Tüm alıştırma soruları

If $n$ is a positive integer such that $2^{n-2} + 2^{n-1} + 2^n + 2^{n+1} = 480$ , what is the value of $n$ ?

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Cevap: 7

Cevap

The value of

n

is 7.

Factoring out

2^{n-2}

converts the sum into

2^{n-2}(1 + 2 + 4 + 8) = 15 \cdot 2^{n-2} = 480

. Dividing 480 by 15 gives

2^{n-2} = 32

. Since

32 = 2^5

, setting

n - 2 = 5

yields

n = 7

.

Adım Adım Çözüm

1

Factor out the lowest power of 2, which is

2^{n-2}

, from all terms on the left side of the equation.

2^{n-2}(1 + 2 + 4 + 8) = 480

Factoring out a common exponential term simplifies the addition of powers into a product of a single exponential term and a constant.

2

Evaluate the constant factor inside the parentheses and solve for the exponential expression

2^{n-2}

.

15 \cdot 2^{n-2} = 480 \implies 2^{n-2} = 32

Summing

1 + 2 + 4 + 8

yields 15. Dividing both sides by 15 isolates the base-2 term.

3

Express 32 as a power with base 2 and equate the exponents.

2^{n-2} = 2^5 \implies n - 2 = 5 \implies n = 7

Since

32 = 2^5

and the bases are identical, the exponents must be equal.

Anahtar Kavram

Factoring sum of exponential terms with common bases

Soru 42Soru →

On the real number line, $x$ is a negative real number such that $|5 - 2x| = 11$ , and $y$ is a positive real number such that $|3y + 1| = 13$ . What is the distance on the real number line between $x$ and $y$ ?

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Cevap: 7

Cevap

The distance between

x

and

y

on the real number line is 7.

Solving

|5 - 2x| = 11

with

x < 0

yields

x = -3

. Solving

|3y + 1| = 13

with

y > 0

yields

y = 4

. The distance between

-3

and

4

on the number line is

|4 - (-3)| = 7

.

Adım Adım Çözüm

1

Solve for the negative real number

x

using the equation

|5 - 2x| = 11

.

x = -3

The equation splits into

5 - 2x = 11

(yielding

x = -3

) and

5 - 2x = -11

(yielding

x = 8

). Since

x

must be negative,

x = -3

is selected.

2

Solve for the positive real number

y

using the equation

|3y + 1| = 13

.

y = 4

The equation splits into

3y + 1 = 13

(yielding

y = 4

) and

3y + 1 = -13

(yielding

y = -\frac{14}{3}

). Since

y

must be positive,

y = 4

is selected.

3

Compute the distance between

x

and

y

on the number line.

7

The distance between two points on the number line is given by

|y - x| = |4 - (-3)| = 7

.

Anahtar Kavram

Distance on the real number line between two points

a

and

b

is given by

|a - b|

, solved by evaluating absolute value equations under given sign constraints.

Alternatif Yöntem

Plot the candidate solutions for

x

(

x = -3

and

x = 8

) and

y

(

y = 4

and

y = -14/3

) on a number line, then directly count units between the valid points

x = -3

and

y = 4

.

Tahmini Süre:1m 15s

Soru 43Soru →

For how many integer values of $n$ in the interval $-15 \le n \le 15$ is the value of the expression $(-1)^{n^2 + n} - (-1)^{3n}$ equal to $2$ ?

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Cevap: 16

Cevap

The correct answer is 16.

Because

n^2 + n = n(n + 1)

is the product of two consecutive integers, it is guaranteed to be even for every integer

n

. Consequently,

(-1)^{n^2 + n} = 1

. Substituting this into the given equation yields

1 - (-1)^{3n} = 2

, which reduces to

(-1)^{3n} = -1

. A power of

-1

equals

-1

if and only if the exponent is odd, so

3n

must be odd, which requires

n

itself to be odd. In the interval

[-15, 15]

, there are 16 odd integers: 8 negative odd integers and 8 positive odd integers.

Adım Adım Çözüm

1

Determine the parity of

n^2 + n

The expression

n^2 + n = n(n + 1)

represents the product of two consecutive integers. Because one of any two consecutive integers is even, their product is always even. Therefore,

(-1)^{n^2 + n} = 1

for all integers

n

.

Simplifying the exponent with a known parity rule reduces the expression to a constant.

2

Isolate

(-1)^{3n}

in the equation

Substituting

1

into the original equation gives

1 - (-1)^{3n} = 2

, which simplifies to

(-1)^{3n} = -1

.

Isolating the exponential term reveals the sign condition required for the equality to hold.

3

Find the parity condition for

n

For

(-1)^{3n}

to equal

-1

, the exponent

3n

must be an odd integer. Since

3

is odd, the product

3n

is odd if and only if

n

is odd.

Applying the product parity rule (

\text{odd} \times \text{odd} = \text{odd}

) relates the condition on

3n

back to

n

.

4

Count the odd integers in the interval

[-15, 15]

The odd integers in the interval are

-15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15

. There are 16 such integers.

Counting all qualifying values within the specified range yields the final numeric answer.

Anahtar Kavram

Parity rules for consecutive integers and exponents of negative numbers

Soru 44Soru →

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 $m$ ?

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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 45Soru →

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 $x$ ?

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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 46Soru →

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

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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 47Soru →

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?

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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 48Soru →

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 49Soru →

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

Soru 50Soru →

A train travels from Station A to Station B, a distance of 180 miles, at a constant speed of $v$ miles per hour. On the return trip from Station B to Station A, the train travels the first half of the distance at a constant speed that is $20\%$ less than $v$ , and the remaining half of the distance at a constant speed that is $25\%$ greater than $v$ . If the total time for the return trip is 6 minutes longer than the total time for the trip from Station A to Station B, what is the value of $v$ ?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The value of

v

is

45

.

The outbound travel time for 180 miles at speed

v

is

\frac{180}{v}

hours. On the return trip, the first 90 miles at speed

0.80v

require

\frac{90}{0.80v} = \frac{112.5}{v}

hours, while the second 90 miles at speed

1.25v

require

\frac{90}{1.25v} = \frac{72}{v}

hours. The total return duration is

\frac{112.5 + 72}{v} = \frac{184.5}{v}

hours. Setting the difference between the return time and outbound time equal to 6 minutes (

0.1

hours) gives

\frac{184.5}{v} - \frac{180}{v} = 0.1

, which simplifies to

\frac{4.5}{v} = 0.1

, yielding

v = 45

.

Adım Adım Çözüm

1

Write the expression for the outbound trip duration in terms of

v

.

T_{\text{outbound}} = \frac{180}{v}

hours.

Time is equal to total distance divided by constant speed.

2

Calculate the duration for each half of the return trip in terms of

v

.

The first 90 miles take

\frac{90}{0.80v} = \frac{112.5}{v}

hours, and the second 90 miles take

\frac{90}{1.25v} = \frac{72}{v}

hours, giving a total return duration of

\frac{184.5}{v}

hours.

The return trip consists of two 90-mile segments driven at

0.80v

and

1.25v

respectively.

3

Equate the difference between return and outbound times to 0.1 hours and solve for

v

.

\frac{184.5}{v} - \frac{180}{v} = 0.1 \implies \frac{4.5}{v} = 0.1 \implies v = 45

.

The time difference of 6 minutes is equal to

\frac{6}{60} = 0.1

hours.

Anahtar Kavram

Distance, Rate, and Time Modeling with Piecewise Speed Changes

Soru 51Soru →

An isosceles triangle has two sides of length $10$ units each and a base of length $12$ units. A line segment parallel to the base cuts through the triangle, creating a smaller top triangle with an area of $12$ square units. What is the perimeter of the resulting trapezoid?

Cevabı ve açıklamayı göster

Cevap: 28

Cevap

The perimeter of the trapezoid is 28.

First, find the height of the original isosceles triangle with sides 10, 10, and base 12 by applying the Pythagorean theorem to half of the base:

h = \sqrt{10^2 - 6^2} = 8

. The area of the original triangle is

\frac{1}{2} \times 12 \times 8 = 48

square units. Because the segment is parallel to the base, the smaller top triangle is similar to the original triangle. The ratio of their areas is

\frac{12}{48} = \frac{1}{4}

, which means the linear scale factor is

\sqrt{\frac{1}{4}} = \frac{1}{2}

. Thus, the top triangle has legs of length

5

and a base of length

6

. The remaining non-parallel sides of the trapezoid each measure

10 - 5 = 5

units, and its bottom base is

12

units. Summing these four side lengths gives

5 + 5 + 6 + 12 = 28

.

Adım Adım Çözüm

1

Calculate the height and area of the original isosceles triangle

The altitude to the base bisects the base into two segments of length

6

. The altitude length is

h = \sqrt{10^2 - 6^2} = 8

. The area of the original triangle is

\frac{1}{2} \times 12 \times 8 = 48

square units.

Splitting the isosceles triangle along its altitude creates two right triangles with hypotenuse 10 and base leg 6.

2

Determine the linear scale factor of the smaller top triangle

The ratio of the area of the smaller triangle to the original triangle is

\frac{12}{48} = \frac{1}{4}

. Taking the square root yields a linear scale factor of

k = \sqrt{\frac{1}{4}} = \frac{1}{2}

.

A line parallel to the base forms a smaller triangle similar to the original triangle, and the ratio of areas of similar triangles is equal to the square of their linear scale factor.

3

Find the side lengths of the smaller triangle and the remaining side segments

The sides of the smaller triangle are

\frac{1}{2} \times 10 = 5

,

\frac{1}{2} \times 10 = 5

, and base

\frac{1}{2} \times 12 = 6

. The non-parallel side segments of the trapezoid are each

10 - 5 = 5

.

Multiplying the dimensions of the original triangle by the linear scale factor gives the side lengths of the top triangle.

4

Calculate the perimeter of the trapezoid

Perimeter =

5 + 5 + 6 + 12 = 28

.

Sum the lengths of the four boundary segments of the trapezoid.

Anahtar Kavram

Properties of isosceles triangles, Pythagorean theorem, area calculations, and similar triangle area ratios

Soru 52Soru →

An urban planning department surveyed a total of $360$ commercial buildings regarding three sustainability certifications: LEED ( $L$ ), BREEAM ( $B$ ), and WELL ( $W$ ). Exactly $50$ of the surveyed buildings held none of the three certifications. The survey revealed the following data:

• $180$ buildings held the LEED certification.
• $140$ buildings held the BREEAM certification.
• $130$ buildings held the WELL certification.
• $45$ buildings held both the LEED and BREEAM certifications.
• $40$ buildings held both the BREEAM and WELL certifications.
• $25$ buildings held all three certifications.

How many of the surveyed buildings held EXACTLY TWO of the three certifications?

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

90

To find the number of buildings with exactly two certifications, we first determine the total number of buildings holding at least one certification:

360 - 50 = 310

. Applying the three-set inclusion-exclusion formula,

310 = 180 + 140 + 130 - (45 + 40 + |L \cap W|) + 25

, which simplifies to

310 = 390 - |L \cap W|

, giving

|L \cap W| = 80

. Next, we isolate the regions holding exactly two certifications by subtracting the 25 triple-certified buildings from each pairwise intersection: LEED & BREEAM only =

45 - 25 = 20

, BREEAM & WELL only =

40 - 25 = 15

, and LEED & WELL only =

80 - 25 = 55

. Summing these three exclusive regions gives

20 + 15 + 55 = 90

.

Adım Adım Çözüm

1

Determine the total size of the union of all three sets

|L \cup B \cup W| = 360 - 50 = 310

Buildings holding at least one certification represent the entire surveyed population minus those holding no certifications.

2

Solve for the unknown overlap of LEED and WELL certifications using the inclusion-exclusion formula

|L \cap W| = 80

Using

|L \cup B \cup W| = |L| + |B| + |W| - |L \cap B| - |B \cap W| - |L \cap W| + |L \cap B \cap W|

, we have

310 = 180 + 140 + 130 - 45 - 40 - |L \cap W| + 25 = 390 - |L \cap W|

.

3

Calculate the number of buildings in each region corresponding to exactly two certifications

LEED and BREEAM only:

45 - 25 = 20

; BREEAM and WELL only:

40 - 25 = 15

; LEED and WELL only:

80 - 25 = 55

Each total pairwise intersection includes the 25 buildings that hold all three certifications, so subtracting 25 isolates those holding exclusively two certifications.

4

Sum the three isolated regions

20 + 15 + 55 = 90

The total number of buildings holding exactly two certifications is the sum of the three non-overlapping regions representing two certifications.

Anahtar Kavram

Three-Set Inclusion-Exclusion Principle and Partitioning Venn Diagrams

Tahmini Süre:2m 0s

Soru 53Soru →

A logistics company packages cargo using three types of containers: small, medium, and large.

- A shipment of $3$ small, $2$ medium, and $1$ large container has a total weight of $130$ kilograms.
- A shipment of $1$ small, $4$ medium, and $2$ large containers has a total weight of $185$ kilograms.
- A shipment of $2$ small, $1$ medium, and $3$ large containers has a total weight of $160$ kilograms.

What is the weight, in kilograms, of one large container?

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

The weight of one large container is 35 kilograms.

Representing the weights of small, medium, and large containers as variables

S

,

M

, and

L

yields the 3x3 system of linear equations:
1)

3S + 2M + L = 130

2)

S + 4M + 2L = 185

3)

2S + M + 3L = 160

Solving for

S

in equation (2) gives

S = 185 - 4M - 2L

. Substituting this into equations (1) and (3) reduces the system to:
-

2M + L = 85

-

7M + L = 210

Subtracting the first equation from the second yields

5M = 125

, so

M = 25

. Substituting

M = 25

into

2M + L = 85

gives

50 + L = 85

, which simplifies to

L = 35

.

Adım Adım Çözüm

1

Set up a system of three linear equations based on the shipment descriptions

3S + 2M + L = 130

,

S + 4M + 2L = 185

, and

2S + M + 3L = 160

Translate the physical constraints of the three shipments into mathematical relationships

2

Isolate variable

S

in the second equation and substitute into the first and third equations

Two equations in two variables:

2M + L = 85

and

7M + L = 210

Reduce the 3x3 system to a 2x2 system to eliminate variable

S

3

Subtract the two simplified equations to solve for

M

5M = 125 \implies M = 25

Eliminate variable

L

to obtain the value of

M

4

Substitute

M = 25

back into

2M + L = 85

to solve for

L

L = 35

Find the requested value for the weight of one large container

Anahtar Kavram

Solving 3x3 Systems of Linear Equations using Substitution and Elimination

Soru 54Soru →

A pharmaceutical laboratory produces a custom solution in a mixing tank using two automated pumps, Pump A and Pump B. Initially, the tank contains $600$ liters of a solution that is $20\%$ active reagent by volume. Pump A adds pure active reagent to the tank at a constant rate of $15$ liters per minute, while Pump B simultaneously adds distilled water ( $0\%$ active reagent) to the tank at a constant rate of $25$ liters per minute. At the exact instant when the mixture in the tank reaches $35\%$ active reagent by volume, Pump B is turned off while Pump A continues to add pure active reagent at $15$ liters per minute. How many total minutes, from the moment both pumps were initially started, does it take for the solution in the tank to reach $50\%$ active reagent by volume?

Cevabı ve açıklamayı göster

Cevap: 174

Cevap

The total elapsed time is 174 minutes.

The problem requires setting up a two-stage rational rate model. In the first stage, initial reagent volume is

120

liters (

20\%

of

600

). With Pump A adding

15

L/min of reagent and Pump B adding

25

L/min of water, the mixture concentration after

t_1

minutes is

\frac{120 + 15t_1}{600 + 40t_1} = 0.35

, yielding

t_1 = 90

minutes. At

t_1 = 90

, the total volume is

4,200

liters and the reagent volume is

1,470

liters. In the second stage, only Pump A is active, so concentration after

t_2

additional minutes is

\frac{1470 + 15t_2}{4200 + 15t_2} = 0.50

, yielding

t_2 = 84

minutes. Summing both intervals gives

90 + 84 = 174

total minutes.

Adım Adım Çözüm

1

Determine initial quantities of active reagent and total solution.

Initial total volume

V_0 = 600

liters. Initial volume of active reagent

A_0 = 0.20 \times 600 = 120

liters.

Establishing the baseline volume and solute concentration before any pumps are turned on.

2

Model Phase 1 where both pumps operate concurrently for

t_1

minutes to reach

35\%

concentration.

Total volume equation:

V(t_1) = 600 + (15 + 25)t_1 = 600 + 40t_1

. Reagent volume equation:

A(t_1) = 120 + 15t_1

. Set concentration to

0.35

:

\frac{120 + 15t_1}{600 + 40t_1} = 0.35 \implies t_1 = 90

minutes.

Both pumps add volume, but only Pump A adds active reagent. Solving

\frac{120 + 15t_1}{600 + 40t_1} = \frac{7}{20}

gives

20(120 + 15t_1) = 7(600 + 40t_1) \implies 2400 + 300t_1 = 4200 + 280t_1 \implies 20t_1 = 1800 \implies t_1 = 90

minutes.

3

Calculate the accumulated total volume and reagent volume at the end of Phase 1.

Total volume

V_1 = 600 + 40(90) = 4,200

liters. Active reagent volume

A_1 = 120 + 15(90) = 1,470

liters.

These amounts serve as the starting parameters for Phase 2.

4

Model Phase 2 where only Pump A operates for

t_2

additional minutes to reach

50\%

concentration.

In Phase 2, volume increases at

15

L/min and reagent increases at

15

L/min. Set concentration to

0.50

:

\frac{1470 + 15t_2}{4200 + 15t_2} = 0.50 \implies t_2 = 84

minutes.

Solving

2(1470 + 15t_2) = 4200 + 15t_2 \implies 2940 + 30t_2 = 4200 + 15t_2 \implies 15t_2 = 1260 \implies t_2 = 84

minutes.

5

Sum the durations of Phase 1 and Phase 2 to find the total time.

Total time

T = t_1 + t_2 = 90 + 84 = 174

minutes.

The question asks for the total minutes from the initial start of both pumps.

Anahtar Kavram

Multi-Phase Algebraic Rate and Mixture Modeling

Soru 55Soru →

In right trapezoid $ABCD$ , segment $AB$ is parallel to segment $CD$ , $\angle DAB = 90^\circ$ , $AD = 12$ , $CD = 15$ , and $BC = 13$ . Point $E$ lies on segment $CD$ such that quadrilateral $ABCE$ is a parallelogram. What is the perimeter of triangle $ADE$ ?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

Decomposing right trapezoid

ABCD

by dropping a perpendicular from

B

to

CD

forms a right triangle with height

12

and hypotenuse

13

. The Pythagorean theorem gives the base of this right triangle as

\sqrt{13^2 - 12^2} = 5

. Subtracting this from

CD = 15

yields

AB = 10

. Because

ABCE

is a parallelogram,

CE = AB = 10

, which leaves

DE = CD - CE = 15 - 10 = 5

. Triangle

ADE

is a right triangle with legs

AD = 12

and

DE = 5

, giving hypotenuse

AE = \sqrt{12^2 + 5^2} = 13

. The perimeter of triangle

ADE

is

12 + 5 + 13 = 30

.

Adım Adım Çözüm

1

Calculate the horizontal projection of segment

BC

onto base

CD

The length of the horizontal projection is

\sqrt{13^2 - 12^2} = 5

Segment

AD = 12

defines the perpendicular distance between parallel lines

AB

and

CD

2

Determine the length of parallel base

AB

AB = 15 - 5 = 10

The total length of base

CD = 15

is the sum of

AB

and the horizontal projection of slant side

BC

3

Calculate the length of segment

DE

DE = 15 - 10 = 5

Quadrilateral

ABCE

is a parallelogram, which implies

CE = AB = 10

4

Compute the hypotenuse

AE

and the total perimeter of triangle

ADE

AE = \sqrt{12^2 + 5^2} = 13

, so

\text{Perimeter} = 12 + 5 + 13 = 30

Triangle

ADE

is a right-angled triangle with right angle at vertex

D

Anahtar Kavram

Trapezoid height decomposition, parallelogram side properties, and Pythagorean theorem application

Soru 56Soru →

When the positive integer $n$ is divided by $7$ , the remainder is $4$ . What is the remainder when $n + 15$ is divided by $7$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Adding 15 to

n

increases the remainder by 15. The original remainder is 4, so the new sum of remainders is

4 + 15 = 19

. Dividing 19 by 7 gives a quotient of 2 and a remainder of 5.

Adım Adım Çözüm

1

Express

n

algebraically based on the given remainder rule.

n = 7k + 4

for some non-negative integer

k

By the division algorithm, any integer divided by 7 with remainder 4 can be written as a multiple of 7 plus 4.

2

Substitute

n

into the expression

n + 15

.

n + 15 = 7k + 4 + 15 = 7k + 19

We need to find the remainder of this new quantity when divided by 7.

3

Extract the largest multiple of 7 from 19.

7k + 19 = 7k + 14 + 5 = 7(k + 2) + 5

Grouping multiples of 7 isolates the remaining constant term.

4

Identify the final remainder.

The remainder is 5.

7(k + 2)

is completely divisible by 7, leaving 5 as the remainder.

Anahtar Kavram

Remainder arithmetic and divisibility properties

Soru 57Soru →

If $k$ is a positive integer less than $100$ such that $k$ is divisible by $6$ but not divisible by $8$ , what is the greatest possible value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

90

The positive multiples of 6 less than 100 in descending order are 96, 90, 84, etc. The largest candidate, 96, is divisible by 8 (

96 = 8 \times 12

), so it is excluded by the conditions. The next largest candidate, 90, is divisible by 6 (

90 = 6 \times 15

) but leaves a remainder of 2 when divided by 8 (

90 = 8 \times 11 + 2

). Thus, 90 is the greatest integer satisfying all conditions.

Adım Adım Çözüm

1

Identify the largest multiples of 6 that are less than 100.

The multiples of 6 less than 100 in descending order are 96, 90, 84, 78, ...

Finding the greatest possible value requires testing candidates starting from the largest possible multiple of 6 below 100.

2

Test 96 against the constraint of not being divisible by 8.

96 / 8 = 12, so 96 is divisible by 8.

The question specifies that k must NOT be divisible by 8, ruling out 96.

3

Test the next candidate, 90, against the constraint.

90 / 8 = 11 R 2, so 90 is not divisible by 8.

90 meets all specified criteria: it is a positive integer less than 100, divisible by 6, and not divisible by 8.

Anahtar Kavram

Properties of Multiples and Divisibility Constraints

Soru 58Soru →

If $k$ is an integer greater than $1$ such that $k^3 - k$ is a multiple of $240$ , what is the least possible value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The least possible value of

k

is 9.

Factoring

k^3 - k

gives

(k-1)k(k+1)

, representing the product of three consecutive integers. The prime factorization of

240

is

2^4 \times 3 \times 5

. To be divisible by

240

, the product

(k-1)k(k+1)

must contain at least four factors of 2, one factor of 3, and one factor of 5. Testing integers

k > 1

: for

k=9

, the product is

(8)(9)(10) = 720

, which is

3 \times 240

. Checking all integers

1 < k < 9

confirms that no smaller integer satisfies the condition.

Adım Adım Çözüm

1

Factor the algebraic expression

k^3 - k

.

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

.

This expresses the polynomial as a product of three consecutive integers.

2

Determine the prime factorization of 240.

240 = 2^4 \times 3 \times 5 = 16 \times 3 \times 5

.

For

(k-1)k(k+1)

to be a multiple of 240, the product must be divisible by 16, 3, and 5 simultaneously.

3

Test candidate values of

k > 1

to find the smallest valid integer.

For

k=9

,

(k-1)k(k+1) = (8)(9)(10) = 720

, which equals

3 \times 240

.

Testing smaller values:

k=5

yields

120

,

k=6

yields

210

,

k=7

yields

336

, and

k=8

yields

504

. None of these are divisible by 240.

Anahtar Kavram

Divisibility rules and prime factor distribution across consecutive integers

Tahmini Süre:2m 0s

Soru 59Soru →

When the positive integer $n$ is divided by $7$ , the remainder is $4$ . What is the remainder when $n + 18$ is divided by $7$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The remainder when

n + 18

is divided by

7

is

1

.

Because

n

leaves a remainder of

4

when divided by

7

,

n

can be represented as

7k + 4

. Adding

18

gives

n + 18 = 7k + 22

. Since

7k

is a multiple of

7

, the remainder of

(7k + 22) \div 7

depends entirely on

22 \div 7

. Dividing

22

by

7

gives a quotient of

3

with a remainder of

1

.

Adım Adım Çözüm

1

Represent the positive integer

n

algebraically based on the given remainder.

n = 7k + 4

for some non-negative integer

k

.

By the division algorithm, any integer

n

divided by

7

with remainder

4

can be written as

7k + 4

.

2

Add

18

to

n

.

n + 18 = (7k + 4) + 18 = 7k + 22

.

Substitute

7k + 4

for

n

in the expression

n + 18

.

3

Determine the remainder when

7k + 22

is divided by

7

.

Since

7k

is divisible by

7

, the remainder is

22 \pmod 7 = 1

.

Dividing

22

by

7

yields a quotient of

3

and a remainder of

1

(

22 = 7 \times 3 + 1

).

Anahtar Kavram

Remainder Properties under Addition

Soru 60Soru →

When the positive integer $n$ is divided by 7, the remainder is 3, and when $n$ is divided by 11, the remainder is 5. If $n$ is a three-digit integer less than 200 that is divisible by 6, what is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 192

Cevap

The value of

n

is 192.

To find

n

, combine the remainder requirements

n \equiv 3 \pmod 7

and

n \equiv 5 \pmod{11}

. The smallest positive integer solution is

38

. Because

\text{lcm}(7, 11) = 77

, all valid integers take the form

n = 77m + 38

. Restricting

n

to three-digit numbers less than 200 gives candidates

n = 115

(

m=1

) and

n = 192

(

m=2

). Among these, only 192 is divisible by 6 (since

192 = 6 \times 32

).

Adım Adım Çözüm

1

Formulate remainder congruences for the given conditions

n \equiv 3 \pmod{7}

and

n \equiv 5 \pmod{11}

Dividing

n

by 7 leaves remainder 3, and dividing by 11 leaves remainder 5.

2

Find the smallest positive integer satisfying both congruences

n = 38

Checking values

11k + 5

:

5, 16, 27, 38

.

38 \div 7 = 5

remainder 3, so 38 satisfies both conditions.

3

Determine the general solution for

n

using the Chinese Remainder Theorem logic

n = 77m + 38

for integer

m \ge 0

Since 7 and 11 are coprime, the solutions repeat every

\text{lcm}(7, 11) = 77

.

4

Evaluate candidate values for

n

such that

100 \le n < 200

For

m=1

,

n = 115

; for

m=2

,

n = 192

These are the only three-digit integers less than 200 of the form

77m + 38

.

5

Apply the final constraint that

n

must be divisible by 6

192

is divisible by 6 (

192 = 6 \times 32

)

115

is odd, so it is not divisible by 6.

192

is even and the sum of its digits (

1+9+2=12

) is a multiple of 3, so it is divisible by 6.

Anahtar Kavram

Simultaneous congruences and combined divisibility rules (Chinese Remainder Theorem)

Tahmini Süre:2m 0s

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