Tüm alıştırma soruları

Soru 101Soru →

In a medical study involving $300$ clinical trial participants, researchers recorded the occurrence of three common side effects: fatigue, nausea, and insomnia. A total of $160$ participants reported fatigue, $140$ reported nausea, and $120$ reported insomnia. Additionally, $60$ participants reported both fatigue and nausea, $50$ reported both fatigue and insomnia, and $55$ reported both nausea and insomnia. If $30$ participants reported none of these three side effects, how many participants reported exactly two of these side effects?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120 participants reported exactly two of the side effects.

To calculate the number of participants with exactly two side effects, first find the total number of participants experiencing at least one side effect (

300 - 30 = 270

). Next, using the inclusion-exclusion formula

270 = 160 + 140 + 120 - (60 + 50 + 55) + x

, we find that

x = 15

participants experienced all three side effects. Finally, subtracting

15

from each pairwise overlap gives the counts for exactly two side effects:

45

(Fatigue & Nausea only),

35

(Fatigue & Insomnia only), and

40

(Nausea & Insomnia only). Summing these regions yields

45 + 35 + 40 = 120

.

Adım Adım Çözüm

1

Calculate the total number of participants who experienced at least one side effect

270 participants

Subtracting the 30 participants who reported no side effects from the total study size of 300 gives

|F \cup N \cup I| = 300 - 30 = 270

.

2

Determine the number of participants who experienced all three side effects

15 participants

Applying inclusion-exclusion gives

270 = 160 + 140 + 120 - (60 + 50 + 55) + x

, simplifying to

270 = 255 + x

, so

x = 15

.

3

Calculate the sum of participants experiencing exactly two side effects

120 participants

The sum of pairwise intersections (

60 + 50 + 55 = 165

) counts individuals with all three side effects three times. Subtracting

3 \times 15 = 45

from

165

yields

120

.

Anahtar Kavram

Three-Set Overlapping Sets and Inclusion-Exclusion Principle

Tahmini Süre:2m 0s

Soru 102Soru →

A set of $7$ positive integers has an arithmetic mean of $16$ , a median of $15$ , a unique mode of $12$ , and a range of $14$ . What is the maximum possible value of the largest integer in this set?

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

The maximum possible value of the largest integer in the set is 26.

The total sum of the 7 positive integers is

7 \times 16 = 112

. Ordering the terms as

a \le b \le c \le d \le e \le f \le g

, the median constraint gives

d = 15

. The range constraint gives

g - a = 14

, or

g = a + 14

. To maximize

g

, we need to maximize

a

. Because 12 is the unique mode of the set, 12 must appear at least twice. Since

d = 15

, the number 12 can only occupy positions

a, b,

or

c

, which means

a \le 12

. Setting

a = 12

gives the maximum value

g = 12 + 14 = 26

. A valid set achieving this is

\{12, 12, 12, 15, 15, 20, 26\}

, which sums to 112 and meets all statistical constraints.

Adım Adım Çözüm

1

Determine the sum of the set

Sum = 112

The mean of 7 positive integers is 16, so the sum is 7 times 16.

2

Apply median and range constraints

d = 15 and a = g - 14

In a sorted set of 7 integers, the 4th element is the median (15), and range is the difference between the largest element g and smallest element a.

3

Bound the smallest element using the mode constraint

a <= 12, so max g = 12 + 14 = 26

Since 12 is the unique mode, it must occur at least twice. Because elements are sorted and median is 15, 12 must be among the first three terms, so the smallest element a cannot exceed 12.

4

Verify existence of a valid set with g = 26

Set {12, 12, 12, 15, 15, 20, 26} satisfies all conditions

The sum is 112, median is 15, mode is 12 (appears 3 times), and range is 26 - 12 = 14.

Anahtar Kavram

Extremal problems involving mean, median, mode, and range constraints

Soru 103Soru →

A logistics firm generates a 4-character tracking code, $c_1 c_2 c_3 c_4$ , for express parcels according to the following rules:
- The first character, $c_1$ , must be a non-zero digit ( $1$ through $9$ ).
- The second character, $c_2$ , must be a vowel chosen from the set $\{\text{A, E, I, O, U}\}$ .
- The third character, $c_3$ , must be an even digit ( $0, 2, 4, 6, 8$ ).
- The fourth character, $c_4$ , must be a consonant chosen from the set $\{\text{B, F, G, K, M, P}\}$ .

If no digit may be repeated within the code, how many different tracking codes can be created?

Cevabı ve açıklamayı göster

Cevap: 1230

Cevap

1230

To find the total number of unique tracking codes, calculate the independent choices for letters and dependent choices for digits. For digits

c_1

and

c_3

, if

c_1

is odd (

1, 3, 5, 7, 9

), there are

5 \times 5 = 25

ways to choose

c_3

. If

c_1

is even (

2, 4, 6, 8

), there are

4 \times 4 = 16

ways to choose

c_3

, giving

41

valid digit pairs. Multiplying by the

5

vowel options for

c_2

and

6

consonant options for

c_4

yields

41 \times 5 \times 6 = 1230

.

Adım Adım Çözüm

1

Determine valid digit choices for

c_1

and

c_3

considering the non-repetition constraint

Split into Case 1 (

c_1

is odd) and Case 2 (

c_1

is even)

The availability of even digits for

c_3

depends on whether

c_1

uses an even or odd digit.

2

Calculate the number of ways to pick

c_1

and

c_3

Case 1 (

c_1 \in \{1, 3, 5, 7, 9\}

):

5 \times 5 = 25

ways. Case 2 (

c_1 \in \{2, 4, 6, 8\}

):

4 \times 4 = 16

ways. Total digit combinations =

25 + 16 = 41

.

If

c_1

is odd, all 5 even digits remain for

c_3

. If

c_1

is even, only 4 even digits remain for

c_3

.

3

Calculate the number of ways to pick letters

c_2

and

c_4

5 choices for

c_2

and 6 choices for

c_4

, giving

5 \times 6 = 30

letter combinations.

Vowels and consonants are chosen independently from distinct sets without restrictions between them.

4

Multiply independent stage outcomes using the Fundamental Counting Principle

41 \times 30 = 1230

The overall number of distinct codes is the product of digit pair combinations and letter pair combinations.

Anahtar Kavram

Fundamental Counting Principle with Conditional Restrictions

Soru 104Soru →

Data set $S$ consists of the four integers $4, 8, 12,$ and $16$ . A fifth integer $k$ , which is greater than the mean of data set $S$ , is added to the set to form data set $T = \{4, 8, 12, 16, k\}$ . If the standard deviation of data set $T$ is equal to the standard deviation of data set $S$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The value of

k

is 15.

The variance of the original set

S = \{4, 8, 12, 16\}

is 20 with a mean of 10. Adding

k

creates a 5-element set

T

with mean

\frac{40+k}{5}

. Setting the variance of set

T

equal to 20 yields the quadratic equation

k^2 - 20k + 75 = 0

, which factors as

(k-5)(k-15) = 0

. Since

k

must be greater than 10, the correct answer is 15.

Adım Adım Çözüm

1

Calculate the mean and variance of data set

S

.

Mean

\mu_S = 10

and variance

\sigma_S^2 = 20

.

Determining the baseline mean and variance for the initial dataset of 4 elements.

2

Express the mean and variance of data set

T = \{4, 8, 12, 16, k\}

algebraically in terms of

k

.

Mean

\mu_T = \frac{40+k}{5}

and variance

\sigma_T^2 = \frac{480 + k^2 - 5\left(\frac{40+k}{5}\right)^2}{5}

.

Formulating the standard deviation equation for the expanded dataset.

3

Equate

\sigma_T^2

to

\sigma_S^2 = 20

and solve for

k

.

The resulting quadratic equation

k^2 - 20k + 75 = 0

yields solutions

k = 5

and

k = 15

.

Finding all potential values of

k

that leave the dataset's standard deviation unchanged.

4

Apply the condition that

k

must be strictly greater than the mean of data set

S

(

\mu_S = 10

).

k = 15

.

Rejecting

k = 5

because it is less than 10.

Anahtar Kavram

Standard deviation measures the average distance of data points from the mean. Adding a value to a dataset shifts the mean and alters variance; finding when the standard deviation remains unchanged requires solving the quadratic relationship generated by the addition of the new element.

Soru 105Soru →

For real numbers $p$ , $q$ , and $r$ , it is given that $p < 0 < q < r$ . If $p^3 q r = -288$ and $\frac{q r}{p^2} = 9$ , what is the value of $p + q r$ ?

Cevabı ve açıklamayı göster

Cevap: 34

Cevap

The value of

p + q r

is 34.

From

\frac{qr}{p^2} = 9

, we get

qr = 9p^2

. Substituting this into

p^3 qr = -288

yields

9p^5 = -288

, so

p^5 = -32

. Taking the fifth root gives

p = -2

, which satisfies

p < 0

. Then

qr = 9(-2)^2 = 36

. Finally,

p + qr = -2 + 36 = 34

.

Adım Adım Çözüm

1

Relate

qr

to

p

using the given quotient equality

qr = 9p^2

Multiplying both sides of

\frac{qr}{p^2} = 9

by

p^2

isolates

qr

.

2

Substitute

qr = 9p^2

into the product equation

p^3 (9p^2) = -288 \implies 9p^5 = -288 \implies p^5 = -32

Replacing

qr

with

9p^2

produces a single-variable polynomial in

p

.

3

Solve for

p

enforcing the sign constraint

p < 0

p = -2

Taking the fifth root of

-32

yields

-2

, which satisfies

p < 0

.

4

Calculate the value of

qr

qr = 9(-2)^2 = 36

Squaring a negative number yields a positive value:

(-2)^2 = 4

, so

9 \times 4 = 36

.

5

Evaluate the expression

p + qr

p + qr = -2 + 36 = 34

Adding

p = -2

and

qr = 36

results in

34

.

Anahtar Kavram

Positive and Negative Number Properties

Soru 106Soru →

A chemical processing plant prepares a batch of solution by mixing two stock liquid compounds, Solution A and Solution B, to form an initial mixture with a total mass of $480\text{ kg}$ . Solution A contains $25\%$ active compound by mass, and Solution B contains $5\%$ active compound by mass. Solution A and Solution B are combined in a mass ratio of $3:1$ , respectively.

To satisfy export specifications, technicians must raise the active compound concentration of the entire mixture to $25\%$ by mass by adding a quantity of pure active compound. How many kilograms of pure active compound must be added to the $480\text{ kg}$ mixture?

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

32 kg of pure active compound must be added to reach a final concentration of 25% active compound by mass.

The initial mixture consists of 360 kg of Solution A and 120 kg of Solution B. This contains a total of 96 kg of active compound (90 kg from A and 6 kg from B). Adding 32 kg of pure active compound brings the total active compound to 128 kg and total mixture mass to 512 kg, yielding an exact concentration of 128 / 512 = 25%.

Adım Adım Çözüm

1

Calculate the mass of Solution A and Solution B in the initial mixture

Solution A = 360 kg, Solution B = 120 kg

The total mass of 480 kg is divided according to the 3:1 ratio (3/4 and 1/4).

2

Calculate the mass of pure active compound in the initial mixture

Initial active compound = 96 kg

Solution A provides 25% of 360 kg (90 kg) and Solution B provides 5% of 120 kg (6 kg).

3

Set up an equation for the final concentration after adding x kg of active compound

(96 + x) / (480 + x) = 1/4

Adding pure active compound increases both the mass of the active compound and the total mass of the mixture by x kg.

4

Solve for x

x = 32 kg

Cross-multiplying gives 4(96 + x) = 480 + x, which simplifies to 3x = 96.

Anahtar Kavram

Weighted average mixtures and algebraic optimization of concentration ratios

Soru 107Soru →

In 2025, an architectural design firm divided its working hours among commercial, residential, and municipal projects. Commercial projects accounted for $0.40$ of the total hours. Of the remaining hours, $\frac{5}{12}$ were spent on residential projects, and the rest were spent on municipal projects.

In 2026, the firm's total working hours increased by $25\%$ . Commercial project hours decreased by $15\%$ , while residential project hours increased by $40\%$ . By what percent did the number of hours spent on municipal projects increase from 2025 to 2026?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The number of hours spent on municipal projects increased by

60\%

.

By setting the total initial hours to a variable

T

(or a convenient constant like

100

), we find that in 2025, commercial projects comprised

0.40T

, residential comprised

\frac{5}{12} \times 0.60T = 0.25T

, and municipal comprised

0.35T

. In 2026, total hours rose to

1.25T

, commercial dropped to

0.34T

, and residential rose to

0.35T

, leaving

0.56T

for municipal. The percent increase in municipal hours is

\frac{0.56T - 0.35T}{0.35T} = \frac{0.21}{0.35} = 60\%

.

Adım Adım Çözüm

1

Determine the baseline breakdown of working hours for 2025 in terms of total hours

T

.

Commercial hours =

0.40T

, Residential hours =

0.25T

, Municipal hours =

0.35T

.

Commercial is given as

0.40T

. The remaining fraction

0.60T

is split such that

\frac{5}{12} \times 0.60T = 0.25T

goes to residential, leaving

0.60T - 0.25T = 0.35T

for municipal.

2

Calculate the updated working hours for each category in 2026.

Total hours =

1.25T

, Commercial hours =

0.34T

, Residential hours =

0.35T

.

A

25\%

total increase yields

1.25T

. A

15\%

decrease in commercial hours yields

0.40T \times 0.85 = 0.34T

. A

40\%

increase in residential hours yields

0.25T \times 1.40 = 0.35T

.

3

Find the municipal project hours for 2026 by subtracting commercial and residential hours from total 2026 hours.

Municipal hours (2026) =

0.56T

.

Municipal hours in 2026 equal

1.25T - 0.34T - 0.35T = 0.56T

.

4

Compute the percent change in municipal project hours from 2025 to 2026.

Percent Increase =

60\%

.

Percent increase is

\frac{\text{Municipal}_{2026} - \text{Municipal}_{2025}}{\text{Municipal}_{2025}} \times 100\% = \frac{0.56T - 0.35T}{0.35T} \times 100\% = \frac{0.21}{0.35} \times 100\% = 60\%

.

Anahtar Kavram

Multi-step percentage change and fraction-decimal conversions

Tahmini Süre:2m 0s

Soru 108Soru →

At the start of a fiscal year, a sovereign wealth fund allocated its capital into three asset classes: Stocks, Bonds, and Commodities. Exactly $0.40$ of the total capital was allocated to Stocks. Of the remaining capital, $\frac{1}{3}$ was allocated to Bonds and the rest was allocated to Commodities. Over the course of the fiscal year, the value of Stocks increased by $15\%$ and the value of Bonds decreased by $10\%$ . If the total value of the fund increased by $5\%$ by the end of the fiscal year, by what percent did the value of Commodities change?

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

The value of Commodities increased by

2.5\%

.

By setting the initial total capital to

V

, the initial values are Stocks =

0.40V

, Bonds =

\frac{1}{3}(0.60V) = 0.20V

, and Commodities =

0.60V - 0.20V = 0.40V

. Applying the percentage changes gives final values of Stocks =

0.46V

, Bonds =

0.18V

, and Total Fund =

1.05V

. The final Commodities value is

1.05V - 0.64V = 0.41V

. The percent change in Commodities is

\frac{0.41V - 0.40V}{0.40V} \times 100\% = 2.5\%

.

Adım Adım Çözüm

1

Determine initial asset allocations as proportions of total capital

V

Stocks (

S_0

) =

0.40V

, Bonds (

B_0

) =

0.20V

, Commodities (

C_0

) =

0.40V

Stocks take

0.40V

, leaving

0.60V

. Bonds take

\frac{1}{3}

of

0.60V = 0.20V

, leaving

0.40V

for Commodities.

2

Compute final values for Stocks, Bonds, and Total Fund after percentage changes

S_1 = 0.46V

,

B_1 = 0.18V

, and Total Fund

V_1 = 1.05V

Stocks increase by

15\%

(

0.40V \times 1.15 = 0.46V

), Bonds decrease by

10\%

(

0.20V \times 0.90 = 0.18V

), and the total fund grows by

5\%

(

1.05V

).

3

Calculate the end-of-year value of Commodities

C_1 = 0.41V

Subtract final Stocks and Bonds from total fund:

C_1 = 1.05V - (0.46V + 0.18V) = 0.41V

.

4

Calculate the percent change of Commodities relative to its initial value

2.5\%

increase

Percent change

= \frac{0.41V - 0.40V}{0.40V} \times 100\% = \frac{0.01}{0.40} \times 100\% = 2.5\%

.

Anahtar Kavram

Weighted percentage change and fractional portion modeling

Tahmini Süre:1m 40s

Soru 109Soru →

A commercial coffee roastery prepares a batch of specialty coffee beans containing Arabica, Robusta, and Liberica beans. Initially, Arabica beans make up $0.50$ of the total weight of the batch, Robusta beans make up $\frac{3}{10}$ of the total weight, and Liberica beans make up the remaining weight. After a specialized roasting process, the weight of the Arabica beans decreases by $10\%$ , the weight of the Robusta beans decreases by $50\%$ , and the weight of the Liberica beans remains unchanged. What percent of the final total weight of the coffee batch is made up of Liberica beans?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25%

To find the percentage of Liberica beans in the final mixture, first determine the initial weight breakdown assuming a total weight of 100 units: Arabica is 50 units, Robusta is 30 units, and Liberica is 20 units. After roasting, Arabica weight decreases by 10% to 45 units, Robusta weight decreases by 50% to 15 units, and Liberica weight stays at 20 units. The new total weight is 45 + 15 + 20 = 80 units. The proportion of Liberica in the final batch is 20 out of 80 units, which equals 1/4 or 25%.

Adım Adım Çözüm

1

Find the initial fractional and percentage composition of the coffee blend.

Arabica accounts for 50% (0.50), Robusta accounts for 30% (3/10), and Liberica accounts for 20% (1 - 0.50 - 0.30 = 0.20).

The sum of all components in the initial blend must equal 1 (or 100%).

2

Assume an initial reference weight of 100 units to represent the batch.

Initial Arabica weight = 50 units, initial Robusta weight = 30 units, and initial Liberica weight = 20 units.

Using a convenient base value like 100 simplifies multi-step percentage change calculations without loss of generality.

3

Calculate the post-roasting weight for each component.

New Arabica weight = 50 * (1 - 0.10) = 45 units. New Robusta weight = 30 * (1 - 0.50) = 15 units. New Liberica weight = 20 units.

Apply the respective percentage decreases to each individual component weight.

4

Determine the new total weight of the batch.

Total final weight = 45 + 15 + 20 = 80 units.

The final total weight is the sum of the remaining weights of all three bean types.

5

Compute the final percentage of Liberica beans.

(20 / 80) * 100% = 25%.

Divide the final weight of Liberica beans by the final total weight of the batch and convert to a percentage.

Anahtar Kavram

Combining initial fractions and percentages to compute multi-step composition changes

Soru 110Soru →

What is the remainder when $17^{17^{17}}$ is divided by $7$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The remainder when

17^{17^{17}}

is divided by

7

is

5

.

First, reduce the base modulo

7

:

17 \equiv 3 \pmod 7

, turning the expression into

3^{17^{17}} \pmod 7

. Next, evaluate the pattern of powers of

3 \pmod 7

:

3^1 \equiv 3

,

3^2 \equiv 2

,

3^3 \equiv 6

,

3^4 \equiv 4

,

3^5 \equiv 5

, and

3^6 \equiv 1

, showing a period of

6

. To find which term of the cycle corresponds to the exponent

17^{17}

, evaluate

17^{17} \pmod 6

. Since

17 \equiv -1 \pmod 6

,

17^{17} \equiv (-1)^{17} = -1 \equiv 5 \pmod 6

. Finally, the

5

th term of the cycle gives

3^5 = 243 \equiv 5 \pmod 7

. Therefore, the remainder is

5

.

Adım Adım Çözüm

1

Reduce the base modulo 7

17 \equiv 3 \pmod 7

, so

17^{17^{17}} \equiv 3^{17^{17}} \pmod 7

.

Modular arithmetic permits replacing the base of an exponential expression with its remainder upon division by the modulus.

2

Determine the cyclicity period of powers of 3 modulo 7

The remainders of

3^n \pmod 7

repeat in a cycle of length 6:

(3, 2, 6, 4, 5, 1)

.

By Fermat's Little Theorem,

3^6 \equiv 1 \pmod 7

, meaning the sequence of remainders repeats every 6 integer powers.

3

Evaluate the exponent

17^{17}

modulo the period length 6

17 \equiv -1 \pmod 6

, so

17^{17} \equiv (-1)^{17} = -1 \equiv 5 \pmod 6

.

The position within the 6-term cyclicity pattern depends on the exponent modulo 6. Using negative remainders simplifies calculating odd powers of

-1

.

4

Calculate the final remainder using the 5th position in the cyclicity pattern

3^5 = 243 = 7 \times 34 + 5 \equiv 5 \pmod 7

.

Since the exponent leaves a remainder of 5 when divided by 6, the overall remainder corresponds to

3^5 \pmod 7

.

Anahtar Kavram

Modular cyclicity and tower exponent remainder reduction

Soru 111Soru →

For all real numbers

x

except

0

and

1

, the function

f

satisfies the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x + 3

What is the value of

f(2)

?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

f(2)

is

7

.

Evaluating the functional equation iteratively at the points of the cycle

x = 2

,

x = -1

, and

x = \frac{1}{2}

produces a system of three linear equations in terms of

f(2)

,

f(-1)

, and

f\left(\frac{1}{2}\right)

. Solving this linear system gives

f(2) = 7

.

Adım Adım Çözüm

1

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) + 3 \implies f(2) + 2f(-1) = 21

This establishes a relationship between

f(2)

and

f(-1)

.

2

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) + 3 \implies f(-1) + 2f\left(\frac{1}{2}\right) = -6

This links

f(-1)

to

f\left(\frac{1}{2}\right)

to build a closed system of transformations under the mapping

g(x) = \frac{1}{1-x}

.

3

Evaluate the functional equation at

x = \frac{1}{2}

f\left(\frac{1}{2}\right) + 2f\left(\frac{1}{1-1/2}\right) = 9\left(\frac{1}{2}\right) + 3 \implies f\left(\frac{1}{2}\right) + 2f(2) = \frac{15}{2}

This completes the cycle since

g\left(\frac{1}{2}\right) = 2

, creating a system of three linear equations with three unknown values.

4

Solve the system of three linear equations for

f(2)

Let

a = f(2)

,

b = f(-1)

, and

c = f\left(\frac{1}{2}\right)

. From equation (3),

c = \frac{15}{2} - 2a

. Substituting

c

into equation (2) gives

b + 2\left(\frac{15}{2} - 2a\right) = -6 \implies b = 4a - 21

. Substituting

b

into equation (1) gives

a + 2(4a - 21) = 21 \implies 9a - 42 = 21 \implies 9a = 63 \implies a = 7

.

Algebraic substitution yields the exact value of

f(2)

.

Anahtar Kavram

Cyclic Functional Equations and System Substitution

Tahmini Süre:2m 30s

Soru 112Soru →

An integer $n$ is selected at random from the set $\{1, 2, 3, \dots, 500\}$ . What is the probability that $n$ is a perfect square that leaves a remainder of $1$ when divided by $3$ ?

Cevabı ve açıklamayı göster

Cevap: 0.03

Cevap

0.03

The sample space consists of 500 equally likely outcomes. Perfect squares in this range are generated by integers

k

from 1 to 22. Evaluating

k^2 \pmod 3

shows that

k^2

has a remainder of 1 when divided by 3 for all integers

k

not divisible by 3. Excluding the 7 multiples of 3 from the 22 possible values of

k

leaves 15 favorable integers. Dividing the 15 favorable outcomes by the 500 total outcomes gives a probability of 0.03.

Adım Adım Çözüm

1

Determine total possible outcomes.

The total number of integers in the set

\{1, 2, 3, \dots, 500\}

is

N = 500

.

Each integer in the set is equally likely to be selected.

2

Find the range of square roots for perfect squares in the set.

Since

22^2 = 484 \le 500

and

23^2 = 529 > 500

,

n = k^2

for

k \in \{1, 2, \dots, 22\}

.

There are 22 perfect squares in the given sample space.

3

Apply remainder constraints to

k^2

.

A square

k^2

leaves a remainder of

1

when divided by

3

if and only if

k

is not divisible by

3

.

If

k

is a multiple of 3,

k^2

is divisible by 3 (remainder 0). If

k

is not a multiple of 3,

k \equiv 1

or

2 \pmod 3

, so

k^2 \equiv 1 \pmod 3

.

4

Count the number of favorable outcomes.

The multiples of 3 up to 22 are

3, 6, 9, 12, 15, 18, 21

(7 numbers). Thus, there are

22 - 7 = 15

values of

k

that satisfy the condition.

Favorable outcomes correspond exactly to integers

k \in \{1, 2, \dots, 22\}

that are not multiples of 3.

5

Compute the single-event probability.

Probability

= \frac{15}{500} = 0.03

.

The probability of a single event is the number of favorable outcomes divided by the total number of outcomes.

Anahtar Kavram

Basic Single-Event Probability with Number Properties

Soru 113Soru →

If $x$ is a real number such that $x^2 - 5x + 3 = 0$ , what is the value of $x^2 + \frac{9}{x^2}$ ?

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

The value of

x^2 + \frac{9}{x^2}

is 19.

Dividing

x^2 - 5x + 3 = 0

by

x

gives

x + \frac{3}{x} = 5

. Squaring both sides yields

\left(x + \frac{3}{x}\right)^2 = x^2 + 2(x)\left(\frac{3}{x}\right) + \frac{9}{x^2} = 25

, which simplifies to

x^2 + 6 + \frac{9}{x^2} = 25

. Subtracting 6 from both sides yields the final answer of 19.

Adım Adım Çözüm

1

Divide the quadratic equation by

x

x + \frac{3}{x} = 5

Since

x = 0

is not a root of

x^2 - 5x + 3 = 0

, dividing by

x

rearranges the terms into

x

and its reciprocal component.

2

Square both sides of

x + \frac{3}{x} = 5

x^2 + 6 + \frac{9}{x^2} = 25

Applying the algebraic identity

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

generates the required expression

x^2 + \frac{9}{x^2}

alongside a constant middle term.

3

Isolate

x^2 + \frac{9}{x^2}

x^2 + \frac{9}{x^2} = 19

Subtracting the constant cross-term 6 from 25 yields the target value.

Anahtar Kavram

Algebraic manipulation of quadratic equations using reciprocal square identities

Tahmini Süre:2m 0s

Soru 114Soru →

What is the sum of all real solutions to the polynomial equation

(x^2 - 3x + 1)^2 - 3(x^2 - 3x + 1) + 1 = x ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The sum of all real solutions to the equation is 6.

The equation

(x^2 - 3x + 1)^2 - 3(x^2 - 3x + 1) + 1 = x

represents

f(f(x)) = x

for

f(x) = x^2 - 3x + 1

. Expanding and setting to zero yields

x^4 - 6x^3 + 8x^2 + 2x - 1 = 0

. Since solutions to

f(x) = x

(i.e.,

x^2 - 4x + 1 = 0

) also satisfy

f(f(x)) = x

, we can factor out

x^2 - 4x + 1

to get

(x^2 - 4x + 1)(x^2 - 2x - 1) = 0

. Both quadratic factors have positive discriminants (

12

and

8

), ensuring four distinct real roots. By Vieta's formulas, the sum of roots from the first quadratic is 4 and from the second is 2, giving a total sum of 6.

Adım Adım Çözüm

1

Express the equation as a polynomial in standard form

Expanding

(x^2 - 3x + 1)^2 - 3(x^2 - 3x + 1) + 1 - x = 0

yields

x^4 - 6x^3 + 8x^2 + 2x - 1 = 0

.

Expanding allows the composite expression to be treated as a degree-4 polynomial.

2

Find a quadratic factor using fixed points of the iteration

Let

f(x) = x^2 - 3x + 1

. The equation has the form

f(f(x)) = x

. Any solution to

f(x) = x

must satisfy this equation. Solving

x^2 - 3x + 1 = x

gives

x^2 - 4x + 1 = 0

.

If

f(x) = x

, then

f(f(x)) = f(x) = x

, so

x^2 - 4x + 1

must be a factor of the 4th-degree polynomial.

3

Factor the degree-4 polynomial completely

Polynomial division of

x^4 - 6x^3 + 8x^2 + 2x - 1

by

x^2 - 4x + 1

yields

(x^2 - 4x + 1)(x^2 - 2x - 1) = 0

.

Factoring reduces the 4th-degree equation into two quadratic equations.

4

Verify real roots and calculate their sum

For

x^2 - 4x + 1 = 0

, discriminant

D_1 = 12 > 0

and sum of roots is 4. For

x^2 - 2x - 1 = 0

, discriminant

D_2 = 8 > 0

and sum of roots is 2. The four roots are all real and distinct, giving a total sum of

4 + 2 = 6

.

Vieta's formulas provide the sum of roots for each quadratic factor without needing to compute individual radical expressions.

Anahtar Kavram

Polynomial Factoring and Composite Quadratic Equations

Tahmini Süre:2m 30s

Soru 115Soru →

An architectural restoration project allocates its workforce across three specialized teams: Masonry, Carpentry, and Glasswork. Initially, the ratio of Masonry workers to Carpentry workers is $5 : 6$ , and the ratio of Carpentry workers to Glasswork workers is $4 : 5$ .

During phase two of the project:
• The number of Masonry workers is increased by $50\%$ .
• The number of Carpentry workers is decreased by $25\%$ .
• The number of Glasswork workers is increased by $X\%$ .

If the new ratio of Masonry workers to the total number of workers across all three teams is $1 : 3$ , what is the value of $X$ ?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The value of

X

is 40.

To find

X

, first combine the initial ratios by equating the common Carpentry term: Masonry : Carpentry = 10 : 12 and Carpentry : Glasswork = 12 : 15, yielding a joint ratio of 10 : 12 : 15. Representing the initial teams as

10k

,

12k

, and

15k

, the updated Masonry team is

10k \times 1.5 = 15k

and Carpentry is

12k \times 0.75 = 9k

. Using the part-to-total ratio

\frac{\text{New Masonry}}{\text{New Total}} = \frac{1}{3}

, we set up

\frac{15k}{15k + 9k + \text{New Glasswork}} = \frac{1}{3}

, which yields

\text{New Glasswork} = 21k

. The percentage increase from

15k

to

21k

is

\frac{21k - 15k}{15k} \times 100\% = 40\%

. Thus,

X = 40

.

Adım Adım Çözüm

1

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

The ratio Masonry : Carpentry : Glasswork is

10 : 12 : 15

.

Carpentry is common to both given ratios (

5 : 6

and

4 : 5

). The least common multiple of 6 and 4 is 12. Scaling

5 : 6

by 2 yields

10 : 12

, and scaling

4 : 5

by 3 yields

12 : 15

.

2

Define initial workforce quantities using a multiplier

k

.

Initial Masonry =

10k

, Carpentry =

12k

, Glasswork =

15k

.

Using an algebraic variable preserves the proportional relationships across all teams.

3

Calculate the updated worker counts for Masonry and Carpentry.

New Masonry =

15k

, New Carpentry =

9k

.

Masonry increases by

50\%

(

10k \times 1.50 = 15k

). Carpentry decreases by

25\%

(

12k \times 0.75 = 9k

).

4

Set up the part-to-total ratio equation to solve for the new Glasswork team size.

New Glasswork worker count =

21k

.

The total workforce is

15k + 9k + \text{New Glasswork} = 24k + \text{New Glasswork}

. Given

\frac{\text{New Masonry}}{\text{New Total}} = \frac{1}{3}

, we have

\frac{15k}{24k + \text{New Glasswork}} = \frac{1}{3} \implies 45k = 24k + \text{New Glasswork} \implies \text{New Glasswork} = 21k

.

5

Compute the percentage increase

X

for the Glasswork team.

X = 40

.

The initial Glasswork count was

15k

and the new count is

21k

. The percentage increase is

\frac{21k - 15k}{15k} \times 100 = \frac{6}{15} \times 100 = 40\%

.

Anahtar Kavram

Combining compound ratios and solving multi-step part-to-whole proportion equations.

Tahmini Süre:2m 30s

Soru 116Soru →

A container holds $20$ liters of a solution that is $15\%$ acid by volume. If $5$ liters of pure water are added to the solution, what is the percentage concentration of acid in the resulting mixture?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The concentration of acid in the resulting mixture is 12%.

The correct concentration is found by taking the invariant amount of pure acid (

3

liters) and dividing it by the total updated volume of the liquid (

25

liters), which yields

\frac{3}{25} = 0.12

, or

12\%

.

Adım Adım Çözüm

1

Determine the amount of solute (pure acid) in the original mixture.

Amount of acid =

0.15 \times 20 = 3

liters.

The solution initially consists of

15\%

acid out of a total volume of

20

liters.

2

Determine the total volume of the new solution after dilution.

New total volume =

20 + 5 = 25

liters.

Adding

5

liters of pure water increases the total liquid volume without adding any additional acid.

3

Compute the new percentage concentration.

New concentration =

\frac{3}{25} \times 100\% = 12\%

.

The concentration is the ratio of pure solute to total solution volume expressed as a percentage.

Anahtar Kavram

Dilution of a mixture by adding pure solvent

Soru 117Soru →

Set $S$ consists of $n$ consecutive integers. The product of the smallest and largest integers in $S$ is $-144$ . If the sum of all elements in $S$ is $91$ , what is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

The number of consecutive integers

n

in Set

S

is 26.

By combining the consecutive set sum formula

n(a + b) = 182

with the term span relation

b - a = n - 1

and the product constraint

ab = -144

, testing the factors of

182

uniquely determines

n = 26

, corresponding to the set of consecutive integers from

-9

to

16

.

Adım Adım Çözüm

1

Relate the number of elements

n

to the smallest element

a

and largest element

b

.

b - a = n - 1

In any set of

n

consecutive integers, the distance between the maximum and minimum elements is

n - 1

.

2

Use the sum formula for an evenly spaced set.

n(a + b) = 182

The sum is

\text{Sum} = n \times \frac{a + b}{2} = 91

, which simplifies to

n(a + b) = 182

.

3

Factor

182

to find integer solutions satisfying

a \cdot b = -144

.

n = 26

,

a = -9

,

b = 16

For

n = 26

,

a + b = 7

and

b - a = 25

, giving

a = -9

and

b = 16

. Their product is

(-9)(16) = -144

.

Anahtar Kavram

Properties of consecutive integer sets, average/median sum formula, and term boundary indexing

Soru 118Soru →

If $x$ is a real number such that $\sqrt[4]{3^x + 3^x + 3^x} = 27$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of

x

is 11.

Summing three terms of

3^x

yields

3 \cdot 3^x = 3^{x+1}

. Expressing the fourth root of

3^{x+1}

as a fractional power gives

3^{\frac{x+1}{4}}

. Equating this to

27 = 3^3

results in

\frac{x+1}{4} = 3

, leading directly to

x = 11

.

Adım Adım Çözüm

1

Combine the repeated addition terms inside the fourth root.

3^x + 3^x + 3^x = 3 \cdot 3^x = 3^{x+1}

Adding three identical terms

3^x

is equivalent to multiplying

3^x

by 3, which increments the exponent by 1.

2

Rewrite the fourth root expression using a fractional exponent.

\sqrt[4]{3^{x+1}} = 3^{\frac{x+1}{4}}

Applying the property

\sqrt[n]{a^m} = a^{m/n}

.

3

Express the constant on the right side of the equation with base 3.

27 = 3^3

Rewriting both sides of the equation with the same base allows equating exponents.

4

Equate the exponents and solve for

x

.

\frac{x+1}{4} = 3 \implies x + 1 = 12 \implies x = 11

Since the bases are equal, the powers must be equal.

Anahtar Kavram

Combining like exponential terms and equating fractional exponents

Tahmini Süre:1m 30s

Soru 119Soru →

A computer program randomly selects a single positive integer $x$ from the set $\{1, 2, 3, \dots, 300\}$ , where each integer has an equal probability of being chosen. What is the probability that $x$ is divisible by either 6 or 9, but not by 4?

Cevabı ve açıklamayı göster

Cevap: 0.14

Cevap

The probability that

x

is divisible by 6 or 9, but not by 4, is

0.14

(or

\frac{7}{50}

).

To calculate the probability, determine the ratio of favorable outcomes to total outcomes (300). First, count integers divisible by 6 (50) or 9 (33). Subtracting the 16 multiples of 18 counted in both yields 67 integers. Next, identify how many of these 67 integers are divisible by 4: there are 25 multiples of 12 and 8 multiples of 36 (all 8 of which are already included in the multiples of 12), resulting in 25 integers divisible by 4. Subtracting 25 from 67 leaves 42 favorable integers. The probability is

42 / 300 = 0.14

.

Adım Adım Çözüm

1

Determine the total sample space size.

The total number of equally likely outcomes is

|S| = 300

.

Single-event probability requires dividing the count of favorable outcomes by the total number of outcomes in the set.

2

Calculate the count of integers divisible by 6 or 9.

Count =

50 + 33 - 16 = 67

.

Apply the Principle of Inclusion-Exclusion: sum the multiples of 6 and 9, then subtract the multiples of

\text{lcm}(6, 9) = 18

to eliminate double counting.

3

Exclude integers from this set that are divisible by 4.

Count of multiples of 4 to exclude =

25 + 8 - 8 = 25

.

Integers divisible by both 6 and 4 are multiples of

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

. Integers divisible by both 9 and 4 are multiples of

\text{lcm}(9, 4) = 36

. Using inclusion-exclusion,

25 + 8 - 8 = 25

integers must be excluded.

4

Find the number of favorable outcomes and compute the probability.

Favorable outcomes

|E| = 67 - 25 = 42

. Probability

P(E) = \frac{42}{300} = 0.14

.

Divide the favorable set size by the total sample space size.

Anahtar Kavram

Basic Single-Event Probability with Set Inclusion-Exclusion and Number Properties

Soru 120Soru →

An archival digitization laboratory uses two high-resolution scanner units, Unit $A$ and Unit $B$ , to process historic document collections. Working alone at its constant rate, Unit $A$ can digitize a standard collection in $24$ hours. Working alone at its constant rate, Unit $B$ can digitize the same collection in $40$ hours. Unit $A$ begins digitizing a standard collection alone. After $8$ hours of operation, Unit $B$ is turned on, and both units work together at their respective constant rates until the entire collection is digitized. How many total hours does it take to digitize the collection from start to finish?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The total time required to digitize the collection from start to finish is 18 hours.

Unit A completes

\frac{1}{3}

of the total job during its 8 solo hours. The remaining

\frac{2}{3}

of the job is completed by Units A and B working together at a combined rate of

\frac{1}{15}

per hour, taking 10 hours. Adding the 8 solo hours to the 10 combined hours yields a total time of 18 hours.

Adım Adım Çözüm

1

Determine the individual work rates per hour for each unit.

Unit A's rate is

\frac{1}{24}

of the collection per hour, and Unit B's rate is

\frac{1}{40}

of the collection per hour.

Work rate is defined as the fraction of the job completed per unit of time (

R = \frac{1}{T}

).

2

Calculate the fraction of the work completed by Unit A alone during the first 8 hours.

Work completed

= 8 \times \frac{1}{24} = \frac{8}{24} = \frac{1}{3}

of the collection.

Work done equals rate multiplied by time (

W = R \times t

).

3

Calculate the remaining fraction of the collection to be digitized.

Remaining work

= 1 - \frac{1}{3} = \frac{2}{3}

of the collection.

The total job represents 1 whole unit.

4

Calculate the combined work rate when both units operate together.

Combined rate

= \frac{1}{24} + \frac{1}{40} = \frac{5}{120} + \frac{3}{120} = \frac{8}{120} = \frac{1}{15}

of the collection per hour.

When working simultaneously, individual rates add together.

5

Find the time required for both units working together to complete the remaining work.

Combined time

= \frac{2/3}{1/15} = \frac{2}{3} \times 15 = 10

hours.

Time equals remaining work divided by the combined rate (

t = \frac{W_{\text{remaining}}}{R_{\text{combined}}}

).

6

Calculate the total time required from start to finish.

Total time

= 8 \text{ hours} + 10 \text{ hours} = 18

hours.

Total time is the sum of the time Unit A worked alone and the time both units worked together.

Anahtar Kavram

Work Rate and Combined Work

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram