuel efficiency x.

204 + x = 11 × 19 = 209, so x = 5 miles per gallon.

Multiplying total items (11) by total mean (19) gives total combined miles per gallon units (209).

Anahtar Kavram

Weighted Average of Multiple Combined Subgroups

Soru 129Soru →

Following a major revision to its regulatory standards, an international financial oversight agency began conducting unannounced cybersecurity audits of commercial banks. To avoid severe regulatory fines for non-compliance, virtually all audited banks substantially increased their annual expenditure on advanced threat-detection software and external consulting over the subsequent two years. Surprisingly, data collected at the end of this two-year period revealed that the average financial loss per successful cyberattack at these audited banks increased by over 40 percent, whereas the average loss per successful cyberattack at unaudited financial institutions remained completely unchanged. Which of the following, if true, most helps to account for the apparent discrepancy described above?

The financial penalties imposed by the oversight agency for failing an audit are significantly higher than the average cost of remediating a standard cyberattack.

Unaudited financial institutions generally process a smaller total volume of high-value electronic fund transfers than audited commercial banks do.

The advanced threat-detection software installed by audited banks successfully intercepts virtually all low-level, low-impact intrusions, leaving only highly sophisticated, catastrophic breaches among the attacks that manage to succeed.

Audited commercial banks that increased spending on threat-detection software simultaneously reduced their budget allocations for physical security at retail branch locations.

Cybercriminals targeted audited banks far more frequently than unaudited institutions because audited banks publicly disclosed their compliance upgrades.

Cevabı ve açıklamayı göster

Cevap: The advanced threat-detection software installed by audited banks successfully intercepts virtually all low-level, low-impact intrusions, leaving only highly sophisticated, catastrophic breaches among the attacks that manage to succeed.

Cevap

The apparent discrepancy is best explained by the fact that advanced threat-detection software intercepts routine, low-impact intrusions, ensuring that only highly sophisticated breaches succeed, thereby raising the average financial loss per successful attack.

The correct answer provides a statistical selection mechanism that reconciles both premises. When audited banks install advanced threat-detection software, they successfully block routine, minor attacks before any damage occurs. Consequently, the only attacks that manage to penetrate the bank's defenses are exceptionally sophisticated, high-impact breaches. Removing minor losses from the pool of 'successful cyberattacks' shifts the composition of that pool upward, raising the average loss per successful attack even though total overall damage across all attempted attacks may have declined.

Adım Adım Çözüm

1

Identify the two seemingly contradictory facts presented in the stimulus.

Fact 1: Audited banks significantly increased spending on advanced threat-detection software. Fact 2: The average financial loss per successful cyberattack at audited banks increased by 40%, whereas it remained unchanged at unaudited banks.

Resolving a paradox requires finding a missing factor that allows both premise facts to be true simultaneously.

2

Analyze the mathematical composition of the metric 'average loss per successful attack'.

The metric is calculated as (Total Financial Losses from Successful Attacks) / (Total Number of Successful Attacks).

Understanding how the ratio is structured helps identify how filtering out minor attacks alters the average without indicating a failure of the security measures.

3

Evaluate the impact of filtering out low-level attacks.

If threat-detection software prevents low-severity attacks from succeeding, those low-dollar losses are removed from the numerator and denominator of the 'successful attacks' calculation. The remaining successful attacks consist exclusively of complex, high-dollar breaches, raising the calculated average.

This explains why security spending worked as intended while simultaneously causing the observed statistical increase in loss per breach.

Anahtar Kavram

Resolving Rate vs. Composition Paradoxes (Denominator and Selection Effects)

Soru 130Soru →

A boutique bookstore sells hardcover books for $\$25$ each and paperback books for $\$12$ each. On a certain day, the store sold a total of $85$ books and generated a total revenue of $\$1,579$ . How many hardcover books were sold on that day?

Cevabı ve açıklamayı göster

Cevap: 43

Cevap

The total number of hardcover books sold on that day is 43.

By setting up the two linear equations

h + p = 85

(quantity) and

25h + 12p = 1579

(revenue), eliminating

p

yields

13h = 559

, which gives

h = 43

.

Adım Adım Çözüm

1

Define variables and construct the system of linear equations

Let

h

be the number of hardcover books and

p

be the number of paperback books. The equations are

h + p = 85

and

25h + 12p = 1579

.

The total quantity of items sold gives a sum equation, and the individual prices multiplied by their quantities yield the total revenue equation.

2

Use substitution or elimination to solve for

h

Multiplying the total books equation by 12 yields

12h + 12p = 1020

. Subtracting this from

25h + 12p = 1579

gives

13h = 559

.

Eliminating

p

directly isolates the variable

h

representing hardcover books.

3

Calculate the value of

h

h = \frac{559}{13} = 43

.

Dividing the remaining total revenue by the coefficient of

h

yields the precise quantity.

Anahtar Kavram

Solving two-variable systems of linear equations using elimination or substitution

Soru 131Soru →

A commuter travels to work along a straight route at a constant speed of $40$ miles per hour. On the return trip along the exact same route, traffic delays reduce the average speed to $24$ miles per hour. If the return trip takes $40$ minutes longer than the trip to work, what is the total distance, in miles, traveled by the commuter for the round trip?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

The total distance traveled by the commuter for the round trip is 80 miles.

The total distance for the round trip is 80 miles. By expressing time as distance divided by speed, the outbound trip takes

\frac{D}{40}

hours and the return trip takes

\frac{D}{24}

hours. Since the return trip takes

40

minutes (

\frac{2}{3}

hours) longer, setting up the equation

\frac{D}{24} - \frac{D}{40} = \frac{2}{3}

yields

D = 40

miles for one way. The round-trip distance is

2 \times 40 = 80

miles.

Adım Adım Çözüm

1

Define variables for the one-way distance and expressions for the travel time of each leg.

Let

D

represent the one-way distance in miles. Outbound time

t_1 = \frac{D}{40}

hours; return time

t_2 = \frac{D}{24}

hours.

Using the relation

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

expresses time in terms of a single unknown variable.

2

Convert the given time difference from minutes to hours.

40 \text{ minutes} = \frac{40}{60} \text{ hours} = \frac{2}{3} \text{ hours}

.

Speeds are given in miles per hour, so time units must be in hours for dimensional consistency.

3

Formulate the equation representing the time difference and solve for

D

.

\frac{D}{24} - \frac{D}{40} = \frac{2}{3} \implies \frac{2D}{120} = \frac{2}{3} \implies \frac{D}{60} = \frac{2}{3} \implies D = 40 \text{ miles}

.

Subtracting the faster outbound time from the slower return time equals the time difference.

4

Compute the total distance for the entire round trip.

\text{Total distance} = 2 \times D = 2 \times 40 = 80 \text{ miles}

.

The question asks for the round-trip distance, which includes both the outbound and return journeys (

2D

).

Anahtar Kavram

Rate, Time, and Distance Problems

Soru 132Soru →

Let $d(k)$ denote the number of positive divisors of a positive integer $k$ . What is the smallest positive integer $n$ that is a multiple of $72$ and satisfies $d(n) = 35$ ?

Cevabı ve açıklamayı göster

Cevap: 5184

Cevap

The smallest positive integer

n

that satisfies all conditions is 5184.

The prime factorization of

72

is

2^3 \times 3^2

. Any multiple

n

of

72

must be written as

2^a \times 3^b \times dots

with

a \ge 3

and

b \ge 2

. The total number of positive divisors

d(n) = (a+1)(b+1) dots = 35

. The integer

35

can be factored into integers greater than

1

only as

7 \times 5

. Thus,

n

must have exactly two distinct prime factors, which must be

2

and

3

. The required exponents are

a, b \in \{6, 4\}

. To minimize

n

, we assign the larger exponent

6

to the smaller base

2

, yielding

n = 2^6 \times 3^4 = 64 \times 81 = 5184

.

Adım Adım Çözüm

1

Express 72 in prime factor form to determine minimum required prime exponents.

72 = 2^3 \times 3^2

, meaning

n

must have prime factorization

2^a \times 3^b \dots

where

a \ge 3

and

b \ge 2

.

Any multiple of 72 must contain at least three factors of 2 and two factors of 3.

2

Analyze the divisor count condition

d(n) = 35

.

Since

35 = 7 \times 5

, the number of prime factors of

n

can be at most 2, with exponent increments

(a+1)(b+1) = 7 \times 5

.

The number of positive divisors for

p_1^{e_1} p_2^{e_2} \dots

is

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

, and 35 factors into integers greater than 1 only as 35 or

7 \times 5

.

3

Determine the required exponents for the prime factors of

n

.

The exponents of the prime factors must be

7-1 = 6

and

5-1 = 4

.

Since

n

must contain both 2 and 3 as prime factors,

n

cannot have only 1 prime factor, so it has exactly two prime factors (2 and 3).

4

Optimize the exponent assignment to minimize

n

.

Assign exponent 6 to base 2 and exponent 4 to base 3, giving

n = 2^6 \times 3^4 = 64 \times 81 = 5184

.

Assigning the larger exponent to the smaller prime base minimizes the total product while meeting the conditions

a \ge 3

and

b \ge 2

.

Anahtar Kavram

Divisor counting formula and prime factor exponent allocation under divisibility constraints

Tahmini Süre:2m 0s

Soru 133Soru →

A research laboratory placed two orders for high-precision sensors. The first order, consisting of $5$ acoustic sensors and $3$ thermal sensors, cost a total of $\$1{}160$ . The second order, consisting of $3$ acoustic sensors and $5$ thermal sensors, cost a total of $\$1{}080$ . If each acoustic sensor costs the same amount and each thermal sensor costs the same amount, what is the combined cost of $1$ acoustic sensor and $1$ thermal sensor?

\$140

\$160

\$280

\$320

\$560

Cevabı ve açıklamayı göster

Cevap: $\$280$

Cevap

The combined cost of 1 acoustic sensor and 1 thermal sensor is

\$280

.

By representing the acoustic sensor price as

a

and thermal sensor price as

t

, we form two equations:

5a + 3t = 1160

and

3a + 5t = 1080

. Adding both equations gives

8a + 8t = 2240

. Dividing the entire equation by

8

isolates

a + t = 280

.

Adım Adım Çözüm

1

Define variables and write the system of equations based on the problem statement.

Let

a

be the price of one acoustic sensor and

t

be the price of one thermal sensor.
Equation 1:

5a + 3t = 1160

Equation 2:

3a + 5t = 1080

Translate word problem conditions into algebraic equations.

2

Add the two equations together to find a direct linear combination.

(5a + 3t) + (3a + 5t) = 1160 + 1080 \implies 8a + 8t = 2240

Since the target is

a + t

, adding symmetric coefficients yields a common factor of

8

, avoiding the need to solve for

a

and

t

individually.

3

Factor out

8

and solve for

(a + t)

.

8(a + t) = 2240 \implies a + t = \frac{2240}{8} = 280

Dividing the combined equation by

8

directly isolates the required quantity

(a + t)

.

Anahtar Kavram

Solving Linear Systems via Algebraic Combinations

Soru 134Soru →

Is the integer $n$ odd?

(1) $n$ is a multiple of 4.
(2) $n > 5$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Cevabı ve açıklamayı göster

Cevap: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (1) tells us that

n

is a multiple of 4, which guarantees that

n

is even. This allows us to answer the question 'Is

n

odd?' with a definitive and unambiguous 'No'. In Data Sufficiency Yes/No questions, a consistent 'No' answer means the statement is sufficient. Statement (2) allows

n

to be 6 (even) or 7 (odd), yielding both 'No' and 'Yes' responses, so it is insufficient. Therefore, the statement stating that Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient, is the correct choice.

Adım Adım Çözüm

1

Understand the decision rule for Yes/No Data Sufficiency questions.

A statement is sufficient if it yields a consistent 'Yes' OR a consistent 'No'. It is insufficient only if it yields 'Yes' for some values and 'No' for others.

Data Sufficiency requires determinacy, not necessarily an affirmative response.

2

Evaluate Statement (1):

n

is a multiple of 4.

Any multiple of 4 can be written as

n = 4k

for some integer

k

, which is divisible by 2 and therefore even. Since

n

is even, the answer to 'Is

n

odd?' is a definitive 'No'.

Since the answer is always 'No', Statement (1) alone is SUFFICIENT.

3

Evaluate Statement (2):

n > 5

.

If

n = 6

,

n

is even (Answer: No). If

n = 7

,

n

is odd (Answer: Yes). Since both 'Yes' and 'No' are possible, Statement (2) alone is INSUFFICIENT.

Statement (2) does not fix the parity of

n

.

Anahtar Kavram

In Yes/No Data Sufficiency questions, a statement that conclusively answers 'No' to the question stem is SUFFICIENT.

Soru 135Soru →

In 2024, a metropolitan public transit authority replaced its flat-fare structure on suburban commuter lines with a distance-based pricing model, resulting in a 25 percent increase in the average price of long-distance single-ride suburban tickets. Transit analysts predicted this fare hike would decrease total single-ride passenger volume on suburban lines. Surprisingly, over the subsequent twelve months, total single-ride passenger trips recorded across the suburban network increased by 15 percent, even though suburban population growth remained static and no new residential developments were constructed. Which of the following, if true, most helps to resolve the apparent discrepancy described above?

The distance-based fare structure significantly reduced ticket prices for short-distance trips within suburban districts, attracting a large volume of local commuters who previously walked or drove.

The transit authority decreased the frequency of suburban train departures during non-peak hours, increasing average rider wait times.

A major regional highway expansion completed two years prior to the fare restructuring permanently reduced vehicular commute times into the city center.

Commuters traveling long distances from suburban stations report higher average household incomes than commuters taking short local trips.

Many long-distance suburban riders reacted to the single-ride price hike by purchasing monthly unlimited passes, which are tracked separately from single-ride trips.

Cevabı ve açıklamayı göster

Cevap: The distance-based fare structure significantly reduced ticket prices for short-distance trips within suburban districts, attracting a large volume of local commuters who previously walked or drove.

Cevap

The apparent discrepancy is best resolved by the fact that the distance-based fare structure lowered prices for short-distance suburban trips, driving a substantial surge in local ridership that increased total single-ride trips overall.

The correct answer resolves the paradox by demonstrating that while long-distance single-ride ticket prices rose, the distance-based system concurrently lowered short-distance single-ride ticket prices. The influx of new short-distance riders created a net increase in total single-ride trips across the suburban network, reconciling the 25% price increase on long-distance tickets with the 15% overall increase in single-ride trip volume.

Adım Adım Çözüm

1

Identify the two conflicting premises in the stimulus

Premise 1: Long-distance single-ride suburban tickets increased in price by 25%, leading analysts to expect a drop in suburban transit ridership. Premise 2: Total single-ride trips on suburban lines actually increased by 15% without any population growth.

Resolving a paradox requires finding an underlying cause that permits both premises to be true simultaneously.

2

Evaluate the scope of the price increase versus the scope of the outcome metric

The price increase affected long-distance single-ride tickets specifically, but the outcome metric measures total single-ride trips across the entire suburban network (which includes short-distance trips).

A change in short-distance fare pricing under the new distance-based model could reconcile the overall net increase.

3

Select the option that supplies the missing reconciling variable

The option showing that short-distance ticket prices decreased significantly accounts for a surge in short-distance single-ride trips that offsets any reduction in long-distance trips.

This bridges both stated facts without disputing either premise.

Anahtar Kavram

Resolving Paradoxes and Apparent Discrepancies

Soru 136Soru →

Under the 2026 Municipal Clean Power Accord, any regional microgrid that sources more than 60 percent of its electricity from solar energy must maintain a battery storage reserve capable of supplying at least 4 hours of continuous power, unless it uses hydro-pumped storage as its primary backup reserve. Every microgrid operating in District 7 sources at least 75 percent of its electricity from solar energy. Microgrid Alpha is located in District 7 and does not maintain a battery storage reserve capable of supplying at least 4 hours of continuous power.

If the statements above are true, which of the following must also be true about Microgrid Alpha?

Microgrid Alpha sources a higher percentage of its electricity from solar energy than any other microgrid in District 7.

Microgrid Alpha fails to comply with the regulatory guidelines established under the 2026 Municipal Clean Power Accord.

Microgrid Alpha uses hydro-pumped storage as its primary backup reserve.

If a microgrid in District 7 uses hydro-pumped storage as its primary backup reserve, it sources less than 60 percent of its electricity from solar energy.

Microgrid Alpha maintains a battery storage reserve capable of supplying power for fewer than 3 hours.

Cevabı ve açıklamayı göster

Cevap: Microgrid Alpha uses hydro-pumped storage as its primary backup reserve.

Cevap

Microgrid Alpha uses hydro-pumped storage as its primary backup reserve.

The passage establishes that any microgrid sourcing over 60% solar power must maintain a battery reserve of at least 4 hours, UNLESS it uses hydro-pumped storage as its primary backup reserve. Microgrid Alpha is in District 7, meaning its solar sourcing is at least 75% (which is strictly greater than 60%). However, Microgrid Alpha does not maintain the 4-hour battery reserve. To remain consistent with the factual premises, Microgrid Alpha must fall under the exception clause. Therefore, it MUST use hydro-pumped storage as its primary backup reserve.

Adım Adım Çözüm

1

Identify the given conditional rules and factual premises.

Rule: If Solar > 60% AND NOT (Hydro-Pumped Storage) → Must have Battery Reserve ≥ 4 hours. Fact 1: District 7 microgrids have Solar ≥ 75%. Fact 2: Microgrid Alpha is in District 7, so its Solar ≥ 75% (> 60%). Fact 3: Microgrid Alpha does NOT have Battery Reserve ≥ 4 hours.

Establishing formal logical relationships allows for precise deduction.

2

Apply Modus Tollens (contrapositive reasoning) to the conditional rule.

Since Microgrid Alpha satisfies the condition (Solar > 60%) but fails to satisfy the requirement (Battery Reserve ≥ 4 hours), it must satisfy the exemption clause.

If a required consequence is absent for a subject that meets the trigger condition, the exemption condition must hold true.

3

Deduce the necessary truth regarding Microgrid Alpha.

Microgrid Alpha must use hydro-pumped storage as its primary backup reserve.

This is the only logically guaranteed outcome that prevents a contradiction with the provided premises.

Anahtar Kavram

Conditional Logic Contrapositive and Valid Deductions

Soru 137Soru →

An investor deposits $\$800$ into a savings account that pays a simple annual interest rate of $5\%$ . If no additional deposits or withdrawals are made, what is the total interest, in dollars, earned on the account after $3$ years?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

The total interest earned after 3 years is 120 dollars.

The total interest earned is computed using the simple interest formula

I = P \times r \times t

. Substituting

P = 800

,

r = 0.05

, and

t = 3

yields

I = 800 \times 0.05 \times 3 = 120

dollars.

Adım Adım Çözüm

1

Identify the principal, rate, and time from the problem stem.

Principal

P = \$800

, interest rate

r = 5\% = 0.05

, and time

t = 3

years.

These values are required to apply the simple interest formula.

2

Calculate simple interest using

I = P \times r \times t

.

I = 800 \times 0.05 \times 3 = 120

.

Simple interest accumulates linearly over time based on the initial principal.

Anahtar Kavram

Simple Interest Calculation

Tahmini Süre:45s

Soru 138Soru →

How many integer values of $n$ satisfy the inequality $|n^2 - 9| \leq 2n + 6$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

There are 6 integer values of

n

that satisfy the given inequality.

Because

|n^2 - 9| \geq 0

for all real numbers,

2n + 6

must be non-negative, requiring

n \geq -3

. At

n = -3

, the inequality reduces to

0 \leq 0

, which is valid. For

n > -3

, the term

n + 3

is positive, so factoring

|n^2 - 9|

into

|n - 3|(n + 3)

and dividing by

n + 3

simplifies the inequality to

|n - 3| \leq 2

. This yields the integer range

1 \leq n \leq 5

, containing 5 integers. Including

n = -3

gives a total of 6 integer solutions.

Adım Adım Çözüm

1

Determine the domain constraint from the non-negativity of the absolute value expression.

2n + 6 \geq 0 \implies n \geq -3

The absolute value expression

|n^2 - 9|

is non-negative for all real

n

, so the right-hand side must also be non-negative.

2

Evaluate the boundary case

n = -3

.

|(-3)^2 - 9| = 0

and

2(-3) + 6 = 0 \implies 0 \leq 0

(True)

When

n = -3

, both sides equal 0, making

n = -3

a valid integer solution.

3

Factor the quadratic inside the absolute value for

n > -3

and simplify.

|n - 3||n + 3| \leq 2(n + 3) \implies |n - 3| \leq 2

Since

n > -3

, the term

n + 3

is strictly positive, allowing division of both sides by

n + 3

without reversing the inequality.

4

Solve the linear absolute value inequality

|n - 3| \leq 2

.

-2 \leq n - 3 \leq 2 \implies 1 \leq n \leq 5

Removing the absolute value creates a compound inequality bounded between

-2

and

2

.

5

List all valid integer solutions and count them.

n \in \{-3, 1, 2, 3, 4, 5\} \implies 6

integer solutions

Combining the boundary root

n = -3

with the five consecutive integers from

1

to

5

gives 6 integer solutions.

Anahtar Kavram

Absolute Value Inequalities with Variable RHS and Boundary Factors

Soru 139Soru →

To reduce municipal water consumption during severe dry spells, the city of Oakhaven plans to install smart water meters for all residential customers and charge a steep surge tariff whenever an individual household's weekly water usage exceeds a baseline threshold. The city council projects that this financial disincentive will prompt residents to curb non-essential outdoor watering, thereby lowering total municipal water demand by 20 percent over the next summer. Which of the following would it be most useful to determine in evaluating whether the city council's plan will achieve its intended goal?

Whether installing the smart water meters will require the municipal utility to incur administrative expenses that exceed initial budget projections.

Whether neighboring municipalities have successfully reduced their overall water consumption by offering rebates for water-efficient appliances.

Whether a substantial proportion of the residential properties that regularly exceed the baseline threshold are rented units whose water bills are paid by property owners rather than by occupants.

Whether residents whose weekly water usage is currently below the baseline threshold will feel motivated to lower their consumption even further.

Whether local environmental organizations endorse the city council's reliance on monetary penalties rather than voluntary conservation campaigns.

Cevabı ve açıklamayı göster

Cevap: Whether a substantial proportion of the residential properties that regularly exceed the baseline threshold are rented units whose water bills are paid by property owners rather than by occupants.

Cevap

Whether a substantial proportion of the residential properties that regularly exceed the baseline threshold are rented units whose water bills are paid by property owners rather than by occupants.

The plan relies on a financial disincentive (the surge tariff) to compel high-volume water users to reduce their consumption. For this mechanism to work, the individuals making daily water-use decisions must directly experience the financial penalty. Determining whether high-usage properties are occupied by renters whose water bills are paid by landlords directly tests this assumption. If landlords pay the bill, tenants face no price signal, neutralizing the tariff and causing the plan to fail.

Adım Adım Çözüm

1

Identify the plan's specific goal and proposed causal mechanism.

Goal: Lower total municipal water demand by 20%. Mechanism: Financial surge tariffs on excessive usage will disincentivize residents from overusing water.

Evaluating a plan requires testing whether the causal chain connecting the proposed action to the intended outcome is logically sound.

2

Formulate a Variance Test to identify critical unstated assumptions.

Determine whether the financial disincentive actually reaches the individuals controlling the water consumption.

If landlords pay the utility bills, tenants using the water face no financial penalty and will not alter their consumption habits, breaking the plan's causal chain.

3

Evaluate the two opposite answers to the key parameter.

If YES (most high-usage units are tenant-occupied with landlord-paid bills), the plan fails. If NO (occupants pay their own bills), the plan is far more likely to succeed.

A question whose opposite answers directly determine the success or failure of the plan identifies the most useful information for evaluation.

Anahtar Kavram

Evaluating Plans: Variance Test & Causal Linkages

Tahmini Süre:2m 0s

Soru 140Soru →

In a four-year ecological survey of Archipelago X, marine biologists monitored five deep-water reefs where commercial trawling was strictly restricted to depths shallower than 200 meters. Over the survey period, benthic invertebrate populations decreased by 35% across all five monitored reefs combined, while sea surface temperatures in Archipelago X remained within historical averages. On the two monitored reefs located within 10 kilometers of active deep-sea mineral dredging operations, benthic invertebrate populations decreased by 60%. On the remaining three monitored reefs, which were situated more than 50 kilometers from any dredging operations, benthic invertebrate populations decreased by 20%.

Which of the following can be properly inferred from the statements above?

The absolute number of individual benthic invertebrates lost on the two reefs situated near dredging operations exceeded the absolute number lost on the three reefs located farther from dredging operations.

The percentage decrease in benthic invertebrate populations recorded on the monitored reefs located within 10 kilometers of dredging operations was greater than the percentage decrease recorded on those monitored reefs more than 50 kilometers away.

Deep-sea mineral dredging operations were the direct primary cause of the benthic invertebrate population declines observed near Archipelago X.

Benthic invertebrate populations on the monitored reefs would not have declined if commercial trawling had been completely prohibited at all water depths.

Commercial trawling conducted at depths shallower than 200 meters contributed to the 20 percent population drop observed on reefs located more than 50 kilometers from dredging sites.

Cevabı ve açıklamayı göster

Cevap: The percentage decrease in benthic invertebrate populations recorded on the monitored reefs located within 10 kilometers of dredging operations was greater than the percentage decrease recorded on those monitored reefs more than 50 kilometers away.

Cevap

The percentage decrease in benthic invertebrate populations recorded on the monitored reefs located within 10 kilometers of dredging operations was greater than the percentage decrease recorded on those monitored reefs more than 50 kilometers away.

The passage explicitly states that benthic invertebrate populations decreased by 60% on the monitored reefs within 10 kilometers of dredging operations and by 20% on the monitored reefs more than 50 kilometers away. Because 60% is mathematically greater than 20%, it must be true that the percentage decrease was greater on the reefs located closer to dredging operations.

Adım Adım Çözüm

1

Analyze the factual premises provided in the passage.

Identified explicit facts: 5 deep-water reefs monitored; trawling restricted < 200m; overall 35% decline; sea surface temps normal; 2 reefs within 10 km of dredging had a 60% decline; 3 reefs > 50 km from dredging had a 20% decline.

GMAT Critical Reasoning inference questions require evaluating statements solely based on what must be strictly true from the provided facts.

2

Distinguish strictly logical deductions from assumptions and causal speculations.

Recognized that comparing 60% to 20% yields a direct mathematical truth (60% > 20%), whereas asserting causation (dredging caused the drop) or absolute counts (total number of organisms) requires unstated facts.

Valid inferences cannot depend on unstated premises, external real-world knowledge, or speculative leaps.

3

Evaluate the answer choices against the strict deduction.

The statement comparing the percentage decreases on reefs near dredging (60%) versus reefs farther away (20%) MUST BE TRUE based solely on the explicit numbers given in the passage.

A 60% decline is strictly greater than a 20% decline, making this comparison a valid inference.

Anahtar Kavram

Distinguishing Inferences from Assumptions and Speculations

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