Tüm alıştırma soruları

Soru 901Soru →

A cargo logistics warehouse handles three types of freight: Standard, Express, and Fragile. At the start of the week, $\frac{2}{5}$ of the total shipments in the warehouse were Standard freight, $0.35$ of the total shipments were Express freight, and the remaining shipments were Fragile freight. During a midweek inventory adjustment, the number of Standard freight shipments increased by $25\%$ , the number of Express freight shipments decreased by $20\%$ , and the number of Fragile freight shipments remained unchanged at $125$ units. What was the total number of shipments in the warehouse after the midweek inventory adjustment?

Cevabı ve açıklamayı göster

Cevap: 515

Cevap

515 shipments

Converting

\frac{2}{5}

to a decimal gives

0.40

. The initial proportion of Fragile freight is

1 - 0.40 - 0.35 = 0.25

. Since Fragile freight equals

125

shipments, the initial total number of shipments is

\frac{125}{0.25} = 500

. Prior to adjustment, the warehouse has

0.40 \times 500 = 200

Standard shipments and

0.35 \times 500 = 175

Express shipments. After the adjustment, Standard shipments increase to

200 \times 1.25 = 250

, Express shipments decrease to

175 \times 0.80 = 140

, and Fragile shipments remain

125

. Summing these post-adjustment counts yields

250 + 140 + 125 = 515

shipments.

Adım Adım Çözüm

1

Determine the initial fraction representing Fragile freight.

Fragile freight constitutes

0.25

(or

25\%

) of the total initial shipments.

Converting

\frac{2}{5}

to a decimal yields

0.40

. Subtracting the Standard (

0.40

) and Express (

0.35

) proportions from the total (

1.00

) leaves

1.00 - 0.40 - 0.35 = 0.25

.

2

Calculate the total initial number of shipments in the warehouse.

Initial total shipments =

500

.

Since

25\%

of the total initial shipments equals

125

Fragile units, the initial total is

\frac{125}{0.25} = 500

.

3

Calculate the initial quantities of Standard and Express shipments.

Initial Standard shipments =

200

, Initial Express shipments =

175

.

Standard shipments =

0.40 \times 500 = 200

, and Express shipments =

0.35 \times 500 = 175

.

4

Apply the inventory percentage changes to each category.

New Standard shipments =

250

, New Express shipments =

140

, Fragile shipments =

125

.

A

25\%

increase in Standard shipments gives

200 \times 1.25 = 250

. A

20\%

decrease in Express shipments gives

175 \times 0.80 = 140

. Fragile shipments remain

125

.

5

Sum the adjusted shipment counts to determine the net total.

Final total shipments =

515

.

Adding the three updated categories gives

250 + 140 + 125 = 515

.

Anahtar Kavram

Fractions, Decimals, and Percents Arithmetic

Tahmini Süre:1m 30s

Soru 902Soru →

A café sells small cold brew coffees for $\$4$ each and large cold brew coffees for $\$6$ each. On Monday morning, the café sold a total of $40$ cold brew coffees for $\$196$ . How many large cold brew coffees were sold?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The café sold 18 large cold brew coffees.

Setting up the system

s + l = 40

and

4s + 6l = 196

, substituting

s = 40 - l

yields

4(40 - l) + 6l = 196

. Simplifying gives

160 + 2l = 196

, so

2l = 36

and

l = 18

.

Adım Adım Çözüm

1

Set up equations based on total count and total cost.

Let

s

= number of small coffees and

l

= number of large coffees. The system is

s + l = 40

and

4s + 6l = 196

.

The problem gives two distinct relationships between the quantity sold and the total dollar value.

2

Substitute

s = 40 - l

into the second equation.

4(40 - l) + 6l = 196 \implies 160 + 2l = 196

.

Substituting reduces the system from two variables to a single equation in

l

.

3

Solve the linear equation for

l

.

2l = 36 \implies l = 18

.

Subtracting 160 from both sides and dividing by 2 yields the number of large cold brews.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution.

Soru 903Soru →

If $k = 2^3 \cdot 3^a \cdot 7^b$ , where $a$ and $b$ are positive integers, and $k$ has exactly 36 positive integer divisors, how many positive integer divisors does $3k$ have?

39

40

48

54

72

Cevabı ve açıklamayı göster

Cevap: 48

Cevap

48

The number of positive integer divisors of a prime-factored number

p_1^{e_1} p_2^{e_2} \cdots p_m^{e_m}

is

(e_1+1)(e_2+1)\cdots(e_m+1)

. For

k = 2^3 \cdot 3^a \cdot 7^b

, the divisor count is

4(a+1)(b+1) = 36

, which simplifies to

(a+1)(b+1) = 9

. Given

a, b \ge 1

, the only integer solution is

a = 2

and

b = 2

. Multiplying

k

by 3 yields

3k = 2^3 \cdot 3^3 \cdot 7^2

, which has

(3+1)(3+1)(2+1) = 48

positive divisors.

Adım Adım Çözüm

1

Use the formula for the number of positive divisors of an integer.

The number of positive divisors of

k = 2^3 \cdot 3^a \cdot 7^b

is given by

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

.

If

N = p_1^{e_1} p_2^{e_2} \cdots p_m^{e_m}

, the total number of positive integer divisors is

(e_1 + 1)(e_2 + 1) \cdots (e_m + 1)

.

2

Set the divisor count equal to 36 and solve for

a

and

b

.

4(a+1)(b+1) = 36 \implies (a+1)(b+1) = 9

.

Dividing both sides by 4 isolates the term containing

a

and

b

.

3

Determine the values of

a

and

b

using the constraint that

a

and

b

are positive integers.

Since

a \ge 1

and

b \ge 1

, we have

a+1 \ge 2

and

b+1 \ge 2

. The only factor pair of 9 where both factors are at least 2 is

3 \times 3

. Thus,

a+1 = 3 \implies a = 2

and

b+1 = 3 \implies b = 2

.

Positive integers must be greater than or equal to 1.

4

Write the prime factorization of

3k

and compute its number of positive divisors.

3k = 3 \cdot (2^3 \cdot 3^2 \cdot 7^2) = 2^3 \cdot 3^3 \cdot 7^2

. The number of divisors is

(3+1)(3+1)(2+1) = 4 \cdot 4 \cdot 3 = 48

.

Multiplying

k

by 3 increases the exponent of 3 from 2 to 3.

Anahtar Kavram

Prime Factorization and Total Divisors Rule

Tahmini Süre:1m 30s

Soru 904Soru →

At an urban hydroponic farm, a monthly harvest consists of three types of crops: leafy greens, herbs, and microgreens. Leafy greens account for $\frac{5}{8}$ of the total harvest by weight, and herbs account for $0.25$ of the total harvest by weight. If the yield of microgreens increases by $40\%$ in the following month while the yields of leafy greens and herbs remain constant, what fraction of the new total harvest by weight will be microgreens?

\frac{7}{40}

\frac{1}{6}

\frac{1}{5}

\frac{7}{22}

\frac{7}{32}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{6}$

Cevap

The fraction of the new total harvest that will be microgreens is

\frac{1}{6}

.

Initially, leafy greens and herbs account for

\frac{5}{8} + 0.25 = \frac{5}{8} + \frac{2}{8} = \frac{7}{8}

of the total harvest weight, leaving microgreens with

1 - \frac{7}{8} = \frac{1}{8}

of the total weight. When microgreens increase by

40\%

, their new yield becomes

\frac{1}{8} \times 1.40 = \frac{7}{40}

of the original total harvest. The total harvest weight increases to

\frac{7}{8} + \frac{7}{40} = \frac{35}{40} + \frac{7}{40} = \frac{42}{40}

. The fraction of microgreens in this new total harvest is

\frac{7/40}{42/40} = \frac{7}{42} = \frac{1}{6}

.

Adım Adım Çözüm

1

Convert all initial crop portions into consistent fractions to find the initial microgreens fraction.

Leafy greens =

\frac{5}{8}

. Herbs =

0.25 = \frac{1}{4} = \frac{2}{8}

. Microgreens fraction =

1 - \left(\frac{5}{8} + \frac{2}{8}\right) = 1 - \frac{7}{8} = \frac{1}{8}

.

The sum of all three crop fractions must equal

1

(the total harvest).

2

Calculate the updated relative weight of microgreens after a

40\%

increase.

New microgreens weight relative to original total =

\frac{1}{8} \times (1 + 0.40) = \frac{1}{8} \times \frac{7}{5} = \frac{7}{40}

.

Increasing a quantity by

40\%

is equivalent to multiplying by

1.40

or

\frac{7}{5}

.

3

Determine the new total harvest weight relative to the original harvest.

New total harvest = Leafy greens + Herbs + New microgreens =

\frac{5}{8} + \frac{2}{8} + \frac{7}{40} = \frac{35}{40} + \frac{7}{40} = \frac{42}{40} = \frac{21}{20}

.

Since the yields of leafy greens and herbs did not change, their combined relative weight remains

\frac{35}{40}

.

4

Compute the fraction of the new total harvest that consists of microgreens.

Fraction =

\frac{\frac{7}{40}}{\frac{42}{40}} = \frac{7}{42} = \frac{1}{6}

.

Dividing the new microgreens weight by the new total harvest weight gives the updated proportion.

Anahtar Kavram

Fractions, Decimals, and Percents Arithmetic

Tahmini Süre:1m 30s

Soru 905Soru →

Set $A$ consists of $k$ consecutive integers, where $k > 1$ . Set $B$ consists of $2k$ consecutive integers. The median of Set $A$ is equal to the least integer in Set $B$ , and the sum of all integers in Set $A$ is equal to the sum of all integers in Set $B$ . If the sum of the integers in Set $B$ is $-153$ , what is the greatest integer in Set $A$ ?

-21

-17

-13

-12

-9

Cevabı ve açıklamayı göster

Cevap: $-13$

Cevap

The greatest integer in Set

A

is

-13

.

The problem establishes a relationship between two sets of consecutive integers using their medians and sums. Equating the sum formulas

S_A = k\left(a + \frac{k-1}{2}\right)

and

S_B = 2k\left(a + \frac{3k-2}{2}\right)

leads to

a = \frac{3-5k}{2}

. Substituting this into the sum expression gives

k - 2k^2 = -153

, yielding

k = 9

. Thus, the smallest element is

a = -21

, and the greatest element is

-21 + 9 - 1 = -13

.

Adım Adım Çözüm

1

Express the sum of Set

A

in terms of

k

and its smallest element

a

.

The median of Set

A

is

M_A = a + \frac{k-1}{2}

, so the sum of Set

A

is

S_A = k \cdot M_A = k\left(a + \frac{k-1}{2}\right)

.

In any set of consecutive integers, the sum equals the number of terms multiplied by the average (median).

2

Express the sum of Set

B

using the given conditions.

The least integer in Set

B

is

M_A = a + \frac{k-1}{2}

. Since Set

B

has

2k

consecutive integers, its median is

M_B = \left(a + \frac{k-1}{2}\right) + \frac{2k-1}{2} = a + \frac{3k-2}{2}

. The sum of Set

B

is

S_B = 2k \cdot M_B = 2k\left(a + \frac{3k-2}{2}\right) = k(2a + 3k - 2)

.

Set

B

starts at the median of Set

A

and contains

2k

terms.

3

Equate

S_A

and

S_B

to solve for

a

in terms of

k

.

Since

S_A = S_B

and

k > 1

, we divide by

k

:

a + \frac{k-1}{2} = 2a + 3k - 2 \implies a = \frac{3 - 5k}{2}

.

The problem states that the sums of the two sets are equal.

4

Substitute

a = \frac{3 - 5k}{2}

into

S_B

and set it equal to

-153

.

S_B = k\left(2\left(\frac{3-5k}{2}\right) + 3k - 2\right) = k(3 - 5k + 3k - 2) = k(1 - 2k) = k - 2k^2

. Setting

k - 2k^2 = -153 \implies 2k^2 - k - 153 = 0

. Factoring gives

(2k + 17)(k - 9) = 0

. Since

k

must be a positive integer greater than 1,

k = 9

.

Finding

k

allows us to calculate the exact elements of Set

A

.

5

Calculate the greatest integer in Set

A

.

With

k = 9

, the smallest integer

a = \frac{3 - 5(9)}{2} = -21

. The greatest integer in Set

A

is

a + k - 1 = -21 + 9 - 1 = -13

.

The

k

-th term of a sequence of consecutive integers starting at

a

is

a + k - 1

.

Anahtar Kavram

Properties of consecutive integer sets: sum equals number of terms times median; indexing terms in evenly spaced sets.

Soru 906Soru →

A financial analyst models a company's total annual revenue, $R(t)$ , and total annual expenses, $E(t)$ , in thousands of dollars, as linear functions of time $t$ , where $t$ represents the number of years since 2020. In 2022 ( $t = 2$ ), the annual revenue was $\$850,000$ and annual expenses were $\$610,000$ . In 2025 ( $t = 5$ ), the annual revenue reached $\$1,120,000$ while annual expenses reached $\$700,000$ . Assuming these linear trends continue, in which calendar year will the company's annual profit (defined as total annual revenue minus total annual expenses) reach exactly $\$540,000$ ?

Cevabı ve açıklamayı göster

Cevap: 2027

Cevap

The company's annual profit will reach $540,000 in the year 2027.

The linear profit function increases at a constant rate of

60,000 per year from a base profit of

240,000 in 2022 (

t = 2

). Solving

240 + 60(t - 2) = 540

yields

t = 7

, which corresponds to the calendar year 2027.

Adım Adım Çözüm

1

Calculate the annual profit in thousands of dollars for the known years 2022 (

t = 2

) and 2025 (

t = 5

).

In 2022 (

t = 2

), profit

P(2) = 850 - 610 = 240

thousand dollars. In 2025 (

t = 5

), profit

P(5) = 1120 - 700 = 420

thousand dollars.

Profit is defined as revenue minus expenses.

2

Determine the linear profit function

P(t) = mt + b

.

The rate of change of profit per year is

m = \frac{420 - 240}{5 - 2} = \frac{180}{3} = 60

thousand dollars per year.

Since both revenue and expenses are linear functions of time

t

, their difference

P(t) = R(t) - E(t)

is also a linear function of

t

.

3

Set up the linear equation for profit using point-slope form.

P(t) - 240 = 60(t - 2) \implies P(t) = 60t + 120

.

Using the point

(2, 240)

and slope

m = 60

establishes the complete linear equation for annual profit.

4

Solve the linear equation for

t

when profit

P(t) = 540

thousand dollars.

60t + 120 = 540 \implies 60t = 420 \implies t = 7

.

Setting the profit equal to 540 gives the value of

t

years after 2020.

5

Convert the value of

t

into the target calendar year.

Calendar Year

= 2020 + 7 = 2027

.

Since

t

represents years elapsed since 2020,

t = 7

corresponds to calendar year 2027.

Anahtar Kavram

Linear Modeling and Linear Equations in Two Variables

Soru 907Soru →

How many integer values of $x$ satisfy both $|x^2 - 10| \le 6$ and $|x - 1| > 2$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4 integer values satisfy both inequalities:

-4

,

-3

,

-2

, and

4

.

Solving

|x^2 - 10| \le 6

gives

4 \le x^2 \le 16

, which yields 6 integer solutions:

\{-4, -3, -2, 2, 3, 4\}

. Applying the condition

|x - 1| > 2

requires

x < -1

or

x > 3

. Only

-4, -3, -2

, and

4

satisfy both conditions, giving exactly 4 valid integer values.

Adım Adım Çözüm

1

Solve the quadratic absolute value inequality

|x^2 - 10| \le 6

.

-6 \le x^2 - 10 \le 6 \implies 4 \le x^2 \le 16

.

Removing the absolute value bars gives the bounded inequality

-6 \le x^2 - 10 \le 6

. Adding 10 across all parts isolates

x^2

.

2

Identify all candidate integer values of

x

for which

4 \le x^2 \le 16

.

x \in \{-4, -3, -2, 2, 3, 4\}

.

Taking the square root yields

-4 \le x \le -2

or

2 \le x \le 4

. The integer solutions are

-4, -3, -2, 2, 3, 4

.

3

Solve the linear absolute value inequality

|x - 1| > 2

.

x > 3

or

x < -1

.

An absolute value distance greater than 2 expands into two disjunctive linear inequalities:

x - 1 > 2

or

x - 1 < -2

.

4

Intersect the candidate integer set

\{-4, -3, -2, 2, 3, 4\}

with the domain

x < -1

or

x > 3

.

The valid integers are

-4, -3, -2,

and

4

. The values

2

and

3

fail the inequality since they fall inside the excluded interval

[-1, 3]

.

Testing each candidate shows that

-4, -3, -2

are strictly less than

-1

, and

4

is strictly greater than

3

.

Anahtar Kavram

Solving compound absolute value inequalities and finding integer solution set intersections.

Soru 908Soru →

What is the remainder when the integer $N = 3^{83} + 7^{83}$ is divided by $100$ ?

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

The remainder when

N = 3^{83} + 7^{83}

is divided by

100

is 70.

Finding the remainder when an integer is divided by

100

is equivalent to finding its last two digits. Powers of

3

repeat their last two digits in cycles of

20

, and

3^{20} \equiv 1 \pmod{100}

. Thus

3^{83} \equiv 3^3 = 27 \pmod{100}

. Powers of

7

repeat their last two digits in cycles of

4

, with

7^4 = 2401 \equiv 1 \pmod{100}

. Thus

7^{83} \equiv 7^3 = 343 \equiv 43 \pmod{100}

. Combining these gives

27 + 43 = 70

.

Adım Adım Çözüm

1

Find the remainder of

3^{83}

modulo

100

by identifying the cyclicity pattern of powers of

3

.

3^{20} \equiv 1 \pmod{100}

, which implies

3^{83} = (3^{20})^4 \cdot 3^3 \equiv 1^4 \cdot 27 = 27 \pmod{100}

.

Since

3^{20} \equiv 1 \pmod{100}

, exponent

83

can be reduced to

83 \pmod{20} = 3

.

2

Find the remainder of

7^{83}

modulo

100

by identifying the cyclicity pattern of powers of

7

.

7^4 = 2401 \equiv 1 \pmod{100}

, which implies

7^{83} = (7^4)^{20} \cdot 7^3 \equiv 1^{20} \cdot 343 \equiv 43 \pmod{100}

.

Since

7^4 \equiv 1 \pmod{100}

, exponent

83

can be reduced to

83 \pmod{4} = 3

.

3

Sum the modular remainders to obtain the final remainder of

N

.

N = 3^{83} + 7^{83} \equiv 27 + 43 = 70 \pmod{100}

.

The remainder of a sum is equal to the sum of the individual remainders modulo

100

.

Anahtar Kavram

Modular Arithmetic and Cyclicity of Last Two Digits

Soru 909Soru →

If $x$ is a real number satisfying the equation $|2x - 3| = 3x + 1$ , what is the value of $x^2 + 2x$ ?

-\frac{16}{25}

\frac{8}{25}

\frac{24}{25}

8

\frac{56}{25}

Cevabı ve açıklamayı göster

Cevap: $\frac{24}{25}$

Cevap

The value of

x^2 + 2x

is

\frac{24}{25}

.

The equation

|2x - 3| = 3x + 1

requires

3x + 1 \ge 0

, or

x \ge -\frac{1}{3}

. Setting

2x - 3 = 3x + 1

gives

x = -4

, which violates

x \ge -\frac{1}{3}

and is extraneous. Setting

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

yields

-2x + 3 = 3x + 1

, which simplifies to

5x = 2

, or

x = \frac{2}{5}

. Since

\frac{2}{5} \ge -\frac{1}{3}

, this is the unique valid solution. Substituting

x = \frac{2}{5}

into

x^2 + 2x

gives

\left(\frac{2}{5}\right)^2 + 2\left(\frac{2}{5}\right) = \frac{4}{25} + \frac{20}{25} = \frac{24}{25}

.

Adım Adım Çözüm

1

Set up the conditions for the absolute value equation

|2x - 3| = 3x + 1

.

Since the absolute value

|2x - 3|

must be non-negative, the right-hand side must satisfy

3x + 1 \ge 0

, which means

x \ge -\frac{1}{3}

.

An absolute value expression cannot equal a negative number.

2

Solve Case 1:

2x - 3 \ge 0

(so

x \ge \frac{3}{2}

).

2x - 3 = 3x + 1 \implies x = -4

.

Remove the absolute value bars directly when the inside expression is non-negative.

3

Check Case 1 candidate solution against constraints.

x = -4

is extraneous because

-4 < \frac{3}{2}

and

3(-4) + 1 = -11 < 0

. Reject

x = -4

.

Plugging

x = -4

into the original equation yields

|-11| = -11

, which is false.

4

Solve Case 2:

2x - 3 < 0

(so

x < \frac{3}{2}

).

-(2x - 3) = 3x + 1 \implies -2x + 3 = 3x + 1 \implies 5x = 2 \implies x = \frac{2}{5}

.

Negate the expression inside the absolute value when it is negative.

5

Check Case 2 candidate solution and evaluate the target expression.

Since

x = \frac{2}{5} \ge -\frac{1}{3}

, it is valid. Then

x^2 + 2x = \left(\frac{2}{5}\right)^2 + 2\left(\frac{2}{5}\right) = \frac{4}{25} + \frac{4}{5} = \frac{24}{25}

.

Substitute the single valid real solution into

x^2 + 2x

.

Anahtar Kavram

Absolute Value Equations and Extraneous Solution Elimination

Soru 910Soru →

What is the smallest positive integer $m$ such that the remainder when $6^m$ is divided by $11$ is equal to $8$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The smallest positive integer

m

is 7.

Evaluating powers of 6 modulo 11 sequentially gives:

6^1 \equiv 6

,

6^2 \equiv 3

,

6^3 \equiv 7

,

6^4 \equiv 9

,

6^5 \equiv 10

,

6^6 \equiv 5

, and

6^7 \equiv 8 \pmod{11}

. Therefore,

m = 7

is the smallest positive integer satisfying the condition.

Adım Adım Çözüm

1

Calculate the remainder of powers of 6 when divided by 11 starting from

m = 1

.

6^1 \equiv 6 \pmod{11}

,

6^2 \equiv 3 \pmod{11}

,

6^3 \equiv 7 \pmod{11}

,

6^4 \equiv 9 \pmod{11}

,

6^5 \equiv 10 \pmod{11}

,

6^6 \equiv 5 \pmod{11}

,

6^7 \equiv 8 \pmod{11}

.

Modular multiplication allows computing powers step-by-step by multiplying the previous remainder by 6 and finding the remainder modulo 11.

2

Identify the smallest exponent

m

that yields a remainder of 8.

m = 7

At

m = 7

,

6^7 = 279,936 = 11 \times 25,448 + 8

, which leaves a remainder of 8.

Anahtar Kavram

Modular Arithmetic and Cyclicity of Powers

Soru 911Soru →

An investment firm allocated a total of $\$120,000$ between two venture capital funds, Fund X and Fund Y. Fund X yielded an annual simple interest rate of $8\%$ , while Fund Y yielded an annual simple interest rate of $12\%$ . If the total interest earned from both funds combined after one year was $\$11,600$ , how much money was invested in Fund X?

\$70,000

\$50,000

\$60,000

\$58,000

\$75,000

Cevabı ve açıklamayı göster

Cevap: $\$70,000$

Cevap

\$70,000

The correct answer is

\$70,000

. By modeling the investment in Fund X as

x

, the investment in Fund Y becomes

120,000 - x

. Setting up the total annual interest equation

0.08x + 0.12(120,000 - x) = 11,600

yields

-0.04x = -2,800

, giving

x = 70,000

.

Adım Adım Çözüm

1

Define variables for the invested amounts

Let

x

be the amount invested in Fund X. The remaining amount invested in Fund Y is

\$120,000 - x

.

The total capital is

\$120,000

, so the investments must sum to this total.

2

Set up the interest equation

0.08x + 0.12(120,000 - x) = 11,600

Total interest is the sum of interest from Fund X (

8\%

of

x

) and Fund Y (

12\%

of

120,000 - x

).

3

Expand and simplify the linear equation

0.08x + 14,400 - 0.12x = 11,600 \Rightarrow -0.04x + 14,400 = 11,600

Distribute

0.12

across

(120,000 - x)

and combine like terms.

4

Isolate the variable

x

-0.04x = 11,600 - 14,400 \Rightarrow -0.04x = -2,800 \Rightarrow x = \frac{-2,800}{-0.04} = 70,000

Subtract

14,400

from both sides and divide by

-0.04

to find the investment in Fund X.

Anahtar Kavram

Linear Equation Modeling for Investment Allocations

Tahmini Süre:1m 40s

Soru 912Soru →

Let $n$ be the smallest positive integer that is a multiple of 15 and leaves a remainder equivalent to $-2$ when divided by 7. What is the value of $n$ ?

30

75

45

105

60

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

75

The correct value is 75. Since

n

is a multiple of 15,

n = 15k

. A remainder of

-2

when divided by 7 means

n \equiv 5 \pmod 7

. Substituting

n = 15k

gives

15k \equiv 5 \pmod 7

, which simplifies to

k \equiv 5 \pmod 7

since

15 \equiv 1 \pmod 7

. The smallest positive integer value for

k

is 5, giving

n = 15 \times 5 = 75

.

Adım Adım Çözüm

1

Express

n

as a multiple of 15

n = 15k

for some positive integer

k

Since

n

is a multiple of 15, it must be of the form

15k

.

2

Convert the negative remainder condition into a standard positive remainder modulo 7

n \equiv -2 \equiv 5 \pmod 7

A remainder of

-2

modulo 7 is equivalent to

7 + (-2) = 5

.

3

Solve the modular congruence for

k

k = 5

Substituting

n = 15k

into

15k \equiv 5 \pmod 7

gives

1k \equiv 5 \pmod 7

because

15 \equiv 1 \pmod 7

. The smallest positive integer solution is

k = 5

.

4

Calculate the value of

n

n = 15 \times 5 = 75

Multiplying 15 by

k = 5

yields 75.

Anahtar Kavram

Modular Arithmetic and Remainder Properties

Tahmini Süre:1m 30s

Soru 913Soru →

For any positive integer $n$ , let $f(n)$ denote the remainder when $13^n + 17^n$ is divided by $10$ . What is the sum of all possible distinct values of $f(n)$ ?

0

8

10

18

20

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The sum of all possible distinct values is 10.

Finding the remainder when a number is divided by

10

is equivalent to finding its units digit. The units digit of

13^n

follows the repeating pattern

[3, 9, 7, 1]

, and the units digit of

17^n

follows the repeating pattern

[7, 9, 3, 1]

. Adding these term-by-term yields units digits of

(3+7) \rightarrow 0

,

(9+9) \rightarrow 8

,

(7+3) \rightarrow 0

, and

(1+1) \rightarrow 2

. The set of distinct remainders is

\{0, 8, 2\}

, and their sum is

0 + 8 + 2 = 10

.

Adım Adım Çözüm

1

Relate division by 10 to units digits

The remainder when any positive integer is divided by

10

is equal to its units digit. Therefore,

f(n)

is the units digit of

13^n + 17^n

, which depends only on the units digits of

3^n

and

7^n

.

Dividing by 10 isolates the ones place of an integer.

2

Determine the units digit pattern for

3^n

and

7^n

The units digits of

3^n

follow a 4-term repeating cycle:

[3, 9, 7, 1]

. The units digits of

7^n

follow a 4-term repeating cycle:

[7, 9, 3, 1]

.

Powers of integers exhibit periodic patterns in their units digits.

3

Evaluate

f(n)

for each term in the 4-step pattern

For

n=1

: units digit of

3 + 7 = 10

is

0

, so

f(1) = 0

.
For

n=2

: units digit of

9 + 9 = 18

is

8

, so

f(2) = 8

.
For

n=3

: units digit of

7 + 3 = 10

is

0

, so

f(3) = 0

.
For

n=4

: units digit of

1 + 1 = 2

is

2

, so

f(4) = 2

.

Test one full period of length 4 to find all possible outputs.

4

Find the sum of all distinct values

The distinct values of

f(n)

are

0

,

8

, and

2

. Their sum is

0 + 8 + 2 = 10

.

The question asks for the sum of distinct possible remainders.

Anahtar Kavram

Units Digit Cyclicity and Modular Arithmetic

Tahmini Süre:1m 45s

Soru 914Soru →

A production facility uses two automated machines, Alpha and Beta, which operate at constant hourly production rates. Machine Alpha produces $x$ units per hour and Machine Beta produces $y$ units per hour. When both machines operate together for 4 hours, followed by Machine Alpha operating alone for 3 hours, the facility produces a total of 310 units. When Machine Beta operates alone for 2 hours, followed by both machines operating together for 5 hours, the facility produces a total of 325 units. If Machine Alpha's hourly rate is increased by $20\%$ and Machine Beta's hourly rate is decreased by $20\%$ , how many total units will both machines produce if they work together for 8 hours at their new rates?

Cevabı ve açıklamayı göster

Cevap: 448

Cevap

The total number of units produced by both machines working together for 8 hours under the modified rates is 448.

Setting up the system of equations based on individual hours worked yields

7x + 4y = 310

and

5x + 7y = 325

. Solving this system gives original rates

x = 30

units/hr and

y = 25

units/hr. Applying the

20\%

increase to Alpha (

36

units/hr) and

20\%

decrease to Beta (

20

units/hr) gives a combined rate of

56

units/hr. Multiplying by

8

hours gives the final answer of

448

units.

Adım Adım Çözüm

1

Translate the given operational scenarios into a system of two linear equations.

Equation 1:

7x + 4y = 310

; Equation 2:

5x + 7y = 325

.

Operating both machines for 4 hours and Alpha alone for 3 hours means Alpha works

4 + 3 = 7

hours while Beta works

4

hours, giving

7x + 4y = 310

. Operating Beta alone for 2 hours and both for 5 hours means Alpha works

5

hours while Beta works

2 + 5 = 7

hours, giving

5x + 7y = 325

.

2

Solve the system of linear equations for variables

x

and

y

.

x = 30

and

y = 25

.

Eliminating

y

by multiplying the first equation by 7 and the second by 4 yields

49x - 20x = 2170 - 1300 \implies 29x = 870 \implies x = 30

. Substituting

x = 30

back into

7x + 4y = 310

gives

210 + 4y = 310 \implies 4y = 100 \implies y = 25

.

3

Calculate the modified production rates after percentage adjustments.

New rate for Alpha is

36

units/hr; new rate for Beta is

20

units/hr.

A

20\%

increase on

x = 30

yields

30 \times 1.20 = 36

. A

20\%

decrease on

y = 25

yields

25 \times 0.80 = 20

.

4

Compute total combined output over 8 hours.

Total units produced =

448

.

Combined modified rate is

36 + 20 = 56

units per hour. Total production over 8 hours is

56 \times 8 = 448

.

Anahtar Kavram

Linear Equations in One and Two Variables

Tahmini Süre:2m 30s

Soru 915Soru →

A logistics company dispatches a fleet of delivery vehicles consisting entirely of Van A models and Van B models for a day's operations. Each Van A carries $40$ packages and consumes $5$ gallons of fuel per route. Each Van B carries $60$ packages and consumes $8$ gallons of fuel per route. On a given day, $18$ vans were dispatched in total, delivering a combined total of $920$ packages. Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The total number of Van B models dispatched was

10

.

The total fuel consumed by all Van A models combined was

50

gallons.

Van B models accounted for more than

60\%

of the total fuel consumed by the fleet.

The ratio of the number of Van A models to Van B models dispatched was

5

to

4

.

The total fuel consumed by the entire fleet was

120

gallons.

Cevabı ve açıklamayı göster

Cevap: The total number of Van B models dispatched was $10$ .; Van B models accounted for more than $60\%$ of the total fuel consumed by the fleet.; The total fuel consumed by the entire fleet was $120$ gallons.

Cevap

The true statements are those stating that 10 Van B models were dispatched, that Van B models accounted for more than 60% of total fleet fuel consumed, and that the total fuel consumed by the entire fleet was 120 gallons.

Solving the linear system

A + B = 18

and

40A + 60B = 920

yields

A = 8

Van A models and

B = 10

Van B models. Evaluating the claims: 10 Van B models were dispatched (true); Van B fuel (

80

gallons) represents

80/120 = 66.67\%

of total fuel, which is greater than

60\%

(true); and total fleet fuel consumption is

40 + 80 = 120

gallons (true).

Adım Adım Çözüm

1

Define variables and construct the system of linear equations.

Let

A

be the number of Van A models and

B

be the number of Van B models. Total vans equation:

A + B = 18

. Total packages equation:

40A + 60B = 920

.

Word problem modeling requires translating fleet count and package delivery totals into algebraic linear equations.

2

Solve the system of equations for

A

and

B

.

Divide the package equation by

20

:

2A + 3B = 46

. Substitute

A = 18 - B

:

2(18 - B) + 3B = 46 \implies 36 + B = 46 \implies B = 10

. Then

A = 18 - 10 = 8

.

Determining exact vehicle quantities is required to evaluate all downstream statements.

3

Calculate total fuel consumption for each van model and the entire fleet.

Van A fuel:

8 \text{ vans} \times 5 \text{ gal/van} = 40 \text{ gallons}

. Van B fuel:

10 \text{ vans} \times 8 \text{ gal/van} = 80 \text{ gallons}

. Total fleet fuel:

40 + 80 = 120 \text{ gallons}

.

Provides exact quantities to test statement validity.

4

Evaluate each statement against the calculated values.

1) Van B count is 10 (True). 2) Van A fuel is 40 gallons, not 50 (False). 3) Van B fuel percentage is

80/120 = 66.67\% > 60\%

(True). 4) Van A to Van B ratio is

8:10 = 4:5

, not

5:4

(False). 5) Total fuel is 120 gallons (True).

Identifies all correct options for multiple selection.

Anahtar Kavram

Linear Equation System Modeling in Applied Contexts

Soru 916Soru →

If $x$ is a real number satisfying the equation $|x + 3| + |x - 5| = 8$ , what is the maximum possible value of $|x^2 - 4x - 12|$ ?

7

9

12

16

25

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

The equation

|x + 3| + |x - 5| = 8

represents the sum of the distances from

x

to

-3

and from

x

to

5

. Since the distance between

-3

and

5

is exactly

8

, this equation holds for all

x

in the interval

[-3, 5]

. Rewriting

x^2 - 4x - 12

as

(x - 2)^2 - 16

, the vertex occurs at

x = 2

, which lies inside

[-3, 5]

. At

x = 2

,

x^2 - 4x - 12 = -16

, so

|x^2 - 4x - 12| = |-16| = 16

. At the endpoints

x = -3

and

x = 5

, the values are

|9| = 9

and

|-7| = 7

, respectively. Thus, the maximum possible value of

|x^2 - 4x - 12|

on the interval is 16.

Adım Adım Çözüm

1

Determine the solution set of the absolute value equation

|x + 3| + |x - 5| = 8

.

The domain of valid

x

values is the continuous closed interval

[-3, 5]

.

By the geometric distance interpretation, the sum of distances from

x

to

-3

and

x

to

5

equals the total distance between

-3

and

5

(which is

8

) if and only if

x

lies between

-3

and

5

inclusive.

2

Express the quadratic expression

g(x) = x^2 - 4x - 12

in vertex form.

g(x) = (x - 2)^2 - 16

.

Completing the square allows straightforward evaluation of the vertex and minimum/maximum values of the quadratic on

[-3, 5]

.

3

Find the range of

g(x) = (x - 2)^2 - 16

for

x \in [-3, 5]

.

The minimum value occurs at the vertex

x = 2

, where

g(2) = -16

. The maximum value occurs at the endpoint

x = -3

, where

g(-3) = 9

. Thus,

-16 \le g(x) \le 9

.

The parabola opens upwards with vertex at

x = 2 \in [-3, 5]

. The distance from

x = 2

to

x = -3

is

5

, while the distance to

x = 5

is

3

.

4

Evaluate the maximum value of

|g(x)| = |x^2 - 4x - 12|

on

[-3, 5]

.

The absolute value

|g(x)|

ranges from

0

to

\max(|-16|, |9|) = 16

.

The absolute value converts negative outputs to positive, so the extreme magnitude

|-16| = 16

at

x = 2

is the maximum value.

Anahtar Kavram

Absolute Value Distance Interpretation and Quadratic Range Optimization

Tahmini Süre:2m 0s

Soru 917Soru →

Urban planners in Metroville recently proposed planting deep-rooted oak trees along city avenues to expand the tree canopy and mitigate the urban heat island effect. Critics argued that the extensive root networks would crack adjacent underground water pipes, incurring massive municipal repair costs. In response, city planners pointed out that the target avenues are lined with modern flexible polymer pipes rather than traditional brittle clay pipes, concluding that the tree-planting initiative will not lead to pipe repair costs. Which of the following, if true, most strengthens the city planners' argument?

Flexible polymer pipes have been shown in municipal engineering tests to bend under pressure from expanding tree roots without fracturing or leaking.

Expanding root systems of oak trees exude chemical compounds that gradually degrade polymer material, causing flexible pipes to rupture faster than clay pipes.

Traditional brittle clay pipes in older municipal districts currently account for over seventy percent of Metroville's annual water utility maintenance budget.

Expanding the urban tree canopy reduces ambient surface temperatures, thereby lowering municipal expenditures on seasonal air conditioning.

Cities that replaced brittle clay pipes with flexible polymer pipes experienced a significant drop in initial line installation costs.

Cevabı ve açıklamayı göster

Cevap: Flexible polymer pipes have been shown in municipal engineering tests to bend under pressure from expanding tree roots without fracturing or leaking.

Cevap

The statement that flexible polymer pipes bend under root pressure without fracturing or leaking most strengthens the argument.

The city planners defend their plan against critics by noting that the target avenues use flexible polymer pipes rather than brittle clay pipes. For this defense to hold, flexible polymer pipes must actually withstand root pressure better than clay pipes. Confirming that polymer pipes bend under root pressure without fracturing directly reinforces the planners' conclusion that pipe repair costs will be avoided.

Adım Adım Çözüm

1

Deconstruct the argument structure

Premise: Target avenues use flexible polymer pipes instead of brittle clay pipes. Conclusion: Planting oak trees will not cause pipe repair costs.

Identifying the premise-to-conclusion gap isolates the unstated assumption.

2

Identify the logical gap

The argument assumes that flexible polymer pipes will not crack or break when exposed to expanding oak tree roots.

Strengthening questions require finding evidence that validates key assumptions or rules out potential failure modes.

3

Evaluate the choices against the logical gap

The option confirming that polymer pipes withstand root expansion by bending without fracturing directly validates the planners' defense.

Providing empirical evidence of durability bridges the premise to the conclusion.

Anahtar Kavram

Strengthening Causal Defenses by Validating Assumptions

Soru 918Soru →

For any positive integer $k$ , what is the units digit of the expression $2^{4k+2} + 3^{4k+3} + 7^{4k+1}$ ?

8

4

6

2

0

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

To find the units digit of

2^{4k+2} + 3^{4k+3} + 7^{4k+1}

, analyze the units digit cyclicity of each base. Powers of 2 have a units digit cycle of [2, 4, 8, 6]. Since

4k+2

leaves a remainder of 2 when divided by 4,

2^{4k+2}

ends in 4. Powers of 3 have a units digit cycle of [3, 9, 7, 1]. Since

4k+3

leaves a remainder of 3 when divided by 4,

3^{4k+3}

ends in 7. Powers of 7 have a units digit cycle of [7, 9, 3, 1]. Since

4k+1

leaves a remainder of 1 when divided by 4,

7^{4k+1}

ends in 7. Adding these units digits gives

4 + 7 + 7 = 18

, so the final units digit is 8.

Adım Adım Çözüm

1

Determine the units digit cyclicity pattern for base 2

The units digits of powers of 2 repeat in a 4-step cycle: 2, 4, 8, 6. For

2^{4k+2}

, the exponent leaves a remainder of 2 when divided by 4, so its units digit is 4.

Units digits of powers follow periodic cycles modulo 10.

2

Determine the units digit cyclicity pattern for base 3

The units digits of powers of 3 repeat in a 4-step cycle: 3, 9, 7, 1. For

3^{4k+3}

, the exponent leaves a remainder of 3 when divided by 4, so its units digit is 7.

The exponent

4k+3

corresponds to the 3rd position in the 4-step cycle.

3

Determine the units digit cyclicity pattern for base 7

The units digits of powers of 7 repeat in a 4-step cycle: 7, 9, 3, 1. For

7^{4k+1}

, the exponent leaves a remainder of 1 when divided by 4, so its units digit is 7.

The exponent

4k+1

corresponds to the 1st position in the 4-step cycle.

4

Sum the units digits and take the units digit of the result

Sum =

4 + 7 + 7 = 18

, which has a units digit of 8.

The units digit of a sum of integers depends only on the sum of their individual units digits.

Anahtar Kavram

Units Digit Cyclicity of Exponents

Soru 919Soru →

A positive integer $n$ has exactly three distinct prime factors. If $n^2$ has exactly $147$ positive divisors, what is the minimum possible value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 1080

Cevap

The minimum possible value of

n

is 1080.

By prime factorizing

n = p^a q^b r^c

, the number of divisors of

n^2

is given by

(2a+1)(2b+1)(2c+1) = 147

. Decomposing 147 into three factors greater than 1 gives

7 \times 7 \times 3

, which determines the exponents

a=3

,

b=3

, and

c=1

. Minimizing

n

by assigning the smallest primes (2, 3, and 5) to these exponents yields

2^3 \times 3^3 \times 5^1 = 1080

.

Adım Adım Çözüm

1

Set up the prime factorization of

n

and

n^2

Let

n = p^a q^b r^c

, where

p < q < r

are prime numbers and

a, b, c \ge 1

. Then

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

.

Any positive integer can be uniquely represented as a product of prime powers.

2

Formulate the divisor counting equation for

n^2

The total number of positive divisors of

n^2

is

(2a+1)(2b+1)(2c+1) = 147

.

If an integer has prime factorization

\prod p_i^{e_i}

, its number of positive divisors is

\prod (e_i + 1)

.

3

Factor 147 into three odd factors greater than 1

The prime factorization of 147 is

3 \times 7^2

. The only factorization into three integers greater than 1 is

7 \times 7 \times 3

.

Since

a, b, c \ge 1

, we have

2a+1 \ge 3

,

2b+1 \ge 3

, and

2c+1 \ge 3

.

4

Solve for the exponents

a, b, c

Setting

2a+1=7

,

2b+1=7

, and

2c+1=3

yields

a=3

,

b=3

, and

c=1

.

Equating the factor values allows us to determine the required exponents of the prime factors of

n

.

5

Minimize

n

by choosing prime bases and assigning exponents

Choose the three smallest primes

p=2, q=3, r=5

and assign exponents

3, 3, 1

to get

n = 2^3 \times 3^3 \times 5^1 = 1080

.

To make the product as small as possible, smaller prime bases must be paired with larger exponents.

Anahtar Kavram

Determining the prime factorization and number of divisors of prime power expressions

Soru 920Soru →

If $5^x = 100^y = 20^z$ for non-zero real numbers $x$ , $y$ , and $z$ , which of the following expressions is equal to $z$ ?

\frac{xy}{x - y}

\frac{xy}{x + y}

\frac{xy}{y - x}

x - y

y - x

Cevabı ve açıklamayı göster

Cevap: $\frac{xy}{x - y}$

Cevap

\frac{xy}{x - y}

By setting

5^x = 100^y = 20^z = K

, we rewrite each base as a power of

K

:

5 = K^{1/x}

,

100 = K^{1/y}

, and

20 = K^{1/z}

. Using the numerical relationship

20 = \frac{100}{5}

, we substitute to obtain

K^{1/z} = \frac{K^{1/y}}{K^{1/x}} = K^{\frac{1}{y} - \frac{1}{x}}

. Equating exponents gives

\frac{1}{z} = \frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy}

. Taking the reciprocal yields the expression stating that

z

equals

\frac{xy}{x - y}

.

Adım Adım Çözüm

1

Set the common exponential expressions equal to a constant

K

.

Let

5^x = 100^y = 20^z = K

, where

K > 0

and

K \neq 1

.

Equating exponential terms to a shared parameter allows conversion from variable powers to fractional exponents.

2

Express the base numbers

5

,

100

, and

20

in terms of

K

.

5 = K^{1/x}

,

100 = K^{1/y}

, and

20 = K^{1/z}

.

Taking the

x

-th,

y

-th, and

z

-th roots of both sides isolates each base number.

3

Relate the base numbers using arithmetic division.

20 = \frac{100}{5} \implies K^{1/z} = \frac{K^{1/y}}{K^{1/x}} = K^{\frac{1}{y} - \frac{1}{x}}

.

Dividing powers with the same base subtracts their exponents.

4

Equate the exponents and solve for

z

.

\frac{1}{z} = \frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy} \implies z = \frac{xy}{x - y}

.

Taking common denominators and inverting the fraction yields

z

in terms of

x

and

y

.

Anahtar Kavram

Relating different bases with variable exponents by setting them equal to a common constant and converting to reciprocal exponents.

Tahmini Süre:2m 0s

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