Tüm alıştırma soruları

Soru 1681Soru →

On a standard number line, the coordinates of three distinct points $A$ , $B$ , and $C$ are represented by the integers $a$ , $b$ , and $c$ , respectively, such that $a < b < c$ . The distance between each point and the origin is represented by its absolute value, and these distances satisfy the inequality $|a| < |b| < |c|$ . If the product of the three coordinates is negative ( $abc < 0$ ) and the sum of their absolute values is $20$ , what is the maximum possible value of the coordinate $b$ ?

6

8

9

10

18

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

The correct answer is 9. By analyzing the signs of the coordinates, we determine that only

a

is negative, so

a < 0 < b < c

. Since

a

is a negative integer, the minimum value of

|a|

is 1. Under the constraint

|a| < |b| < |c|

and

|a| + |b| + |c| = 20

, we let

x = -a

,

y = b

, and

z = c

. We have

x + y + z = 20

with

1 \le x < y < z

. Since

z > y

, we get

20 = x + y + z > 1 + 2y

, which implies

2y < 19

, or

y < 9.5

. The maximum integer value for

y

(which is

b

) is 9, achieved when the coordinates are -1, 9, and 10.

Adım Adım Çözüm

1

Determine the signs of the coordinates based on the given constraints.

The sign configuration must be

a < 0 < b < c

.

Since the product

abc < 0

, either all three coordinates are negative or exactly one is negative. If all three were negative (

a < b < c < 0

), their absolute values would satisfy

|a| > |b| > |c|

, which contradicts

|a| < |b| < |c|

. Thus, exactly one coordinate (

a

) is negative, and the other two (

b

and

c

) are positive.

2

Translate the absolute values and sum constraint into algebraic terms.

-a + b + c = 20

, where

a < 0

and

b, c > 0

.

For a negative number

a

, the absolute value

|a| = -a

. For positive numbers

b

and

c

,

|b| = b

and

|c| = c

. Therefore, the sum

|a| + |b| + |c| = 20

simplifies to

-a + b + c = 20

.

3

Set up inequalities to bound the coordinate of the middle point.

1 \le -a < b < c

.

Since

a

is a non-zero negative integer, the smallest possible value for its absolute value

|a| = -a

is

1

. The condition

|a| < |b| < |c|

then becomes

1 \le -a < b < c

.

4

Use the constraints to find the maximum value of

b

.

The maximum possible value of

b

is

9

.

Let

x = -a

,

y = b

, and

z = c

. We have

x + y + z = 20

with

1 \le x < y < z

. Since

z > y

, we have

20 = x + y + z > x + 2y \ge 1 + 2y

. This simplifies to

19 > 2y

, or

y < 9.5

. Since

y

must be an integer, the maximum possible value for

y

(which is

b

) is

9

. We can verify this maximum by setting

a = -1

,

b = 9

, and

c = 10

, which perfectly satisfies all conditions.

Anahtar Kavram

Analyzing coordinate signs and absolute value inequalities on a number line to perform optimization under integer constraints.

Soru 1682Soru →

Let $S$ be the set of all integers $n$ that satisfy the inequality $|n - 3| + |n + 5| \leq 12$ . What is the sum of the absolute values of all integers in set $S$ ?

43

-13

12

28

37

Cevabı ve açıklamayı göster

Cevap: 43

Cevap

43

The correct answer is 43. Solving the inequality

|n - 3| + |n + 5| \leq 12

by casework on the number line yields the solution set of integers

S = \{-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}

. Summing the absolute values of these elements gives

7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 4 + 5 = 43

.

Adım Adım Çözüm

1

Divide the number line into intervals to remove the absolute value bars based on critical points

n = 3

and

n = -5

.

Three cases to analyze: Case 1 (

n \geq 3

), Case 2 (

-5 \leq n < 3

), and Case 3 (

n < -5

).

Evaluating absolute values requires knowing the signs of the expressions inside them.

2

Solve the inequality

|n - 3| + |n + 5| \leq 12

for each case.

For Case 1 (

n \geq 3

):

(n - 3) + (n + 5) \leq 12 \Rightarrow 2n + 2 \leq 12 \Rightarrow n \leq 5

, yielding integers

\{3, 4, 5\}

. For Case 2 (

-5 \leq n < 3

):

-(n - 3) + (n + 5) \leq 12 \Rightarrow 8 \leq 12

(always true), yielding integers

\{-5, -4, -3, -2, -1, 0, 1, 2\}

. For Case 3 (

n < -5

):

-(n - 3) - (n + 5) \leq 12 \Rightarrow -2n - 2 \leq 12 \Rightarrow -2n \leq 14 \Rightarrow n \geq -7

, yielding integers

\{-7, -6\}

.

Determining the integer values of

n

that satisfy the inequality in each segment of the number line.

3

Combine the intervals to construct the complete set

S

.

S = \{-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}

.

The union of all valid cases gives the entire set of solutions.

4

Compute the sum of the absolute values of the integers in

S

.

|-7| + |-6| + |-5| + |-4| + |-3| + |-2| + |-1| + |0| + |1| + |2| + |3| + |4| + |5| = 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 + 1 + 2 + 3 + 4 + 5 = 43

.

Adding the absolute values of each element in the solution set.

Anahtar Kavram

Solving multi-interval absolute value inequalities with integers and finding absolute values

Tahmini Süre:2m 0s

Soru 1683Soru →

Every autumn, millions of monarch butterflies embark on a perilous three-thousand-mile journey to the oyamel fir forests of central Mexico. This annual migration occurs every year, and it draws scientists and tourists alike who marvel at the insects' precision.

Which choice most effectively eliminates redundancy from the underlined portion of the sentence?

This annual migration occurs every year, and it draws

This annual migration draws them, it draws

This annual migration draws

This annual migration, which draws

Cevabı ve açıklamayı göster

Cevap: This annual migration draws

Cevap

The correct choice is 'This annual migration draws' because it removes the redundant phrase 'every year' while maintaining the correct grammatical structure of the sentence.

The correct choice is the option that states 'This annual migration draws' because it removes the redundant phrase 'every year' (since 'annual' already conveys that the migration happens once a year) and provides a main verb 'draws' to form a grammatically complete sentence.

Adım Adım Çözüm

1

Identify redundant phrases in the original sentence.

The adjective 'annual' and the phrase 'every year' both mean occurring once a year, making one of them redundant.

Redundancy occurs when the same information is repeated unnecessarily, and standardized tests require selecting the most concise grammatically correct option.

2

Evaluate the grammatical role of the underlined portion.

The underlined portion begins a new sentence after a period, meaning it must contain a subject and a verb to form a complete independent clause.

Eliminating redundancy must not introduce new grammatical errors like fragments or comma splices.

3

Test each option for both conciseness and grammatical correctness.

The choice 'This annual migration draws' successfully eliminates the redundancy while providing the main verb 'draws' to form a complete sentence.

This determines the final correct option that is both concise and grammatically sound.

Anahtar Kavram

Eliminating Redundancy

Soru 1684Soru →

A certain cloud database stores a total of $3.6 \times 10^{15}$ bytes of data. If this data is distributed equally among $8.0 \times 10^4$ servers, how many megabytes of data are stored on each server? (Note: $1\text{ megabyte} = 10^6\text{ bytes}$ )

4.5 \times 10^3

4.5 \times 10^4

4.5 \times 10^5

4.5 \times 10^{10}

2.88 \times 10^{20}

Cevabı ve açıklamayı göster

Cevap: $4.5 \times 10^4$

Cevap

4.5 \times 10^4

megabytes

The correct answer is the value that correctly represents the data per server in megabytes. Dividing the total data of

3.6 \times 10^{15}

bytes by the

8.0 \times 10^4

servers gives

4.5 \times 10^{10}

bytes per server. Converting this to megabytes by dividing by

10^6

gives

4.5 \times 10^4

megabytes.

Adım Adım Çözüm

1

Calculate the bytes of data stored on each server by dividing the total data by the number of servers.

0.45 \times 10^{11}\text{ bytes} = 4.5 \times 10^{10}\text{ bytes}

Dividing the total bytes (

3.6 \times 10^{15}

) by the total number of servers (

8.0 \times 10^4

) gives the share of data per server. Mathematically,

\frac{3.6}{8.0} = 0.45

, and

\frac{10^{15}}{10^4} = 10^{15-4} = 10^{11}

.

2

Convert the result from bytes to megabytes using the conversion factor

1\text{ megabyte} = 10^6\text{ bytes}

.

4.5 \times 10^4\text{ megabytes}

To convert bytes to megabytes, divide the number of bytes by

10^6

. Since

\frac{4.5 \times 10^{10}}{10^6} = 4.5 \times 10^{10-6} = 4.5 \times 10^4

.

Anahtar Kavram

Division of numbers in scientific notation and unit conversion.

Soru 1685Soru →

A store sells a winter coat that is originally priced at $120. During a weekend clearance sale, the coat's price is reduced by 25%. What is the sale price of the coat, in dollars?

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

The sale price of the coat is $90.

The correct answer is

90. To find the sale price, we compute the discount by taking 25% of

120, which is

30, and then subtract that discount from the original price of

120.

Adım Adım Çözüm

1

Find 25% of $120 to determine the discount amount.

$30

A 25% reduction means we calculate

120 multiplied by 0.25, which equals

30.

2

Subtract the discount from the original price.

$90

The sale price is the original price minus the discount:

120 -

30 = $90.

Anahtar Kavram

Calculating a percentage markdown to determine a final price

Tahmini Süre:45s

Soru 1686Soru →

Let $x = 2^{100}$ , $y = 3^{75}$ , and $z = 5^{50}$ . Which of the following inequalities correctly represents the relationship among the values of $x$ , $y$ , and $z$ ?

x < y < z

x < z < y

z < y < x

y < z < x

z < x < y

Cevabı ve açıklamayı göster

Cevap: $x < z < y$

Cevap

x < z < y

To find the correct relationship, rewrite

x = 2^{100}

,

y = 3^{75}

, and

z = 5^{50}

with a common exponent. The greatest common divisor of

100

,

75

, and

50

is

25

. Using the rule

(a^m)^n = a^{m \cdot n}

, rewrite the terms:

x = (2^4)^{25} = 16^{25}

,

y = (3^3)^{25} = 27^{25}

, and

z = (5^2)^{25} = 25^{25}

. Comparing the bases shows

16 < 25 < 27

, which means

16^{25} < 25^{25} < 27^{25}

, so

x < z < y

.

Adım Adım Çözüm

1

Find the greatest common divisor (GCD) of the exponents of the three expressions.

The exponents are

100

,

75

, and

50

. The greatest common divisor of these numbers is

25

.

Finding a common exponent allows us to rewrite each expression with the same power so we can compare their bases directly.

2

Rewrite each expression using the power of a power rule:

(a^m)^n = a^{m \cdot n}

.

x = 2^{100} = (2^4)^{25}

,

y = 3^{75} = (3^3)^{25}

, and

z = 5^{50} = (5^2)^{25}

.

This expresses all three values in the form

b^{25}

, where

b

is the base to be evaluated.

3

Evaluate the bases inside the parentheses.

2^4 = 16

, so

x = 16^{25}

;

3^3 = 27

, so

y = 27^{25}

;

5^2 = 25

, so

z = 25^{25}

.

Evaluating the base values simplifies each expression to a single number raised to the power of

25

.

4

Compare the evaluated bases and write the resulting inequality.

Since

16 < 25 < 27

, it follows that

16^{25} < 25^{25} < 27^{25}

. This corresponds to

x < z < y

.

Since the exponent

25

is positive, raising larger positive bases to this exponent results in larger values.

Anahtar Kavram

Comparing exponential expressions by rewriting them with a common exponent using exponent rules.

Soru 1687Soru →

Passage

In 1931, engineering draftsman Harry Beck designed a new map for the London Underground. Previously, transit maps of the railway system had attempted to depict geographic distance accurately, which made the layout in the city center look cluttered and difficult to read. Consequently, Beck proposed a simplified, color-coded schematic diagram that prioritized network connections over geographic accuracy.

Which choice provides the most logical transition from the preceding sentence?

NO CHANGE

However,

For instance,

Moreover, consequently,

Cevabı ve açıklamayı göster

Cevap: NO CHANGE

Cevap

The choice 'NO CHANGE' is correct because the transition 'Consequently' appropriately establishes a cause-and-effect relationship between the cluttered, geographically accurate maps and Beck's proposal of a simplified schematic diagram.

The transition 'Consequently' is the most logical choice because it correctly establishes a cause-and-effect relationship: because the geographically accurate maps were cluttered and confusing, Beck designed a simplified schematic diagram as a solution.

Adım Adım Çözüm

1

Analyze the logical relationship between the preceding sentence and the sentence containing the transition.

The preceding sentence describes a problem (cluttered, hard-to-read transit maps). The following sentence describes Beck's response to that problem (proposing a simplified diagram).

Understanding the semantic connection between the two ideas is necessary to select the correct transition.

2

Evaluate the relationship type.

Since Beck's action was a direct result of the previous maps being confusing, the relationship is cause-and-effect.

This relationship type dictates which category of transition words to look for.

3

Evaluate the choices against this cause-and-effect relationship.

'Consequently' is the only option that correctly establishes a cause-and-effect relationship. 'However' indicates contrast, 'For instance' indicates illustration, and 'Moreover, consequently' is redundant.

Selecting the option that matches the identified relationship ensures the passage is cohesive and logically sound.

Anahtar Kavram

Selecting logical transitions that accurately connect ideas between sentences.

Soru 1688Soru →

The table below shows the number of books read by 7 students during their summer vacation:

Student	Books Read
Amy	16
Ben	8
Chris	3
Diana	12
Ethan	14
Fiona	6
Gabe	11

What is the median number of books read by these students?

8

10

11

12

13

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

To find the median of a data set, the values must first be arranged in order from least to greatest: 3, 6, 8, 11, 12, 14, 16. Since there are 7 values, the median is the single middle value, which is 11.

Adım Adım Çözüm

1

Sort the data set from least to greatest.

3, 6, 8, 11, 12, 14, 16

The median of a data set is the middle value when the numbers are ordered.

2

Identify the middle value of the sorted list.

11

Since there are 7 data points (an odd number), the median is the 4th value, which has exactly 3 values below it and 3 values above it.

Anahtar Kavram

Finding the median of a set of data

Soru 1689Soru →

A writer is editing an essay about historical agricultural science and has drafted the following paragraph:

Russian botanist Nikolay Vavilov dedicated his life to ending famine by gathering seeds from crop wild relatives across five continents. By the 1930s, he had established the world’s largest seed collection in Leningrad. During the devastating 872-day Siege of Leningrad in World War II, Vavilov's assistants chose to starve to death rather than eat the unique samples of rice, wheat, and potatoes they protected. Svalbard, a remote Norwegian archipelago, now houses a global seed vault designed to withstand natural and man-made disasters. Ultimately, the scientists' sacrifice ensured the preservation of vital genetic plant diversity that helped restore agriculture after the war.

The writer is considering deleting the underlined sentence. Should the sentence be kept or deleted?

Keep it, because it establishes the global significance of Vavilov's work by linking it to modern preservation efforts.

Keep it, because it provides essential geographical details that clarify where Vavilov conducted his primary research.

Delete it, because it interrupts the paragraph's focus on Vavilov's Leningrad collection and the wartime sacrifice of his scientists.

Delete it, because it contradicts the assertion that the scientists saved the seed collection during the Siege of Leningrad.

Cevabı ve açıklamayı göster

Cevap: Delete it, because it interrupts the paragraph's focus on Vavilov's Leningrad collection and the wartime sacrifice of his scientists.

Cevap

Delete it, because it interrupts the paragraph's focus on Vavilov's Leningrad collection and the wartime sacrifice of his scientists.

The correct option correctly determines that the sentence should be deleted because it is irrelevant to the specific narrative. The paragraph deals with Vavilov's historical work and the sacrifices made by his team in Leningrad during World War II. Introducing details about a modern facility in Svalbard, Norway, disrupts the chronological flow and introduces a tangential topic.

Adım Adım Çözüm

1

Analyze the paragraph to determine its main focus.

The paragraph focuses on Nikolay Vavilov establishing a massive seed collection in Leningrad, his assistants sacrificing their lives to protect it during the Siege of Leningrad in World War II, and the post-war agricultural impact of that sacrifice.

Understanding the paragraph's main theme is necessary to evaluate the relevance of any single sentence.

2

Evaluate the underlined sentence against the main focus.

The underlined sentence discusses Svalbard, a modern global seed vault in Norway designed to survive disasters. This is a modern development that is tangential to the historical narrative of the Leningrad Siege.

Identifying if a sentence adds value or introduces a distraction helps decide whether to keep or delete it.

3

Select the option that correctly identifies the decision to delete the sentence and provides the proper rhetorical justification.

The option to delete the sentence because it interrupts the historical focus on Vavilov's Leningrad collection and his scientists' sacrifice is the correct choice.

Only one option contains both the correct decision (delete) and the correct reasoning (focus interruption).

Anahtar Kavram

Evaluating Relevance for Adding or Deleting Content

Tahmini Süre:1m 0s

Soru 1690Soru →

A positive integer $N$ has a prime factorization of the form $p^2 \times q$ , where $p$ and $q$ are distinct prime numbers. If the sum of all the positive factors of $N$ (including $1$ and $N$ ) is $78$ , what is the value of $N$ ?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The value of

N

is 45.

By applying the sum of divisors formula, the sum of factors of

N = p^2 \times q

is represented as

(1 + p + p^2)(1 + q) = 78

. Factoring 78 into two integers that satisfy the prime constraints of

p \ge 2

and

q \ge 2

yields

13 \times 6 = 78

. Solving

1 + p + p^2 = 13

gives

p = 3

, and

1 + q = 6

gives

q = 5

. Since 3 and 5 are distinct primes,

N = 3^2 \times 5 = 45

.

Adım Adım Çözüm

1

Express the sum of factors of

N

algebraically

(1 + p + p^2)(1 + q) = 78

The sum of all positive factors of a number with prime factorization

p^a q^b

is given by the product of the sums of the powers of each prime factor.

2

Determine the constraints on

p

and

q

based on them being prime numbers

1 + p + p^2 \ge 7

and

1 + q \ge 3

The smallest prime number is 2, so

p \ge 2

and

q \ge 2

.

3

Find the factor pairs of 78 that satisfy the constraints

(1 + p + p^2, 1 + q) \in \{(13, 6), (26, 3)\}

The factors of 78 are 1, 2, 3, 6, 13, 26, 39, 78. We pair them such that one factor is at least 7 and the other is at least 3.

4

Solve for

p

and

q

for each possible factor pair

p = 3

and

q = 5

If

1 + p + p^2 = 13

, then

p^2 + p - 12 = 0

, which solves to

p = 3

(since

p > 0

). This leaves

1 + q = 6 \implies q = 5

. Both 3 and 5 are distinct primes. The other case,

1 + p + p^2 = 26

, has no integer solution for

p

.

5

Calculate the value of

N

N = 3^2 \times 5 = 45

Substitute the prime values back into the expression for

N

.

Anahtar Kavram

Sum of Divisors Formula and Prime Factorization

Tahmini Süre:1m 30s

Soru 1691Soru →

If $x = -3$ and $y = 2$ , what is the value of the expression $x^2 - y(x - y^3)$ ?

-77

-1

13

27

31

Cevabı ve açıklamayı göster

Cevap: 31

Cevap

31

Substituting

x = -3

and

y = 2

into the expression gives

x^2 - y(x - y^3) = (-3)^2 - 2(-3 - 2^3)

. Evaluating exponents first, we get

9 - 2(-3 - 8)

. Simplifying inside the parentheses gives

9 - 2(-11)

. Multiplying before subtracting yields

9 - (-22) = 9 + 22 = 31

.

Adım Adım Çözüm

1

Substitute the given values into the expression

(-3)^2 - 2((-3) - 2^3)

To evaluate the expression for the specific numbers

x = -3

and

y = 2

.

2

Evaluate the exponents outside and inside the parentheses

9 - 2(-3 - 8)

Exponents must be calculated before multiplication and subtraction, according to PEMDAS rules. Note that

(-3)^2 = 9

and

2^3 = 8

.

3

Perform the operations inside the parentheses

9 - 2(-11)

Parentheses have the highest priority in the order of operations.

4

Multiply the terms

9 - (-22) = 9 + 22

Multiplication must be performed before subtraction.

5

Perform final addition

31

Adding 22 to 9 yields the final simplified value.

Anahtar Kavram

Order of operations (PEMDAS) and substitution of negative numbers in exponents.

Soru 1692Soru →

A supercomputer simulation requires a total of $1.62 \times 10^{14}$ operations. The simulation is divided equally among 12 identical processors running in parallel. If each processor can perform $4.5 \times 10^9$ operations per second, how many seconds will it take to complete the simulation?

6.075 \times 10^{22}

3.6 \times 10^4

3.0 \times 10^3

4.32 \times 10^5

3.0 \times 10^4

Cevabı ve açıklamayı göster

Cevap: $3.0 \times 10^3$

Cevap

3.0 \times 10^3

The correct answer is

3.0 \times 10^3

. To find the total time to complete the simulation, first compute the total operations the 12 parallel processors can perform per second:

12 \times (4.5 \times 10^9) = 5.4 \times 10^{10}

operations/second. Next, divide the total number of required operations by this combined rate:

\frac{1.62 \times 10^{14}}{5.4 \times 10^{10}} = 0.3 \times 10^{14-10} = 0.3 \times 10^4 = 3.0 \times 10^3

seconds.

Adım Adım Çözüm

1

Determine the combined processing rate of the 12 processors working in parallel.

Combined Rate =

12 \times (4.5 \times 10^9) = 5.4 \times 10^{10}

operations per second.

Since the 12 processors run in parallel, their individual speeds are added together to find the total rate of operations performed per second.

2

Divide the total number of operations required by the combined rate of all processors to find the time in seconds.

Time =

\frac{1.62 \times 10^{14}}{5.4 \times 10^{10}}

Time equals the total workload divided by the combined rate of work.

3

Perform the division and convert the result to standard scientific notation.

Time =

0.3 \times 10^{14 - 10} = 0.3 \times 10^4 = 3.0 \times 10^3

seconds.

Subtracting the exponents during division gives

10^4

. The coefficient

0.3

is rewritten in standard scientific notation as

3.0 \times 10^{-1}

, so

0.3 \times 10^4 = 3.0 \times 10^3

.

Anahtar Kavram

Scientific notation operations and rate division

Soru 1693Soru →

On a standard number line, the coordinate of point $P$ is an integer $x$ . If the sum of the distances from $P$ to $-2$ and from $P$ to $4$ is equal to the distance from $P$ to $10$ , what is the sum of all possible values of $x$ ?

Cevabı ve açıklamayı göster

Cevap: -4

Cevap

The sum of all possible values of the integer

x

is

-4

.

The correct answer is

-4

. The distance between any two coordinates

a

and

b

on a number line is defined as

|a - b|

. Translating the problem gives the equation

|x + 2| + |x - 4| = |x - 10|

. Breaking the number line into intervals around the critical points

x = -2

,

x = 4

, and

x = 10

yields two valid integer solutions:

x = -8

and

x = 4

. The sum of these values is

-8 + 4 = -4

.

Adım Adım Çözüm

1

Express the distances on the number line using absolute values.

The distance from

P(x)

to

-2

is

|x - (-2)| = |x + 2|

. The distance from

P(x)

to

4

is

|x - 4|

. The distance from

P(x)

to

10

is

|x - 10|

. The equation is

|x + 2| + |x - 4| = |x - 10|

.

The absolute value

|a - b|

represents the distance between points

a

and

b

on a standard number line.

2

Solve the equation by testing the intervals defined by the critical points

x = -2

,

x = 4

, and

x = 10

.

We analyze the four intervals:
- For

x < -2

:

-(x + 2) - (x - 4) = -(x - 10) \implies -2x + 2 = -x + 10 \implies x = -8

. Since

-8 < -2

, this is a valid solution.
- For

-2 \leq x < 4

:

(x + 2) - (x - 4) = -(x - 10) \implies 6 = -x + 10 \implies x = 4

. Since

4

is not in

[-2, 4)

, there is no solution in this interval.
- For

4 \leq x < 10

:

(x + 2) + (x - 4) = -(x - 10) \implies 2x - 2 = -x + 10 \implies 3x = 12 \implies x = 4

. Since

4

is in

[4, 10)

, this is a valid solution.
- For

x \geq 10

:

(x + 2) + (x - 4) = x - 10 \implies 2x - 2 = x - 10 \implies x = -8

. Since

-8 < 10

, there is no solution in this interval.

Absolute value terms change sign at their critical points, requiring case-by-case evaluation.

3

Sum all valid integer solutions.

The valid values for

x

are

-8

and

4

. Their sum is

-8 + 4 = -4

.

The problem asks for the sum of all possible values of

x

.

Anahtar Kavram

Representing geometric distances on a number line using absolute value equations and solving them using interval analysis.

Soru 1694Soru →

A red blood cell has a diameter of approximately $0.000008$ meters. When this number is written in scientific notation as $a \times 10^n$ , where $1 \leq a < 10$ , what is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: -6

Cevap

The value of the exponent is

-6

.

To write the number

0.000008

in scientific notation, we shift the decimal point 6 places to the right to get

8

. Since the original value is less than 1, the exponent is negative, giving

8 \times 10^{-6}

. The value of

n

is therefore

-6

.

Adım Adım Çözüm

1

Locate the decimal point and determine how many places it must be shifted to obtain a coefficient between 1 and 10.

The decimal point must be shifted 6 places to the right to get the number

8

.

Scientific notation requires the lead coefficient

a

to satisfy

1 \leq a < 10

.

2

Determine the sign and value of the exponent based on the decimal shift.

The exponent

n

is

-6

.

Moving the decimal point to the right to write a decimal value less than 1 results in a negative exponent equal to the number of shifts.

Anahtar Kavram

Scientific Notation for Decimals Less Than One

Soru 1695Soru →

The writer wants to add the following sentence to the paragraph:

"By studying the patterns of dates on these monuments, she realized they actually recorded the lifespans and historical lineages of Maya rulers."

To make the paragraph most logical, where should this sentence be placed in the paragraph?

[1] For decades, scholars believed that Maya hieroglyphs were purely symbolic, representing abstract concepts rather than spoken language. [2] However, in the 1950s, Tatiana Proskouriakoff revolutionized the field by analyzing stelae at the site of Piedras Negras. [3] These monuments, which featured dates paired with depictions of figures, were previously assumed to be astronomical calendars. [4] Her breakthroughs paved the way for modern epigraphers to translate the rich historical records carved into stone.

before Sentence 1

after Sentence 2

after Sentence 3

after Sentence 4

Cevabı ve açıklamayı göster

Cevap: after Sentence 3

Cevap

after Sentence 3

Placing the sentence after Sentence 3 is correct because it logically connects a previous assumption about the monuments to the researcher's discovery of their true purpose, while also providing a clear antecedent for the phrase 'Her breakthroughs' in Sentence 4.

Adım Adım Çözüm

1

Identify pronouns and demonstrative adjectives in the new sentence that require antecedents.

The new sentence contains 'she' (referring to a female individual) and 'these monuments' (referring to specific monuments).

This establishes that the sentence must follow a sentence that introduces a female researcher and mentions monuments.

2

Scan the paragraph to locate where these referents are introduced and discussed.

Tatiana Proskouriakoff is introduced in Sentence 2. Monuments (referred to as 'stelae' in Sentence 2 and 'These monuments' in Sentence 3) are also introduced and described there.

This narrows down the placement to somewhere after Sentence 2 or Sentence 3.

3

Analyze the logical transition between the previous interpretation of the monuments and Proskouriakoff's new discovery.

Sentence 3 establishes what these monuments were 'previously assumed' to be. The new sentence states what she realized they 'actually recorded,' contrasting with the previous assumption. Sentence 4 then summarizes the impact of 'Her breakthroughs.'

Placing the realization after the previous assumption (after Sentence 3) and before the summary of its impact (Sentence 4) creates a seamless, logical flow of ideas.

Anahtar Kavram

Logical sentence sequencing using pronoun reference and contrast transitions.

Tahmini Süre:1m 0s

Soru 1696Soru →

What is the value of the expression $\frac{3^4 \cdot 3^2}{3^3}$ ?

8

9

243

729

27

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

27

Evaluating the numerator using the product rule of exponents yields

3^6

. Then, applying the quotient rule of exponents to divide

3^6

by

3^3

yields

3^3

. Evaluating

3^3

gives

27

.

Adım Adım Çözüm

1

Simplify the numerator using the product rule of exponents:

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

.

3^4 \cdot 3^2 = 3^{4+2} = 3^6

When multiplying exponential terms with the same base, add their exponents.

2

Simplify the fraction using the quotient rule of exponents:

\frac{a^m}{a^n} = a^{m-n}

.

\frac{3^6}{3^3} = 3^{6-3} = 3^3

When dividing exponential terms with the same base, subtract the exponent in the denominator from the exponent in the numerator.

3

Evaluate the simplified exponential expression.

3^3 = 3 \cdot 3 \cdot 3 = 27

Evaluate the base raised to the power of 3 by multiplying it by itself three times.

Anahtar Kavram

Simplifying expressions using the product and quotient rules of exponents

Tahmini Süre:45s

Soru 1697Soru →

Compiler James Murray oversaw the creation of the first Oxford English Dictionary, a monumental task requiring decades of meticulous analysis. Murray and his team systematically cataloged millions of word usage slips submitted by global contributors. Recognizing the project's unprecedented scale, the editors worked tirelessly to ensure the accuracy of every entry.

Which of the following choices best maintains the tone established in the passage?

NO CHANGE

to make sure they didn't mess up

to guarantee the absolute, unblemished verisimilitude of

to ensure the accuracy on

Cevabı ve açıklamayı göster

Cevap: NO CHANGE

Cevap

NO CHANGE

The original phrasing 'to ensure the accuracy of' maintains the objective, academic tone established by words like 'monumental,' 'meticulous,' and 'systematically.' It also correctly utilizes the idiomatic preposition 'of.'

Adım Adım Çözüm

1

Analyze the surrounding context of the underlined portion to determine the established tone of the passage.

The passage uses formal and academic vocabulary, such as 'monumental,' 'meticulous analysis,' and 'systematically cataloged,' establishing a formal, professional tone.

Correct word choice must align with the established tone register of the text.

2

Evaluate the choices to see which option matches this formal register and uses correct grammar and idiomatic usage.

The phrase 'to ensure the accuracy of' matches the academic tone. The informal option 'to make sure they didn't mess up' is too casual, the pretentious option 'to guarantee the absolute, unblemished verisimilitude of' is overly formal, and 'to ensure the accuracy on' is unidiomatic.

Only the correct choice is grammatically accurate, idiomatic, and consistent with the established tone.

Anahtar Kavram

Tone Consistency

Tahmini Süre:1m 0s

Soru 1698Soru →

Echoes of the Past

Archaeoacoustics is a relatively new interdisciplinary field that combines archaeology, ethnomusicology, and acoustic engineering to analyze the soundscapes of ancient sites. Researchers in this field believe that to truly understand ancient civilizations, we must consider not only what they saw, but also what they heard. By studying how sound behaves in prehistoric structures, scholars hope to gain insight into the ritualistic, social, and spiritual lives of ancient people.

At Stonehenge, for example, researchers used 3D laser scanning and acoustic modeling to reconstruct the monument's original layout, including the stones that are now missing or fallen. They discovered that the megaliths acted as a giant chamber, amplifying speech and reverberating musical instruments while blocking out external noise from the surrounding plains. To anyone standing inside the stone circle, the acoustic reflections would have created a sense of isolation and focus, enhancing the dramatic effect of rituals.

Indeed, the acoustic properties of prehistoric structures were rarely accidental. From the subterranean chambers of Malta to the Mayan pyramids of Chichen Itza, ancient builders deliberately shaped their environments to control sound. The acoustic design of Chichen Itza's El Castillo pyramid, for instance, produces a chirping echo resembling the sacred quetzal bird when visitors clap at the base of the stairs.

Given that all the choices are true, which choice most effectively concludes the paragraph and the passage by reinforcing the central idea that ancient acoustic design was intentional and suggesting its ongoing impact on modern scholarship?

As researchers continue to unlock these ancient auditory secrets, archaeoacoustics will undoubtedly expand our understanding of how our ancestors experienced their world.

In addition to sound, ancient civilizations also paid close attention to astronomical alignments, constructing temples that aligned with the winter solstice.

These ancient builders developed sophisticated techniques to manipulate sound, they did not leave written records of their work.

Ultimately, these builders deliberately and intentionally designed these spaces on purpose to alter auditory hearing experiences.

Cevabı ve açıklamayı göster

Cevap: As researchers continue to unlock these ancient auditory secrets, archaeoacoustics will undoubtedly expand our understanding of how our ancestors experienced their world.

Cevap

As researchers continue to unlock these ancient auditory secrets, archaeoacoustics will undoubtedly expand our understanding of how our ancestors experienced their world.

The correct option successfully closes the passage by connecting back to the main topic of archaeoacoustics and its exploration of ancient acoustic design. It highlights how these 'auditory secrets' were deliberate and explains how the field will continue to expand our understanding of history, fulfilling both parts of the prompt's request.

Adım Adım Çözüm

1

Analyze the prompt's requirements for the concluding sentence.

The correct option must conclude both the paragraph and the entire passage, reinforce the central idea that ancient acoustic design was intentional, and suggest its future/ongoing impact on modern scholarship.

This establishes the criteria for evaluating each answer choice.

2

Evaluate each option against the established criteria.

The option beginning with 'As researchers continue to unlock' successfully summarizes the deliberate nature of these acoustic properties ('auditory secrets') and mentions how it will expand modern understanding ('expand our understanding of how our ancestors experienced their world'). The option mentioning 'astronomical alignments' introduces irrelevant details. The option ending with 'written records of their work' contains a comma splice. The option with 'deliberately and intentionally' is excessively redundant.

Eliminating incorrect distractors identifies the most effective conclusion.

Anahtar Kavram

Selecting effective concluding sentences that maintain paragraph and passage unity.

Tahmini Süre:1m 30s

Soru 1699Soru →

In a local park, $\frac{2}{5}$ of the trees are maple trees and $\frac{1}{3}$ of the trees are oak trees. The remaining trees are pine trees. What fraction of the trees in the park are pine trees?

\frac{3}{8}

\frac{5}{8}

\frac{4}{15}

\frac{11}{15}

\frac{14}{15}

Cevabı ve açıklamayı göster

Cevap: $\frac{4}{15}$

Cevap

\frac{4}{15}

To find the fraction of pine trees, first calculate the total fraction occupied by maple and oak trees. Finding a common denominator of

15

for

\frac{2}{5}

and

\frac{1}{3}

gives

\frac{6}{15} + \frac{5}{15} = \frac{11}{15}

. Subtracting this combined fraction from

1

(represented as

\frac{15}{15}

) yields the remaining fraction of pine trees, which is

\frac{4}{15}

.

Adım Adım Çözüm

1

Find the sum of the fractions representing maple and oak trees.

\frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}

Before finding the remaining fraction, we need to know the combined fraction of the park occupied by maple and oak trees.

2

Subtract the combined fraction of maple and oak trees from the whole.

1 - \frac{11}{15} = \frac{15}{15} - \frac{11}{15} = \frac{4}{15}

Since the remaining trees are pine trees, subtracting the combined fraction of other trees from the total of

1

yields the fraction of pine trees.

Anahtar Kavram

To find a remaining fractional part of a whole, first sum the known fractional parts using a common denominator, and then subtract this sum from 1.

Soru 1700Soru →

At the start of the fiscal year, a city's municipal budget is divided among three departments: Education, Healthcare, and Infrastructure. Education receives $\frac{3}{8}$ of the total budget, Healthcare receives $40\%$ of the total budget, and Infrastructure receives the remainder. Midyear, the total municipal budget is increased by $25\%$ . The budget allocated to Education is increased by $20\%$ of its original amount, and the budget allocated to Healthcare is increased by $15\%$ of its original amount. What percent of the new total municipal budget is allocated to Infrastructure?

Cevabı ve açıklamayı göster

Cevap: 27.2

Cevap

27.2

The correct answer is

27.2\%

. By representing the initial total budget as

1

, we find the initial Infrastructure allocation is

1 - (0.375 + 0.40) = 0.225

. After the budget increase of

25\%

, the new total budget is

1.25

. The new Education allocation is

0.375 \times 1.20 = 0.45

, and the new Healthcare allocation is

0.40 \times 1.15 = 0.46

. The remaining amount for Infrastructure is

1.25 - 0.45 - 0.46 = 0.34

. Expressing

0.34

as a percentage of the new total budget

1.25

gives

\frac{0.34}{1.25} \times 100\% = 27.2\%

.

Adım Adım Çözüm

1

Convert the initial budget allocations to decimal shares of the original total budget.

Education share is

0.375

, Healthcare share is

0.40

, and Infrastructure share is

0.225

.

Expressing all initial shares as decimals relative to a total budget of

1.0

makes successive calculations straightforward.

2

Calculate the updated budget amounts for the entire city, Education, and Healthcare.

The new total budget is

1.25

, the new Education allocation is

0.375 \times 1.20 = 0.45

, and the new Healthcare allocation is

0.40 \times 1.15 = 0.46

.

Adjust each allocation by its respective percentage change to find its share relative to the initial total.

3

Compute the remaining budget share left for Infrastructure.

New Infrastructure share is

1.25 - (0.45 + 0.46) = 0.34

of the initial budget.

The sum of the three department budgets must equal the new total budget of

1.25

.

4

Find the percentage of the new total budget represented by the new Infrastructure allocation.

\frac{0.34}{1.25} \times 100\% = 27.2\%

Divide the new Infrastructure share by the new total budget to find the new ratio, then convert to a percentage.

Anahtar Kavram

Converting between fractions, decimals, and percentages, and tracking changes across multiple base values.

Alternatif Yöntem

Instead of using

1.0

as the base, you can assume an initial total budget of

800

dollars (since

800

is a multiple of

8

, which simplifies

\frac{3}{8}

). Initial Education =

300

, Healthcare =

320

, Infrastructure =

180

. The new total budget is

800 \times 1.25 = 1000

. New Education =

300 \times 1.20 = 360

. New Healthcare =

320 \times 1.15 = 368

. New Infrastructure =

1000 - (360 + 368) = 272

. Percentage =

\frac{272}{1000} \times 100\% = 27.2\%

.

Tahmini Süre:3m 0s

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