Tüm alıştırma soruları

Soru 141Soru →

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

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Cevap: 81

Cevap

81

The total sum of the

8

integers is

8 \times 15 = 120

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

7

integers must be minimized. Since

12

is the unique mode,

12

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

5

positions without creating another mode are

1, 2, 3, 4,

and

5

. Placing

12

in the

6\text{th}

and

7\text{th}

positions minimizes the sum of the first

7

elements to

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

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

120 - 39 = 81

.

Adım Adım Çözüm

1

Calculate the total sum of the 8 positive integers.

Total Sum =

8 \times 15 = 120

.

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

2

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

Minimum sum of the first 7 integers =

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

.

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

a_6 = 12

and

a_7 = 12

) and selecting the smallest distinct positive integers (

1, 2, 3, 4, 5

) for the first 5 terms minimizes their total sum.

3

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

Maximum integer =

120 - 39 = 81

.

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

Anahtar Kavram

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

Soru 142Soru →

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

7x + 4y = 113

3x + 6y = 87

what is the value of $x + y$ ?

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Cevap: 20

Cevap

The value of

x + y

is 20.

Adding the two equations yields

10x + 10y = 200

. Dividing the entire equation by 10 directly gives

x + y = 20

. Alternatively, solving the system yields

x = 11

and

y = 9

, whose sum is

11 + 9 = 20

.

Adım Adım Çözüm

1

Add the two equations together to combine like terms.

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

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

x

and

y

allows a direct solution for the sum

(x + y)

.

2

Divide the combined equation by 10.

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

Isolating

(x + y)

directly avoids the extra computation of solving for

x

and

y

individually.

Anahtar Kavram

Solving Systems of Linear Equations by Combination

Tahmini Süre:1m 30s

Soru 143Soru →

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

Cevabı ve açıklamayı göster

Cevap: -10

Cevap

The minimum possible value of

x - y

is

-10

.

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

Adım Adım Çözüm

1

Determine the signs of variables x and y

x < 0 and y > 0

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

2

Transform the inequality without altering its sign direction

y² < -4x

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

3

Evaluate candidate integer pairs to minimize x - y

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

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

Anahtar Kavram

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

Soru 144Soru →

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

Cevabı ve açıklamayı göster

Cevap: 1680

Cevap

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

To calculate net profit, first determine the unit cost:

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

. The marked-up regular price is

\$80 \times 1.50 = \$120

. Selling

40

kits at

\$120

yields

\$4,800

. The remaining

20

kits are sold at a

30\%

discount off

\$120

, giving a clearance price of

\$120 \times 0.70 = \$84

per kit and revenue of

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

. Total revenue is

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

. Subtracting the total cost of

\$4,800

yields a net profit of

\$1,680

.

Adım Adım Çözüm

1

Determine the cost price per unit.

Cost per kit is

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

.

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

2

Calculate the regular retail price per unit.

Regular retail price is

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

.

The

50\%

markup is applied directly to the cost price of

\$80

.

3

Calculate revenue from the full-price sales segment.

Revenue from

40

kits is

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

.

Multiply quantity sold at full price by regular retail price.

4

Determine clearance price and revenue for remaining units.

Clearance price is

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

. Revenue from

20

kits is

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

.

The clearance discount of

30\%

is taken off the regular retail price, not cost.

5

Calculate total revenue and net profit.

Total revenue is

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

. Net profit is

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

.

Net profit equals total revenue minus total cost.

Anahtar Kavram

Profit, Loss, and Markup

Soru 145Soru →

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

Cevabı ve açıklamayı göster

Cevap: 84

Cevap

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

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

Adım Adım Çözüm

1

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

Let

v

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

(v + 3)

mph. Due to a

25\%

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

0.75v

mph, making the effective upstream speed

(0.75v - 3)

mph.

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

2

Set up distance equations for both directions using

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

.

Downstream distance:

D = 4(v + 3)

. Upstream distance:

D = 8(0.75v - 3)

.

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

3

Equate the two distance expressions and solve for

v

.

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

mph.

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

v

.

4

Calculate the total distance

D

.

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

miles.

Substituting

v = 18

back into the downstream distance formula yields the exact distance.

Anahtar Kavram

Rate, Time, and Distance with Currents and Variable Speeds

Tahmini Süre:2m 30s

Soru 146Soru →

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

Cevabı ve açıklamayı göster

Cevap: 56

Cevap

The average annual rental rate for Building C is

56

dollars per square foot.

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

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

sq ft and an overall weighted average rate of

\$52

/sq ft, the total revenue is

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

dollars. Subtracting the revenue generated by Building A (

20,000 \times 45 = 900,000

dollars) and Building B (

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

dollars) leaves

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

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

50,000

square feet yields an average rate of

56

dollars per square foot.

Adım Adım Çözüm

1

Determine total combined rentable area and total combined revenue

Total area

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

sq ft. Total revenue

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

dollars.

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

2

Calculate known annual revenues from Building A and Building B

Building A revenue

= 20,000 \times 45 = 900,000

dollars. Building B revenue

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

dollars. Combined A and B revenue

= 2,400,000

dollars.

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

3

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

Building C revenue

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

dollars. Building C rate

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

dollars.

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

Anahtar Kavram

Weighted Average = Total Weighted Sum / Total Weight Sum

Soru 147Soru →

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

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Cevap: 30

Cevap

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

Formulating the equations

3d + 150m = 180

and

5d + 200m = 270

and solving for the daily rate

d

yields

d = 30

.

Adım Adım Çözüm

1

Define variables and formulate the system of linear equations.

Let

d

be the fixed daily rate (in dollars) and

m

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

3d + 150m = 180

Equation 2:

5d + 200m = 270

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

2

Simplify Equation 1 to express

d

in terms of

m

.

Dividing Equation 1 by 3 yields

d + 50m = 60

, so

d = 60 - 50m

.

Simplifying equations reduces computation error when using substitution.

3

Substitute the expression for

d

into Equation 2 to solve for

m

.

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

Substitution eliminates variable

d

, leaving a linear equation in one variable.

4

Calculate the fixed daily rate

d

.

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

Substituting

m = 0.60

gives the exact daily fixed rate.

Anahtar Kavram

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

Soru 148Soru →

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

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Cevap: 1620

Cevap

The smallest positive integer

n

is 1620.

By writing 180 as

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

and 450 as

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

, we can analyze the exponents needed for

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

. To make

180n

a perfect square,

a

and

b

must be even and

c

must be odd. To make

450n

a perfect cube,

1+a

,

2+b

, and

2+c

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

a = 2

,

b = 4

, and

c = 1

. Thus,

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

.

Adım Adım Çözüm

1

Express 180 and 450 in terms of their prime factorizations.

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

and

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

Prime factorization exposes the exponent requirements for perfect powers.

2

Determine the constraints on exponents of

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

.

For

180n

to be a square,

a

must be even,

b

must be even, and

c

must be odd. For

450n

to be a cube,

1+a

,

2+b

, and

2+c

must be multiples of 3.

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

3

Find the minimal values for

a, b, c

.

a = 2, b = 4, c = 1

a=2

is even and makes

1+2=3

;

b=4

is even and makes

2+4=6

;

c=1

is odd and makes

2+1=3

.

4

Compute

n

.

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

Multiplying the prime powers together yields the smallest integer

n

.

Anahtar Kavram

Prime Factorization and Exponent Rules for Perfect Powers

Soru 149Soru →

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

Cevabı ve açıklamayı göster

Cevap: 760

Cevap

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

Let

C

represent the wholesale cost of a generator. A

40\%

markup yields an original retail price of

1.40C

. Applying a

15\%

discount to this retail price produces a promotional selling price of

0.85 \times 1.40C = 1.19C

. Setting

1.19C = 4,760

gives

C = 4,000

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

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

.

Adım Adım Çözüm

1

Define the relationship between wholesale cost and original retail price.

Original Retail Price =

1.40 \times C

A

40\%

markup on wholesale cost

C

increases the price to

140\%

of

C

.

2

Apply the discount to calculate the promotional selling price.

Selling Price =

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

A

15\%

discount off the retail price leaves

85\%

of the retail price.

3

Solve for the wholesale cost

C

.

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

Given that the final selling price is

\$4,760

, set

1.19C = 4,760

and divide.

4

Calculate the profit made on the sale.

Profit =

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

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

Anahtar Kavram

Successive markup and discount calculations with distinct base values.

Soru 150Soru →

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

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

a + b + c

is

6

.

For an integer with prime factorization

p^a q^b r^c

, the number of positive divisors is

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

. For

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

, the number of positive divisors is

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

. Factoring

105

into three odd terms greater than

1

yields

3 \times 5 \times 7

. Matching these terms in increasing order gives

2a+1=3 \implies a=1

,

2b+1=5 \implies b=2

, and

2c+1=7 \implies c=3

. Checking

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

confirms these values. Summing

a+b+c

yields

1+2+3=6

.

Adım Adım Çözüm

1

Write the formula for the number of positive factors of

N

and

N^2

The number of positive factors of

N = p^a q^b r^c

is

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

. Since

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

, the number of positive factors of

N^2

is

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

.

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

2

Factor

105

into three odd factors greater than

1

The prime factorization of

105

is

3 \times 5 \times 7

. The only way to express

105

as a product of three integers greater than

1

is

3 \times 5 \times 7

.

Since

a, b, c

are positive integers, each term

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

must be an odd integer greater than

1

.

3

Assign the factors using the inequality condition

a < b < c

Since

a < b < c

, it follows that

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

. Therefore:

2a+1 = 3 \implies a = 1

;

2b+1 = 5 \implies b = 2

;

2c+1 = 7 \implies c = 3

.

Matching the ordered values of

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

with the sorted factors

3, 5, 7

uniquely determines

a, b,

and

c

.

4

Verify with the factor count for

N

and calculate the sum

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

, which matches the given condition. The sum

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

.

Verification confirms the solution satisfies all constraints.

Anahtar Kavram

Prime Factorization and Divisor Counting Formula

Tahmini Süre:1m 30s

Soru 151Soru →

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

Cevabı ve açıklamayı göster

Cevap: 3.8

Cevap

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

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

Adım Adım Çözüm

1

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

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

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

2

Calculate the total parts and weight of each shift.

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

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

3

Formulate the weighted average equation for the overall defect rate.

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

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

4

Solve for the unknown defect rate r_Z of Shift Z.

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

Algebraic simplification yields the exact defect rate for Shift Z.

Anahtar Kavram

Weighted Average of Combined Sets

Tahmini Süre:2m 0s

Soru 152Soru →

A project manager must schedule 5 distinct client presentations—for clients A, B, C, D, and E—on 5 consecutive days from Monday through Friday, with exactly one presentation per day. If the presentations for client A and client B cannot be scheduled on consecutive days, how many different presentation schedules are possible?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

The total number of valid presentation schedules is 72.

The correct result is found by subtracting the number of restricted arrangements (where presentation A and presentation B are scheduled on consecutive days) from the total number of unrestricted arrangements of 5 presentations. The total unrestricted arrangements equal

5! = 120

. Treating A and B as a single block leaves 4 items to arrange in

4! = 24

ways, with

2! = 2

internal orderings for A and B, yielding

24 \times 2 = 48

consecutive schedules. Subtracting 48 from 120 results in 72 valid schedules.

Adım Adım Çözüm

1

Calculate total unrestricted linear arrangements of the 5 presentations.

5! = 120

Without restrictions, 5 distinct items can be arranged in 5 distinct positions in

5!

ways.

2

Calculate the number of invalid arrangements where presentations A and B are on consecutive days.

2! \times 4! = 48

Grouping A and B into a single unit results in 4 items to arrange (

4! = 24

), and A and B can swap positions inside the block in

2! = 2

ways.

3

Subtract the invalid arrangements from total arrangements.

120 - 48 = 72

Complementary counting gives the total number of arrangements where A and B are not on consecutive days.

Anahtar Kavram

Permutations with Adjacency Restrictions (Complementary Counting)

Soru 153Soru →

A game features two boxes of tokens. Box X contains $2$ red tokens and $3$ blue tokens. Box Y contains $4$ red tokens and $1$ blue token. A fair coin is flipped to determine which box to draw from: if the coin lands on heads, Box X is chosen; if it lands on tails, Box Y is chosen. Two tokens are then drawn sequentially without replacement from the chosen box. What is the probability that both drawn tokens are red?

Cevabı ve açıklamayı göster

Cevap: 0.35

Cevap

The probability that both drawn tokens are red is 0.35.

The total probability combines the independent choice of box with dependent draws without replacement. Box X yields two red tokens with probability

0.10

, and Box Y yields two red tokens with probability

0.60

. Weighting each by the

0.5

probability of choosing that box gives

(0.5 \times 0.10) + (0.5 \times 0.60) = 0.35

.

Adım Adım Çözüm

1

Determine the conditional probability of drawing two red tokens from Box X without replacement.

The probability is

\frac{2}{5} \times \frac{1}{4} = 0.10

.

Because draws are dependent (without replacement), the number of remaining red tokens decreases to 1 and total tokens to 4 after the first red draw.

2

Determine the conditional probability of drawing two red tokens from Box Y without replacement.

The probability is

\frac{4}{5} \times \frac{3}{4} = 0.60

.

Box Y initially contains 4 red out of 5 total tokens; drawing one red leaves 3 red out of 4 total tokens.

3

Combine the independent box selection probabilities with the dependent drawing probabilities.

Overall probability is

(0.5 \times 0.10) + (0.5 \times 0.60) = 0.05 + 0.30 = 0.35

.

The initial coin flip selects Box X or Box Y with equal, independent probability of 0.5.

Anahtar Kavram

Combining independent events (environment selection) with dependent events (sampling without replacement) using the Law of Total Probability.

Tahmini Süre:2m 0s

Soru 154Soru →

A museum curator is arranging $5$ distinct marble statues and $2$ distinct bronze statues in a single row along a gallery wall. If the $2$ bronze statues must not be placed next to each other, how many different linear arrangements of all $7$ statues are possible?

Cevabı ve açıklamayı göster

Cevap: 3600

Cevap

3,600

Using complementary counting, the total unrestricted arrangements of

7

distinct statues is

7! = 5,040

. The number of arrangements where the

2

bronze statues are placed together is determined by treating them as a single block:

6! \times 2! = 1,440

. Subtracting these forbidden arrangements from the total yields

5,040 - 1,440 = 3,600

valid linear arrangements.

Adım Adım Çözüm

1

Find total arrangements without restriction.

7! = 5,040

There are 7 distinct statues in total to arrange in a line.

2

Find arrangements where the 2 bronze statues are adjacent.

6! × 2! = 1,440

Grouping the 2 bronze statues into 1 block yields 6 items to order (6!), and the 2 bronze statues can swap positions inside the block (2!).

3

Apply complementary counting to find non-adjacent arrangements.

5,040 - 1,440 = 3,600

Subtracting the adjacent arrangements from total arrangements gives all valid arrangements.

Anahtar Kavram

Linear arrangements with non-adjacency restrictions using complementary counting.

Tahmini Süre:1m 30s

Soru 155Soru →

A crate in a warehouse contains $15$ functional electronic components and $10$ defective components. If two components are selected at random from the crate one after another without replacement, what is the probability that both selected components are functional?

Cevabı ve açıklamayı göster

Cevap: 0.35

Cevap

The probability that both selected components are functional is 0.35.

Because selection is made without replacement, the outcome of the second draw depends on the outcome of the first draw. The probability of selecting a functional component first is 15/25 = 3/5. With 14 functional components left among 24 total components, the probability of selecting a second functional component is 14/24 = 7/12. Multiplying these probabilities gives (3/5) * (7/12) = 21/60 = 0.35.

Adım Adım Çözüm

1

Calculate the probability of drawing a functional component on the first selection.

P(First Functional) = 15 / 25 = 3/5 = 0.6.

There are 15 functional components in the total pool of 25 components.

2

Calculate the conditional probability of drawing a second functional component given that the first component drawn was functional.

P(Second Functional | First Functional) = 14 / 24 = 7/12.

Because sampling is without replacement, the total count decreases to 24 and the functional count decreases to 14.

3

Multiply the two dependent probabilities to find the overall joint probability.

P(Both Functional) = (3/5) * (7/12) = 21/60 = 0.35.

By the multiplication rule for dependent events, P(A and B) = P(A) * P(B|A).

Anahtar Kavram

Probability of Dependent Events (Sampling Without Replacement)

Soru 156Soru →

During a quality control inspection, five manufactured items were measured and found to have lengths of $44\text{ mm}$ , $48\text{ mm}$ , $50\text{ mm}$ , $52\text{ mm}$ , and $56\text{ mm}$ . What is the standard deviation, in millimeters, of the lengths of these five items?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The standard deviation of the lengths of the five items is 4 millimeters.

The arithmetic mean of the five measurements is 50 mm. The sum of the squared deviations from 50 is 36 + 4 + 0 + 4 + 36 = 80. Dividing 80 by 5 yields a variance of 16. Taking the principal square root of 16 gives a standard deviation of 4 mm.

Adım Adım Çözüm

1

Calculate the arithmetic mean of the dataset

Mean = 50

The standard deviation measures dispersion relative to the mean.

2

Find the squared difference of each data point from the mean

Squared deviations are 36, 4, 0, 4, and 36

Squaring ensures all deviations are non-negative and penalizes larger deviations.

3

Compute the mean of the squared deviations (variance)

Variance = 80 / 5 = 16

Variance is the average squared distance from the mean.

4

Take the non-negative square root of the variance

Standard deviation = sqrt(16) = 4

Standard deviation converts variance back to the original unit of measurement.

Anahtar Kavram

Standard Deviation Calculation for a Data Set

Soru 157Soru →

An investor deposited a principal sum of money into an account that earns simple annual interest at a rate of $r\%$ . At the end of $4$ years, the total accumulated balance in the account (principal plus interest) was $\$7,200$ . If the account had instead earned simple annual interest at a rate of $(r + 2)\%$ over the same $4$ -year period, the total accumulated balance would have been $\$7,680$ . What was the original principal amount deposited, in dollars?

Cevabı ve açıklamayı göster

Cevap: 6000

Cevap

The original principal amount deposited was

6,000

dollars.

The difference in accumulated balances over

4

years is

\$7,680 - \$7,200 = \$480

. Since simple interest is calculated strictly on the original principal

P

, an annual rate increase of

2\%

yields an additional

2\%

of

P

each year. Over

4

years, this total rate increase is

4 \times 2\% = 8\%

. Therefore,

8\%

of

P

equals

\$480

, giving

0.08P = 480

, which yields

P = 6,000

dollars.

Adım Adım Çözüm

1

Calculate the difference between the two final balances.

\$7,680 - \$7,200 = \$480

The additional

\$480

represents the total extra simple interest earned over

4

years due to the higher interest rate.

2

Determine the cumulative percentage rate increase over the 4-year period.

4 \text{ years} \times 2\% \text{ per year} = 8\%

Simple interest is calculated solely on the original principal for each year.

3

Set up and solve the equation for the principal amount

P

.

8\% \text{ of } P = \$480 \implies 0.08P = 480 \implies P = 6,000

Dividing the additional dollar amount by the additional percentage yields the original base principal.

Anahtar Kavram

Simple Interest and Rate Sensitivity

Tahmini Süre:1m 30s

Soru 158Soru →

An electronics manufacturing plant produces two types of circuit modules: Module X and Module Y. Producing one Module X requires $3$ minutes on Machine A and $5$ minutes on Machine B. Producing one Module Y requires $4$ minutes on Machine A and $2$ minutes on Machine B. During a certain shift, Machine A was operated for a total of $230$ minutes and Machine B was operated for a total of $220$ minutes. If both machines operated at full capacity with no idle time, how many Module X units were produced?

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Cevap: 30

Cevap

30

Formulating equations from the machine times gives

3x + 4y = 230

for Machine A and

5x + 2y = 220

for Machine B. Multiplying the second equation by

2

yields

10x + 4y = 440

. Subtracting

3x + 4y = 230

from

10x + 4y = 440

eliminates

y

and yields

7x = 210

, which solves to

x = 30

.

Adım Adım Çözüm

1

Define variables for the unknown quantities.

Let

x

be the number of Module X units produced, and let

y

be the number of Module Y units produced.

Representing unknown quantities with algebraic variables allows for model formulation.

2

Formulate a system of linear equations from the machine time constraints.

Machine A equation:

3x + 4y = 230

Machine B equation:

5x + 2y = 220

The total operational time on each machine equals the sum of times spent producing each module type.

3

Multiply the Machine B equation by

2

to enable elimination of

y

.

2 \times (5x + 2y) = 2 \times 220 \implies 10x + 4y = 440

Matching the coefficient of

y

with the first equation (

4y

) allows elimination via subtraction.

4

Subtract the Machine A equation from the modified Machine B equation and solve for

x

.

(10x + 4y) - (3x + 4y) = 440 - 230 \implies 7x = 210 \implies x = 30

Subtracting eliminates

y

directly, leaving a single linear equation in terms of

x

.

Anahtar Kavram

Solving a system of two linear equations in two variables using elimination

Tahmini Süre:1m 30s

Soru 159Soru →

What is the smallest positive integer greater than $100$ that is divisible by $9$ ?

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Cevap: 108

Cevap

The smallest positive integer greater than

100

that is divisible by

9

is

108

.

An integer is divisible by 9 if it is an integer multiple of 9. Multiples of 9 around 100 are

9 \times 11 = 99

and

9 \times 12 = 108

. Since 99 is not greater than 100, the smallest integer meeting the condition is 108.

Adım Adım Çözüm

1

Divide 100 by 9 to locate the adjacent multiples of 9.

100 = 9 \times 11 + 1

This shows that

9 \times 11 = 99

is the largest multiple of 9 that is less than or equal to 100.

2

Determine the next multiple of 9 by adding 9 to 99 (or evaluating

9 \times 12

).

99 + 9 = 108

Since 99 is less than 100, the next consecutive multiple of 9 must be the smallest multiple strictly greater than 100.

Anahtar Kavram

Divisibility and Multiples of Integers

Soru 160Soru →

What is the smallest positive integer $n$ such that $n$ is divisible by both $12$ and $18$ , and $n$ has exactly $12$ positive divisors?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

The smallest positive integer satisfying all given conditions is 72.

To satisfy the divisibility requirements,

n

must be a multiple of

\text{LCM}(12, 18) = 36 = 2^2 \times 3^2

. Thus,

n = 2^a \times 3^b \times \dots

where

a \geq 2

and

b \geq 2

. The total number of positive divisors of

n

is given by

(a+1)(b+1)\dots = 12

. Since

a+1 \geq 3

and

b+1 \geq 3

,

n

cannot have any additional prime factors, as that would yield at least

3 \times 3 \times 2 = 18

divisors. Factoring 12 into two integers each at least 3 gives

3 \times 4

or

4 \times 3

. Setting

(a+1, b+1) = (3, 4)

yields

a = 2, b = 3

, giving

n = 2^2 \times 3^3 = 108

. Setting

(a+1, b+1) = (4, 3)

yields

a = 3, b = 2

, giving

n = 2^3 \times 3^2 = 72

. The smallest value is 72.

Adım Adım Çözüm

1

Determine the prime factor constraints on

n

n

must be divisible by

2^2

and

3^2

, so

n = 2^a \times 3^b \times \dots

with

a \geq 2

and

b \geq 2

.

For

n

to be divisible by

12 = 2^2 \times 3^1

and

18 = 2^1 \times 3^2

, its prime factorization must contain at least the maximum power of each prime appearing in either 12 or 18.

2

Set up the divisor count equation

(a+1)(b+1) = 12

with

a+1 \geq 3

and

b+1 \geq 3

, and no additional prime factors.

The number of divisors of

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

is

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

. Since

(a+1) \geq 3

and

(b+1) \geq 3

, adding another prime factor would make the product at least

3 \times 3 \times 2 = 18 > 12

.

3

Find all possible pairs of exponents

(a, b)

and calculate candidate values for

n

Candidate 1:

a=2, b=3 \implies n = 2^2 \times 3^3 = 108

. Candidate 2:

a=3, b=2 \implies n = 2^3 \times 3^2 = 72

.

The only factor pairs of 12 into integers

\geq 3

are

3 \times 4

and

4 \times 3

.

4

Identify the minimum value

n = 72

72 < 108

, so 72 is the smallest positive integer fulfilling all criteria.

Anahtar Kavram

Divisor Count Formula & Least Common Multiple constraints

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