Tüm alıştırma soruları

Consider the following system of linear equations:

\begin{aligned} x + 2y &= 10 \\ 3x - y &= 9 \end{aligned}

Which of the following statements are true? Select all that apply.

Geçerli olan tümünü seçin

The value of

x

is

4

.

The value of

x + y

is

7

.

The value of

y

is

4

.

The value of

2x + y

is

11

.

The value of

x - y

is

5

.

Cevabı ve açıklamayı göster

Cevap: The value of $x$ is $4$ .; The value of $x + y$ is $7$ .; The value of $2x + y$ is $11$ .

Cevap

The true statements are those asserting that

x = 4

, that

x + y = 7

, and that

2x + y = 11

.

Solving the system of linear equations yields the unique solution

x = 4

and

y = 3

. Substituting these values into the given choices demonstrates that the statements claiming

x = 4

,

x + y = 7

, and

2x + y = 11

are all mathematically correct.

Adım Adım Çözüm

1

Express

y

in terms of

x

using the second equation.

y = 3x - 9

Isolating

y

allows for simple substitution into the first equation.

2

Substitute

y = 3x - 9

into the first equation

x + 2y = 10

.

x + 2(3x - 9) = 10 \implies x + 6x - 18 = 10 \implies 7x = 28 \implies x = 4

This yields a single linear equation in terms of

x

.

3

Calculate

y

using

x = 4

.

y = 3(4) - 9 = 3

Substituting

x = 4

back gives the unique solution for

y

.

4

Evaluate the given statements with

(x, y) = (4, 3)

.

x = 4

is true;

x + y = 4 + 3 = 7

is true;

y = 4

is false (

y = 3

);

2x + y = 2(4) + 3 = 11

is true;

x - y = 4 - 3 = 1

is false.

Testing each condition determines which options are valid.

Anahtar Kavram

Solving a 2x2 system of linear equations using substitution to evaluate linear expressions.

Tahmini Süre:1m 0s

Soru 322Soru →

In the $xy$ -plane, line $k$ has a slope of $\frac{2}{5}$ and passes through the point $(5, 8)$ . Line $m$ is perpendicular to line $k$ and has the same $y$ -intercept as line $k$ . What is the $x$ -intercept of line $m$ ?

Cevabı ve açıklamayı göster

Cevap: 2.4

Cevap

2.4

To find the

x

-intercept of line

m

, first determine the equation of line

k

. Using slope-intercept form

y = mx + b

with

m = \frac{2}{5}

and point

(5, 8)

, we get

8 = \frac{2}{5}(5) + b \implies b = 6

. Thus, the

y

-intercept of line

k

(and line

m

) is

(0, 6)

. Next, line

m

is perpendicular to line

k

, so its slope is the negative reciprocal of

\frac{2}{5}

, which is

-\frac{5}{2}

. The equation for line

m

is

y = -\frac{5}{2}x + 6

. Setting

y = 0

to solve for the

x

-intercept yields

0 = -\frac{5}{2}x + 6 \implies \frac{5}{2}x = 6 \implies x = \frac{12}{5} = 2.4

.

Adım Adım Çözüm

1

Find the equation and

y

-intercept of line

k

Line

k

has equation

y = \frac{2}{5}x + 6

, with

y

-intercept at

(0, 6)

.

Using point-slope form

y - y_1 = m(x - x_1)

with

m = \frac{2}{5}

and

(x_1, y_1) = (5, 8)

.

2

Calculate the slope of line

m

The slope of line

m

is

-\frac{5}{2}

.

Perpendicular lines have negative reciprocal slopes:

-\frac{1}{2/5} = -\frac{5}{2}

.

3

Construct the equation of line

m

Line

m

has equation

y = -\frac{5}{2}x + 6

.

Line

m

shares the

y

-intercept

(0, 6)

with line

k

.

4

Solve for the

x

-intercept of line

m

x = 2.4

Set

y = 0

in

y = -\frac{5}{2}x + 6

to get

0 = -\frac{5}{2}x + 6 \implies \frac{5}{2}x = 6 \implies x = 2.4

.

Anahtar Kavram

Perpendicular Slopes and Line Intercepts

Soru 323Soru →

An environmental monitoring station recorded the daily nitrogen dioxide ( $\text{NO}_2$ ) concentrations, in parts per billion (ppb), for 6 consecutive days: $19, 28, 30, 35, 42,$ and $42$ . On the 7th day, a measurement of $x$ ppb was recorded, where $x > 45$ . If the arithmetic mean of the 7 daily concentrations is equal to their median, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 49

Cevap

The value of

x

is

49

.

Sorting the first 6 recorded concentrations gives

19, 28, 30, 35, 42, 42

. Because

x > 45

,

x

is strictly greater than all existing values, so the full set in ascending order is

19, 28, 30, 35, 42, 42, x

. The median of a 7-element set is the 4th value, which is

35

. The sum of the 7 concentrations is

19 + 28 + 30 + 35 + 42 + 42 + x = 196 + x

, making the arithmetic mean

\frac{196 + x}{7}

. Setting the mean equal to the median gives

\frac{196 + x}{7} = 35 \implies 196 + x = 245 \implies x = 49

.

Adım Adım Çözüm

1

Order the first 6 data points in ascending order

The sorted list is

19, 28, 30, 35, 42, 42

.

Establishing ordered positions is necessary to determine the median.

2

Determine the median of the 7-element dataset

Since

x > 45

, the complete ordered dataset is

19, 28, 30, 35, 42, 42, x

, making the 4th element,

35

, the median.

The median of an odd number of sorted values is the middle element.

3

Express the arithmetic mean in terms of

x

and set it equal to the median

\frac{196 + x}{7} = 35

The problem states that the arithmetic mean equals the median.

4

Solve the linear equation for

x

196 + x = 245 \implies x = 49

Multiplying both sides by 7 and subtracting 196 isolates

x

.

Anahtar Kavram

Measures of Central Tendency (Mean and Median)

Soru 324Soru →

Based on the structural contrast and continuation signals in the passage below, fill in the blank with the word that best completes the context.

Aşağıdaki boşlukları doldurun

While nineteenth-century financial commentators widely held that market panics were purely irrational phenomena driven by mass delusion, monetary theorist Walter Bagehot posited that such crises were, on the contrary, entirely ; they unfolded not as arbitrary surges of panic, but as a logical reaction to systemic liquidity shortages.

Cevabı ve açıklamayı göster

Cevap

rational (or acceptable synonyms such as logical, intelligible, or comprehensible)

The sentence relies on dual structural signals: a concessive contrast setup ('While nineteenth-century financial commentators widely held that market panics were purely irrational...') combined with the pivot phrase 'on the contrary'. This indicates that Bagehot's view directly opposes the idea of panics being 'irrational'. Furthermore, the clause following the colon elaborates on this point by characterizing panics as a 'logical reaction'. Thus, the word completing the blank must mean rational, logical, or intelligible.

Adım Adım Çözüm

1

Identify structural transition signals in the sentence

Found the opening contrast signal 'While...', the explicit reversal phrase 'on the contrary', and the colon elaboration '; they unfolded not... but...'

Transition signals dictate the logical direction of the missing term relative to the surrounding context.

2

Analyze the target concept contrasted by the pivot markers

The initial clause describes panics as 'purely irrational phenomena driven by mass delusion'.

The pivot marker 'on the contrary' requires an antonymic relationship with 'irrational'.

3

Verify using the colon elaboration following the blank

The explanatory clause states that panics are 'a logical reaction to systemic liquidity shortages'.

A colon introduces an explanation or elaboration of the preceding statement, confirming that the blank must mean logical or rational.

Anahtar Kavram

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Tahmini Süre:1m 0s

Soru 325Soru →

If $x$ is a solution to the quadratic equation $x^2 - 11x + 28 = 0$ such that $x > 5$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x

is 7.

Factoring

x^2 - 11x + 28 = 0

yields

(x - 7)(x - 4) = 0

, giving solutions

x = 7

and

x = 4

. Since the problem specifies that

x > 5

, the only valid value for

x

is 7.

Adım Adım Çözüm

1

Factor the quadratic expression

(x - 7)(x - 4) = 0

Find two numbers that multiply to 28 and sum to -11, which are -7 and -4.

2

Find the roots of the equation

x = 7

or

x = 4

Set each linear factor equal to 0 and solve for

x

.

3

Apply the given constraint

x > 5

x = 7

Out of the two roots 4 and 7, only 7 is strictly greater than 5.

Anahtar Kavram

Factoring quadratic equations of the form

x^2 + bx + c = 0

Soru 326Soru →

Consider three non-zero integers $x$ , $y$ , and $z$ that satisfy all of the following conditions:

I. $(-1)^{x^2 y + z} = -1$
II. $x y z^2 < 0$
III. $x + y$ is an even integer

Which of the following expressions MUST be an odd integer?

(x + z)(y + z)

x y z

x + y + z

x(y + z)

x z + y

Cevabı ve açıklamayı göster

Cevap: $(x + z)(y + z)$

Cevap

The expression

(x + z)(y + z)

MUST be an odd integer.

Condition I dictates that

x^2 y + z

is odd. Condition III states that

x + y

is even, meaning

x

and

y

share the same parity. If

x

and

y

are both even,

x^2 y

is even, forcing

z

to be odd. If

x

and

y

are both odd,

x^2 y

is odd, forcing

z

to be even. Consequently,

z

always has the opposite parity of both

x

and

y

. Therefore,

(x + z)

is always odd and

(y + z)

is always odd. The product of two odd integers,

(x + z)(y + z)

, is guaranteed to be odd.

Adım Adım Çözüm

1

Analyze Condition I for exponent parity

x^2 y + z

must be an odd integer

For

(-1)^k = -1

, the exponent

k

must be odd.

2

Analyze Condition III for shared parity of

x

and

y

x

and

y

are either both even or both odd

The sum of two integers is even if and only if they share the same parity.

3

Deduce parity relationship for

z

across cases

In Case 1 (

x, y

even),

x^2 y

is even, so

z

must be odd. In Case 2 (

x, y

odd),

x^2 y

is odd, so

z

must be even.

To satisfy

x^2 y + z = \text{odd}

,

x^2 y

and

z

must have opposite parities.

4

Evaluate the parity of

(x + z)(y + z)

(x + z)

is odd and

(y + z)

is odd, so their product is odd

In both cases,

z

has opposite parity to both

x

and

y

. Adding two integers of opposite parity always yields an odd integer, and the product of two odd integers is always odd.

Anahtar Kavram

Parity rules under exponentiation and algebraic combination

Tahmini Süre:2m 0s

Soru 327Soru →

If $x$ is a negative real number, which of the following expressions are equivalent to $\sqrt{-x^3}$ ? Select all such expressions.

Geçerli olan tümünü seçin

-x\sqrt{-x}

x\sqrt{-x}

|x|\sqrt{-x}

(-x)^{3/2}

-x^{3/2}

Cevabı ve açıklamayı göster

Cevap: $-x\sqrt{-x}$ ; $|x|\sqrt{-x}$ ; $(-x)^{3/2}$

Cevap

The expressions equivalent to

\sqrt{-x^3}

are

-x\sqrt{-x}

,

|x|\sqrt{-x}

, and

(-x)^{3/2}

.

Because

x

is negative,

-x

is a positive quantity. We can express

-x^3

as

(-x)^3 = (-x)^2 \cdot (-x)

. Taking the principal square root yields

\sqrt{(-x)^2 \cdot (-x)} = \sqrt{(-x)^2} \cdot \sqrt{-x} = -x\sqrt{-x}

. Because

|x| = -x

for negative numbers, the expression

|x|\sqrt{-x}

is identical to

-x\sqrt{-x}

. Furthermore, converting to rational exponents gives

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

. Thus, all three of these expressions are mathematically equivalent to the original radical expression.

Adım Adım Çözüm

1

Analyze the sign of the base and inside of the radical

Since

x < 0

, the quantity

-x

is strictly positive (

-x > 0

). Consequently,

-x^3 = (-x)^3 > 0

, ensuring

\sqrt{-x^3}

is a real, non-negative number.

Principal square roots require a non-negative radicand and yield a non-negative result in real arithmetic.

2

Simplify the radical expression using perfect squares

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

.

Because

-x > 0

,

\sqrt{(-x)^2} = -x

.

3

Evaluate equivalent representations using absolute value and rational exponents

Since

x < 0

,

|x| = -x

, so

|x|\sqrt{-x} = -x\sqrt{-x}

. Also,

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

.

Both rewrite rules preserve both magnitude and non-negative sign for all

x < 0

.

Anahtar Kavram

Simplifying radicals and fractional exponents with negative variable bases

Soru 328Soru →

A pharmaceutical facility uses two liquid compounding lines, Line $A$ and Line $B$ , to produce a standardized saline solution. Line $A$ operates at a constant flow rate of $120$ liters per hour, producing a solution with active compound and purified water in a ratio of $1 : 3$ by volume. Line $B$ operates at a constant flow rate of $180$ liters per hour, producing a solution with active compound and purified water in a ratio of $2 : 3$ by volume. If Line $A$ runs for $4$ hours and Line $B$ runs for $5$ hours, and all output is collected into a single storage tank, what is the ratio of active compound to purified water in the storage tank?

8 : 15

8 : 23

13 : 27

1 : 2

38 : 31

Cevabı ve açıklamayı göster

Cevap: $8 : 15$

Cevap

The ratio of active compound to purified water in the storage tank is

8 : 15

.

The correct answer is derived by first converting the flow rates and operating hours into total volumes (

480

L for Line A and

900

L for Line B). Using the part-to-whole fraction conversion, Line A contributes

120

L of active compound and

360

L of water, while Line B contributes

360

L of active compound and

540

L of water. Adding these quantities yields

480

L of active compound and

900

L of water, which simplifies to the ratio

8 : 15

.

Adım Adım Çözüm

1

Calculate total output volume and component quantities from Line A

Total Line A Volume =

120 \text{ L/hr} \times 4 \text{ hrs} = 480 \text{ liters}

. Active Compound =

480 \times \frac{1}{1+3} = 120 \text{ liters}

. Purified Water =

480 - 120 = 360 \text{ liters}

.

Line A produces a total volume of 480 liters, and a

1 : 3

ratio means active compound constitutes

\frac{1}{4}

of the total volume.

2

Calculate total output volume and component quantities from Line B

Total Line B Volume =

180 \text{ L/hr} \times 5 \text{ hrs} = 900 \text{ liters}

. Active Compound =

900 \times \frac{2}{2+3} = 360 \text{ liters}

. Purified Water =

900 - 360 = 540 \text{ liters}

.

Line B produces a total volume of 900 liters, and a

2 : 3

ratio means active compound constitutes

\frac{2}{5}

of the total volume.

3

Sum the components from both lines in the storage tank

Total Active Compound =

120 + 360 = 480 \text{ liters}

. Total Purified Water =

360 + 540 = 900 \text{ liters}

.

Combining the output of both lines adds the respective volumes of active compound and purified water.

4

Compute and simplify the final ratio of active compound to purified water

Ratio =

\frac{480}{900} = \frac{8}{15}

, expressed as

8 : 15

.

Dividing both numerator and denominator by their greatest common divisor, 60, yields the simplest integer ratio.

Anahtar Kavram

Weighted mixture ratios combining multi-stage rates and part-to-whole fraction conversions

Soru 329Soru →

A specialty tea shop sells five distinct sizes of loose-leaf tea canisters. The table below shows the weight per canister and the number of canisters sold during a promotional week:

Canister Size	Weight per Canister (ounces)	Number of Canisters Sold
Sampler	5	16
Small	12	18
Medium	15	$n$
Large	20	6
Executive	25	4

If the mean weight of all canisters sold during the week is $13$ ounces, what is the median weight, in ounces, of the canisters sold?

12

ounces

13

ounces

15

ounces

15.4

ounces

16

ounces

Cevabı ve açıklamayı göster

Cevap: $15$ ounces

Cevap

15 ounces

The correct answer is 15 ounces. Calculating the total weight as a function of

n

gives

516 + 15n

, and the total number of canisters as

44 + n

. Setting the mean

\frac{516 + 15n}{44 + n} = 13

yields

n = 28

. The total number of canisters sold is

44 + 28 = 72

. For 72 items sorted in ascending order, the median is the average of the 36th and 37th values. Cumulative frequencies show that the 1st through 16th items are 5 ounces, the 17th through 34th items are 12 ounces, and the 35th through 62nd items are 15 ounces. Thus, both the 36th and 37th items weigh 15 ounces, making the median 15 ounces.

Adım Adım Çözüm

1

Set up an equation for the total number of canisters and total weight.

Total canisters =

16 + 18 + n + 6 + 4 = 44 + n

. Total weight =

5(16) + 12(18) + 15(n) + 20(6) + 25(4) = 516 + 15n

ounces.

The mean of a weighted dataset is the total sum of all values divided by the total number of items.

2

Solve for the missing frequency

n

using the given mean of 13 ounces.

\frac{516 + 15n}{44 + n} = 13 \implies 516 + 15n = 13(44 + n) \implies 516 + 15n = 572 + 13n \implies 2n = 56 \implies n = 28

.

Equating the weighted average expression to 13 allows us to determine the exact number of Medium canisters sold.

3

Determine the total number of items and locate the position of the median.

Total canisters =

44 + 28 = 72

. The median is the average of the

36^{\text{th}}

and

37^{\text{th}}

values when ordered.

For an even number of data points

N = 72

, the median is the arithmetic mean of the

(N/2)^{\text{th}}

and

(N/2 + 1)^{\text{th}}

items.

4

Compute the cumulative frequency distribution to find the values at the

36^{\text{th}}

and

37^{\text{th}}

positions.

5 oz: positions 1 to 16; 12 oz: positions 17 to 34; 15 oz: positions 35 to 62. Both

36^{\text{th}}

and

37^{\text{th}}

positions have a weight of 15 ounces, so the median is 15 ounces.

Cumulative counting reveals that items in positions 35 through 62 all correspond to 15-ounce canisters.

Anahtar Kavram

Weighted mean and finding the median of frequency distribution data

Soru 330Soru →

The distribution of scores on Exam X is normally distributed with a mean of $70$ and a standard deviation of $8$ . The distribution of scores on Exam Y is normally distributed with a mean of $150$ and a standard deviation of $20$ . Candidate Score $A$ is on Exam X and has a z-score of $+1.5$ . Candidate Score $B$ is on Exam Y and corresponds to the 84th percentile of Exam Y.

Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

Candidate Score

A

is equal to

82

.

The proportion of test-takers on Exam Y scoring higher than Candidate Score

B

is approximately

16\%

.

A score of

62

on Exam X corresponds to a higher percentile rank than Candidate Score

B

on Exam Y.

A score of

86

on Exam X corresponds to the 99.7th percentile of Exam X.

A score of

110

on Exam Y has a z-score of

-2.0

.

Cevabı ve açıklamayı göster

Cevap: Candidate Score $A$ is equal to $82$ .; The proportion of test-takers on Exam Y scoring higher than Candidate Score $B$ is approximately $16\%$ .; A score of $110$ on Exam Y has a z-score of $-2.0$ .

Cevap

The statements asserting that Candidate Score

A

is equal to

82

, that approximately

16\%

of test-takers score higher than Candidate Score

B

, and that a score of

110

on Exam Y has a z-score of

-2.0

are all correct.

The correct statements correctly apply the z-score formula and standard normal curve benchmarks: Candidate Score

A

is

82

because

70 + 1.5(8) = 82

; the proportion of scores above the 84th percentile is

100\% - 84\% = 16\%

; and a score of

110

on Exam Y is

40

points below the mean of

150

, giving a z-score of

\frac{-40}{20} = -2.0

.

Adım Adım Çözüm

1

Calculate Candidate Score

A

on Exam X using the z-score formula.

A = \mu + z \cdot \sigma = 70 + (1.5)(8) = 82

.

A z-score of

+1.5

means the score is

1.5

standard deviations above the mean.

2

Analyze Candidate Score

B

at the 84th percentile of Exam Y.

Percentile rank of 84th means

84\%

score below or equal to

B

, so

100\% - 84\% = 16\%

score higher than

B

. Under the standard normal curve (empirical rule), the 84th percentile corresponds to

z = +1.0

, which gives

B = 150 + 1.0(20) = 170

.

Percentile rank measures the proportion of values below a given point.

3

Evaluate the percentile rank of a score of

62

on Exam X.

z = \frac{62 - 70}{8} = -1.0

. A z-score of

-1.0

corresponds to the 16th percentile (

50\% - 34\% = 16\%

).

Since the 16th percentile is less than the 84th percentile (Candidate Score

B

), this statement is false.

4

Evaluate the percentile rank of a score of

86

on Exam X.

z = \frac{86 - 70}{8} = +2.0

. A z-score of

+2.0

corresponds to the 97.5th percentile (

50\% + 34\% + 13.5\% = 97.5\%

).

The 99.7th percentile corresponds to

z = +3.0

(score of

94

), making this statement false.

5

Calculate the z-score for a score of

110

on Exam Y.

z = \frac{110 - 150}{20} = -2.0

.

The raw score

110

is

40

units below the mean of

150

, which is exactly

2

standard deviations below the mean.

Anahtar Kavram

Normal Distribution, Z-Scores, and Percentile Benchmarks

Tahmini Süre:2m 0s

Soru 331Soru →

If $x$ is a negative real number such that $x^2 + 5x - 14 = 0$ , what is the value of $x + 3$ ?

-10

-7

-4

2

5

Cevabı ve açıklamayı göster

Cevap: $-4$

Cevap

The value of

x + 3

is

-4

.

Factoring

x^2 + 5x - 14 = 0

yields

(x + 7)(x - 2) = 0

, giving solutions

x = -7

and

x = 2

. Given that

x

must be negative,

x = -7

. Substituting this into the target expression gives

-7 + 3 = -4

.

Adım Adım Çözüm

1

Factor the quadratic equation

x^2 + 5x - 14 = 0

.

(x + 7)(x - 2) = 0

Find two numbers that multiply to

-14

and add to

5

, which are

7

and

-2

.

2

Solve for the possible values of

x

.

x = -7

or

x = 2

Set each linear factor equal to zero.

3

Apply the condition that

x

is a negative real number.

x = -7

The value

x = 2

is positive and must be discarded.

4

Evaluate the target expression

x + 3

.

-7 + 3 = -4

Substitute

x = -7

into

x + 3

.

Anahtar Kavram

Factoring Quadratic Equations and Evaluating Expressions

Soru 332Soru →

An artisan bakery produces a signature flour blend by combining wheat, rye, and oat flour. Initially, the ratio of wheat flour to rye flour by weight is $5:3$ , and the ratio of rye flour to oat flour by weight is $4:1$ . The baker prepares an initial batch of this mixture weighing exactly $140$ pounds. To adjust the recipe for a special order, pure oat flour is added to the batch until oat flour accounts for exactly $20\%$ of the total weight of the new mixture. How many pounds of pure oat flour must the baker add?

12

16

20

28

32

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20 pounds

To solve this problem, first express the ratios of wheat, rye, and oat flour with a common term for rye. Since Wheat : Rye = 5 : 3 (or 20 : 12) and Rye : Oat = 4 : 1 (or 12 : 3), the combined ratio Wheat : Rye : Oat is 20 : 12 : 3, giving 35 total parts. In a 140-pound batch, each part equals 4 pounds, meaning the batch contains 12 pounds of oat flour and 128 pounds of non-oat flour. When additional oat flour is added, the non-oat weight stays fixed at 128 pounds. For oat flour to be 20% of the new mixture, non-oat flour must be 80%. Dividing 128 by 0.80 gives a new total weight of 160 pounds. The new oat weight is 20% of 160, which is 32 pounds. Subtracting the initial 12 pounds yields 20 pounds of added oat flour.

Adım Adım Çözüm

1

Combine the two given ratios into a single three-part ratio.

Wheat : Rye =

5 : 3 = 20 : 12

, and Rye : Oat =

4 : 1 = 12 : 3

. Thus, Wheat : Rye : Oat =

20 : 12 : 3

.

Scaling the ratios so that the common element (Rye) has equal parts allows unification into a single ratio.

2

Calculate the initial weight of each component in the 140-pound batch.

Total ratio parts =

20 + 12 + 3 = 35

parts. Each part is

140 / 35 = 4

pounds. Initial Oat flour =

3 \times 4 = 12

pounds. Non-oat flour (Wheat + Rye) =

(20 + 12) \times 4 = 128

pounds.

Determining the weight of the constant non-oat portion is key to solving mixture adjustment problems.

3

Set up an equation for the new mixture where oat flour makes up 20% of the total weight.

Since non-oat flour remains unchanged at

128

pounds, it represents

80\%

of the new total weight. New Total Weight =

128 / 0.80 = 160

pounds.

If oat flour is 20% of the new total, the non-oat components must constitute the remaining 80%.

4

Calculate the weight of oat flour added.

New Oat Weight =

20\%

of

160 = 32

pounds. Added Oat Flour =

32 - 12 = 20

pounds.

Subtracting the initial oat weight from the final required oat weight yields the amount added.

Anahtar Kavram

Combining multi-part ratios and solving ratio adjustment problems by holding unchanged components constant.

Soru 333Soru →

Two events $A$ and $B$ within a sample space have probabilities $P(A) = 0.60$ and $P(B) = 0.30$ . Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

If events

A

and

B

are independent, the probability that at least one of the two events occurs is

0.72

.

If events

A

and

B

are mutually exclusive, the joint probability

P(A \text{ and } B)

is

0

.

Events

A

and

B

can be both mutually exclusive and independent.

If events

A

and

B

are independent, the probability that both events occur is

0.90

.

If events

A

and

B

are mutually exclusive, the conditional probability

P(A \mid B)

is

0.30

.

Cevabı ve açıklamayı göster

Cevap: If events $A$ and $B$ are independent, the probability that at least one of the two events occurs is $0.72$ .; If events $A$ and $B$ are mutually exclusive, the joint probability $P(A \text{ and } B)$ is $0$ .

Cevap

The correct statements are that if events are independent, the probability that at least one occurs is 0.72, and if events are mutually exclusive, their joint probability is 0.

The statement regarding independent events is correct because

P(A \cup B) = P(A) + P(B) - P(A)P(B) = 0.60 + 0.30 - 0.18 = 0.72

. The statement regarding mutually exclusive events is correct because by definition, mutually exclusive events cannot occur together, meaning

P(A \text{ and } B) = 0

.

Adım Adım Çözüm

1

Analyze independence condition for

P(A \cup B)

P(A \cap B) = 0.60 \times 0.30 = 0.18

, so

P(A \cup B) = 0.60 + 0.30 - 0.18 = 0.72

.

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

2

Analyze mutual exclusivity definition

P(A \cap B) = 0

.

Mutually exclusive events cannot occur simultaneously.

3

Evaluate simultaneous independence and mutual exclusivity

Since

P(A)P(B) = 0.18 \neq 0

, the events cannot satisfy both conditions simultaneously.

Independence requires

P(A \cap B) = 0.18

, while mutual exclusivity requires

P(A \cap B) = 0

.

Anahtar Kavram

Probability rules for independent and mutually exclusive events

Soru 334Soru →

The frequency distribution table below shows the daily electricity consumption, in kilowatt-hours ( $\text{kWh}$ ), recorded for a sample of $40$ small retail stores over a monitoring period.

Daily Electricity Consumption ( $\text{kWh}$ )	Number of Stores
$10 \le e < 30$	$6$
$30 \le e < 50$	$14$
$50 \le e < 70$	$12$
$70 \le e < 90$	$8$

Based on the table, what is the estimated mean daily electricity consumption, in $\text{kWh}$ , per store, using the midpoint of each interval to represent all data within that interval?

Cevabı ve açıklamayı göster

Cevap: 51

Cevap

51

To estimate the mean from grouped frequency data, each interval is represented by its midpoint. The midpoints for the four intervals are

20

,

40

,

60

, and

80

. Multiplying each midpoint by its frequency gives

20(6) = 120

,

40(14) = 560

,

60(12) = 720

, and

80(8) = 640

. The sum of these products is

120 + 560 + 720 + 640 = 2040

. Dividing this sum by the total frequency of

40

stores gives an estimated mean of

51\text{ kWh}

.

Adım Adım Çözüm

1

Determine the class midpoint for each interval of electricity consumption.

Midpoints:

\frac{10+30}{2} = 20

,

\frac{30+50}{2} = 40

,

\frac{50+70}{2} = 60

, and

\frac{70+90}{2} = 80

.

When working with grouped frequency data, each interval is represented by its midpoint value to estimate total sums.

2

Multiply each midpoint by the number of stores in that interval to find the estimated sum of consumption per group.

20 \times 6 = 120

,

40 \times 14 = 560

,

60 \times 12 = 720

,

80 \times 8 = 640

.

Weighting each midpoint by its interval frequency yields the total estimated consumption across all stores in that bin.

3

Sum the total estimated consumption across all intervals.

120 + 560 + 720 + 640 = 2040 \text{ kWh}

.

This calculation aggregate gives the total estimated electricity consumed by all 40 stores combined.

4

Divide the total estimated electricity consumption by the total number of stores.

\frac{2040}{40} = 51 \text{ kWh}

.

Dividing the aggregate total by the sample size gives the estimated arithmetic mean per store.

Anahtar Kavram

Calculating the estimated mean of grouped data using class midpoints and weighted frequencies.

Tahmini Süre:1m 30s

Soru 335Soru →

For all non-zero real numbers

a

and

b

such that

a \neq b

and

a \neq -b

, which of the following expressions is equivalent to

\frac{a^{-2} - b^{-2}}{a^{-1} - b^{-1}} \div \frac{a^2 + ab + b^2}{a^3 - b^3}?

\frac{a^2 - b^2}{ab}

\frac{b^2 - a^2}{ab}

\frac{a - b}{a + b}

\frac{a + b}{ab(a - b)}

\frac{(a - b)^2}{ab}

Cevabı ve açıklamayı göster

Cevap: $\frac{a^2 - b^2}{ab}$

Cevap

\frac{a^2 - b^2}{ab}

Simplifying

\frac{a^{-2} - b^{-2}}{a^{-1} - b^{-1}}

yields

\frac{a + b}{ab}

after canceling

(b - a)

from both numerator and denominator. Simplifying

\frac{a^2 + ab + b^2}{a^3 - b^3}

yields

\frac{1}{a - b}

using the difference of cubes identity. Dividing

\frac{a + b}{ab}

by

\frac{1}{a - b}

gives

\frac{a + b}{ab} \cdot (a - b) = \frac{a^2 - b^2}{ab}

.

Adım Adım Çözüm

1

Simplify the first rational expression

\frac{a^{-2} - b^{-2}}{a^{-1} - b^{-1}}

\frac{\frac{1}{a^2} - \frac{1}{b^2}}{\frac{1}{a} - \frac{1}{b}} = \frac{\frac{b^2 - a^2}{a^2 b^2}}{\frac{b - a}{ab}} = \frac{(b - a)(b + a)}{a^2 b^2} \cdot \frac{ab}{b - a} = \frac{a + b}{ab}

Convert negative exponents to fractions, find common denominators, and cancel the common non-zero factor

(b - a)

.

2

Simplify the second rational expression

\frac{a^2 + ab + b^2}{a^3 - b^3}

\frac{a^2 + ab + b^2}{(a - b)(a^2 + ab + b^2)} = \frac{1}{a - b}

Factor the denominator using the difference of cubes identity

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

.

3

Divide the simplified first expression by the simplified second expression

\frac{a + b}{ab} \div \frac{1}{a - b} = \frac{a + b}{ab} \cdot (a - b) = \frac{(a + b)(a - b)}{ab} = \frac{a^2 - b^2}{ab}

Multiply by the reciprocal of the second expression and apply the difference of squares identity.

Anahtar Kavram

Simplifying complex fractions and factoring using difference of squares and difference of cubes identities.

Soru 336Soru →

A financial forecasting model predicts monthly sales revenue using a sequence where the initial revenue in month 1 is $R_1 = \$80{}000$ , and the revenue for each subsequent month $n$ (for $n \ge 2$ ) is calculated using the recursive formula $R_n = 0.75 R_{n-1} + 5{}000$ . If each monthly revenue figure $R_n$ is rounded to the nearest hundred dollars before summing, what is the estimated total revenue, in dollars, for the first 4 months combined?

Cevabı ve açıklamayı göster

Cevap: 244100

Cevap

The estimated total revenue for the first 4 months combined is 244,100 dollars.

By evaluating the recursive formula

R_n = 0.75 R_{n-1} + 5{}000

for months 1 through 4, we obtain

R_1 = 80{}000

,

R_2 = 65{}000

,

R_3 = 53{}750

, and

R_4 = 45{}312.50

. Rounding each to the nearest hundred yields

80{}000

,

65{}000

,

53{}800

, and

45{}300

, respectively. Summing these four rounded figures gives

244{}100

.

Adım Adım Çözüm

1

Find the first term

R_1

and its rounded value.

R_1 = 80{}000

, which rounds to

80{}000

.

Given initial term.

2

Calculate the second term

R_2

using the recursive formula and round it.

R_2 = 0.75(80{}000) + 5{}000 = 65{}000

, which rounds to

65{}000

.

Apply

R_2 = 0.75 R_1 + 5{}000

.

3

Calculate the third term

R_3

using unrounded

R_2

and round to the nearest hundred.

R_3 = 0.75(65{}000) + 5{}000 = 53{}750

, which rounds to

53{}800

.

Apply

R_3 = 0.75 R_2 + 5{}000

and round to the nearest 100.

4

Calculate the fourth term

R_4

using unrounded

R_3

and round to the nearest hundred.

R_4 = 0.75(53{}750) + 5{}000 = 45{}312.50

, which rounds to

45{}300

.

Apply

R_4 = 0.75 R_3 + 5{}000

and round to the nearest 100.

5

Add the four rounded terms together.

80{}000 + 65{}000 + 53{}800 + 45{}300 = 244{}100

.

Compute total estimated revenue as specified.

Anahtar Kavram

Estimation, Rounding, and Sequences

Soru 337Soru →

A survey of $200$ financial executives analyzed their personal investment portfolios across three asset classes: Equities ( $E$ ), Real Estate ( $R$ ), and Commodities ( $C$ ). Exactly $20$ executives hold no investments in any of these three asset classes. Among the remaining executives, $115$ hold Equities, $95$ hold Real Estate, and $80$ hold Commodities. Furthermore, $45$ executives hold both Equities and Real Estate, $35$ hold both Real Estate and Commodities, and $40$ hold both Equities and Commodities. How many of the surveyed executives hold investments in exactly two of these three asset classes?

80

90

100

110

120

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

90

To find the number of executives holding investments in exactly two asset classes, we first determine the total number of executives invested in at least one asset class:

200 - 20 = 180

. Applying the three-set inclusion-exclusion formula,

180 = 115 + 95 + 80 - (45 + 35 + 40) + |E \cap R \cap C|

, which simplifies to

180 = 170 + |E \cap R \cap C|

, giving

|E \cap R \cap C| = 10

. The number of executives holding exactly two asset classes is calculated by subtracting the triple intersection (10) from each pairwise overlap:

(45 - 10) + (35 - 10) + (40 - 10) = 35 + 25 + 30 = 90

.

Adım Adım Çözüm

1

Calculate the total number of executives holding at least one asset class.

The total number in the union

|E \cup R \cup C| = 200 - 20 = 180

.

Subtracting executives holding no investments gives the cardinality of the union.

2

Use the Principle of Inclusion-Exclusion for three sets to find the triple intersection

|E \cap R \cap C|

.

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

, so

180 = 115 + 95 + 80 - (45 + 35 + 40) + |E \cap R \cap C|

, yielding

180 = 290 - 120 + |E \cap R \cap C| = 170 + |E \cap R \cap C|

, hence

|E \cap R \cap C| = 10

.

The inclusion-exclusion formula relates the union, individual set sizes, pairwise overlaps, and the triple overlap.

3

Determine the number of executives in each pairwise intersection that belong ONLY to those two sets.

Equities and Real Estate only:

45 - 10 = 35

; Real Estate and Commodities only:

35 - 10 = 25

; Equities and Commodities only:

40 - 10 = 30

.

Subtracting the triple intersection count from each pairwise intersection isolates those holding exactly two asset classes.

4

Sum the counts for executives holding exactly two asset classes.

35 + 25 + 30 = 90

.

Adding these disjoint sets yields the total number of executives in exactly two categories.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Decomposition

Tahmini Süre:2m 0s

Soru 338Soru →

An investor divides a total of $\$15,000$ between two accounts, Account P and Account Q. Account P earns an annual simple interest rate of $6\%$ , while Account Q earns an annual simple interest rate of $8\%$ . If the total interest earned from both accounts combined after one year is $\$1,020$ , how much more money was invested in Account P than in Account Q?

\$1,500

\$3,000

\$4,500

\$6,000

\$9,000

Cevabı ve açıklamayı göster

Cevap: $\$3,000$

Cevap

\$3,000

Setting up the system of linear equations

p + q = 15,000

and

0.06p + 0.08q = 1,020

yields

p = 9,000

and

q = 6,000

. Subtracting the amount in Account Q from Account P gives

9,000 - 6,000 = 3,000

. Thus,

\$3,000

is the correct answer.

Adım Adım Çözüm

1

Define variables and set up the system of linear equations

Let

p

be the amount invested in Account P and

q

be the amount invested in Account Q. The equations are

p + q = 15,000

and

0.06p + 0.08q = 1,020

.

The first equation represents the total investment amount, and the second equation represents the total annual interest earned.

2

Simplify the interest equation and solve for one variable using elimination

Multiply

0.06p + 0.08q = 1,020

by 100 to obtain

6p + 8q = 102,000

, which simplifies to

3p + 4q = 51,000

. Multiplying the first equation by 3 yields

3p + 3q = 45,000

. Subtracting this from

3p + 4q = 51,000

gives

q = 6,000

.

Eliminating

p

allows for direct calculation of the amount invested in Account Q.

3

Calculate the amount invested in Account P

p = 15,000 - 6,000 = 9,000

.

Substitute the value of

q

back into the total investment equation.

4

Determine the requested difference

p - q = 9,000 - 6,000 = 3,000

.

The question specifically asks how much more money was invested in Account P than in Account Q.

Anahtar Kavram

Solving Systems of 2x2 Linear Equations for Word Problems

Soru 339Soru →

Which of the following values are solutions to the equation $(x + 4)^2 = 49$ ? Select all that apply.

Geçerli olan tümünü seçin

-11

-3

3

7

\sqrt{33}

Cevabı ve açıklamayı göster

Cevap: $-11$ ; $3$

Cevap

The correct solutions are

-11

and

3

.

Taking the square root of both sides of

(x + 4)^2 = 49

yields

x + 4 = 7

or

x + 4 = -7

. Solving these two linear equations gives

x = 3

and

x = -11

. Both values satisfy the original quadratic equation.

Adım Adım Çözüm

1

Apply the square root property to both sides of the quadratic equation.

x + 4 = \pm \sqrt{49} = \pm 7

Taking the square root of both sides accounts for both the positive and negative roots of

49

.

2

Solve the first linear equation for the positive case.

x + 4 = 7 \implies x = 7 - 4 = 3

Subtract

4

from both sides when the expression equals

+7

.

3

Solve the second linear equation for the negative case.

x + 4 = -7 \implies x = -7 - 4 = -11

Subtract

4

from both sides when the expression equals

-7

.

Anahtar Kavram

Square Root Property of Quadratic Equations

Soru 340Soru →

In an industrial mechanical system, Gear $A$ has $16$ teeth, Gear $B$ has $24$ teeth, and Gear $C$ has $40$ teeth. Gear $A$ is meshed directly with Gear $B$ , and Gear $B$ is meshed directly with Gear $C$ . If Gear $A$ rotates at a constant speed of $150$ revolutions per minute ( $\text{rpm}$ ), what is the rotational speed, in revolutions per minute ( $\text{rpm}$ ), of Gear $C$ ?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The rotational speed of Gear C is 60 rpm.

For meshed gears, the linear speed of teeth at the point of contact must be identical. Thus, the product of the number of teeth and rotational speed remains constant (

N_A S_A = N_C S_C

). Substituting the known values gives

16 \times 150 = 40 \times S_C

, yielding

2,400 = 40 S_C

, so

S_C = 60

rpm.

Adım Adım Çözüm

1

Determine the inverse proportional relationship between number of gear teeth and rotational speed.

The product of teeth count and rotational speed is constant across directly meshed gears:

N_A \times S_A = N_B \times S_B = N_C \times S_C

.

Directly meshed gears engage tooth for tooth, meaning they pass the same total number of teeth per unit time.

2

Calculate the total tooth displacement rate per minute from Gear A.

16 \times 150 = 2,400

teeth per minute.

Gear A has 16 teeth and completes 150 revolutions per minute.

3

Calculate the rotational speed of Gear C.

\text{Speed}_C = \frac{2,400}{40} = 60

rpm.

Gear C has 40 teeth, so dividing the total tooth displacement rate by 40 yields its revolutions per minute.

Anahtar Kavram

Inverse Proportionality in Gear Rates

Tahmini Süre:1m 15s

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