Arithmetic Alıştırma Soruları — GRE General Test — Sayfa 11

Soru 201Soru →

Pipes $A$ and $B$ , operating together at their respective constant rates, can fill an empty storage tank in $6$ hours. Pipes $B$ and $C$ , operating together at their constant rates, can fill the same tank in $8$ hours. The rate at which Pipe $A$ fills the tank is $1.5$ times the rate at which Pipe $C$ fills the tank. If all three pipes operate together for $3$ hours to fill the empty tank, after which Pipe $B$ is closed, how many additional hours will it take for Pipes $A$ and $C$ together to finish filling the tank?

\frac{6}{5}

\frac{12}{5}

\frac{18}{5}

\frac{21}{5}

3

Cevabı ve açıklamayı göster

Cevap: $\frac{6}{5}$

Cevap

\frac{6}{5}

hours

The rate equations established from the problem yield individual rates of

\frac{1}{8}

tank/hr for Pipe

A

,

\frac{1}{24}

tank/hr for Pipe

B

, and

\frac{1}{12}

tank/hr for Pipe

C

. Working together for

3

hours at a combined rate of

\frac{1}{4}

tank/hr, all three pipes fill

\frac{3}{4}

of the tank, leaving

\frac{1}{4}

of the capacity to be filled. After Pipe

B

closes, Pipes

A

and

C

work at a combined rate of

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

tank/hr. Dividing the remaining

\frac{1}{4}

tank by

\frac{5}{24}

tank/hr gives

\frac{6}{5}

hours.

Adım Adım Çözüm

1

Express given conditions in terms of work rates per hour (

r_A, r_B, r_C

).

r_A + r_B = \frac{1}{6}

,

r_B + r_C = \frac{1}{8}

, and

r_A = 1.5 r_C = \frac{3}{2} r_C

.

Filling a full tank in

H

hours means completing

\frac{1}{H}

of the tank per hour.

2

Solve the system of equations for individual rates

r_A, r_B, r_C

.

Substituting

r_B = \frac{1}{8} - r_C

into

r_A + r_B = \frac{1}{6}

yields

\frac{3}{2} r_C + \frac{1}{8} - r_C = \frac{1}{6} \implies \frac{1}{2} r_C = \frac{1}{24} \implies r_C = \frac{1}{12}

. Consequently,

r_A = \frac{1}{8}

and

r_B = \frac{1}{24}

.

Finding individual unit rates allows calculating any combined work scenario.

3

Calculate the fraction of the tank filled in the first 3 hours with all three pipes open.

Combined rate

r_A + r_B + r_C = \frac{1}{8} + \frac{1}{24} + \frac{1}{12} = \frac{3+1+2}{24} = \frac{6}{24} = \frac{1}{4}

of the tank per hour. In

3

hours,

3 \times \frac{1}{4} = \frac{3}{4}

of the tank is filled, leaving

1 - \frac{3}{4} = \frac{1}{4}

of the tank empty.

Work completed equals rate multiplied by time.

4

Determine the additional time needed for Pipes

A

and

C

to fill the remaining

\frac{1}{4}

of the tank.

Combined rate of

A

and

C

is

r_A + r_C = \frac{1}{8} + \frac{1}{12} = \frac{5}{24}

per hour. Additional time

t = \frac{\frac{1}{4}}{\frac{5}{24}} = \frac{1}{4} \times \frac{24}{5} = \frac{6}{5}

hours.

Time required equals remaining work divided by active rate.

Anahtar Kavram

Combined Work Rates and Systems of Rate Equations

Soru 202Soru →

If $x$ and $y$ are positive integers such that $2^x - 2^y = 1920$ , what is the value of $x + y$ ?

11

15

18

22

25

Cevabı ve açıklamayı göster

Cevap: $18$

Cevap

The correct value is 18.

Factoring

2^y

from

2^x - 2^y

yields

2^y(2^{x-y} - 1)

. Factoring 1920 as

2^7 \times 15

allows us to uniquely match the power-of-two factor

2^y = 2^7

(

y = 7

) and the odd factor

2^{x-y} - 1 = 15

(

x - y = 4 \implies x = 11

). Summing

x

and

y

gives

11 + 7 = 18

.

Adım Adım Çözüm

1

Factor out the smaller power of 2 from the expression

2^y(2^{x-y} - 1) = 1920

Since

x

and

y

are positive integers and

2^x - 2^y > 0

, we know

x > y

. Factoring

2^y

separates the even power of 2 component from an odd component.

2

Determine the prime factorization of 1920 into a power of 2 and an odd integer

1920 = 128 \times 15 = 2^7 \times 15

Repeatedly dividing 1920 by 2 gives

1920 = 2^7 \times 15

, where 15 is an odd integer.

3

Equate the even and odd components

2^y = 2^7 \implies y = 7

and

2^{x-y} - 1 = 15 \implies 2^{x-y} = 16 = 2^4 \implies x - y = 4

Because

x - y \ge 1

, the quantity

2^{x-y} - 1

must be an odd integer, forcing it to equal 15 and the power of 2 factor to equal

2^7

.

4

Solve for

x

and calculate the final sum

x + y

x = 7 + 4 = 11

, so

x + y = 11 + 7 = 18

Adding

x = 11

and

y = 7

gives the required value.

Anahtar Kavram

Factoring exponential expressions by pulling out the common base power and matching unique prime factorizations.

Soru 203Soru →

An investor allocates a total principal of $\$10,000$ between two investment options, Account X and Account Y. Account X earns simple interest at an annual rate of $8\%$ . Account Y earns interest at an annual rate of $8\%$ compounded semi-annually. Neither account receives additional deposits or withdrawals for a period of $2$ years. Which of the following statements MUST be true?

Select all such statements.

Geçerli olan tümünü seçin

If the total principal is divided equally between Account X and Account Y, the total interest earned after

2

years is greater than

\$1,640

.

Allocating a greater portion of the total principal to Account Y strictly increases the total interest earned after

2

years.

If the entire

\$10,000

principal is placed in Account X, the total interest earned after

2

years is

\$1,664

.

Cevabı ve açıklamayı göster

Cevap: If the total principal is divided equally between Account X and Account Y, the total interest earned after $2$ years is greater than $\$1,640$ .; Allocating a greater portion of the total principal to Account Y strictly increases the total interest earned after $2$ years.

Cevap

The correct statements are the statement regarding equal division yielding total interest greater than $1,640, and the statement that allocating a greater portion of principal to Account Y increases the total interest earned.

The statement regarding equal principal division is true because

5,000 placed in Account X yields

800 in simple interest and

5,000 placed in Account Y yields

849.29 under semi-annual compounding, giving a sum of

1,649.29 which exceeds

1,640. The statement regarding shifting allocation to Account Y is true because Account Y's effective two-year yield of 16.99% is higher than Account X's two-year simple yield of 16.00%.

Adım Adım Çözüm

1

Calculate the effective 2-year return for Account X under simple interest.

Simple interest over 2 years at an annual rate of 8% is

2 \times 8\% = 16\%

. Total return factor is

0.16

.

Simple interest accumulates linearly as

I = P \cdot r \cdot t

.

2

Calculate the effective 2-year return for Account Y under semi-annual compound interest.

Semi-annual rate is

\frac{8\%}{2} = 4\% = 0.04

. The number of compounding periods in 2 years is

2 \times 2 = 4

. Total compound multiplier is

(1.04)^4 = 1.16985856

. Thus, the total interest earned is

16.985856\%

of the principal.

Compound interest uses the period rate raised to the power of total compounding periods:

(1 + r_{period})^n

.

3

Evaluate the statement for equal division of $10,000.

With

5,000 in Account X: interest =

5,000 \times 0.16 =

800. With

5,000 in Account Y: interest =

5,000 \times 0.16985856 \approx

849.29. Total interest =

800 +

849.29 =

1,649.29. Since

1,649.29 > $1,640, this statement is TRUE.

Summing the interest from both accounts confirms the total exceeds $1,640.

4

Evaluate the statement comparing allocation shifts.

Since Account Y earns approximately

16.99\%

over 2 years and Account X earns

16.00\%

, the yield of Y exceeds X by approximately

0.99\%

. Thus, increasing the allocation to Account Y increases overall interest earned. This statement is TRUE.

Reallocating principal to an option with a higher effective yield strictly increases overall earnings.

5

Evaluate the statement for placing all funds in Account X.

If all

10,000 is in Account X, interest =

10,000 \times 0.16 =

1,600. The claim of

1,664 is FALSE.

1,664 results from calculating annual compound interest (

10,000 \times (1.08^2 - 1)$), which does not apply to simple interest.

Anahtar Kavram

Simple Interest vs. Compound Interest

Tahmini Süre:2m 0s

Soru 204Soru →

A rectangular floor measuring $126\text{ cm}$ by $180\text{ cm}$ is to be completely covered with identical square tiles of the largest possible side length, without cutting any tiles or leaving gaps. What is the total number of square tiles required to cover the floor?

18

35

70

140

306

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

The total number of square tiles required is 70.

To cover the rectangular floor with the largest possible identical square tiles, the side length of the square tile must be the greatest common divisor of

126

and

180

. Factoring both numbers gives

126 = 2 \times 3^2 \times 7

and

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

. The greatest common divisor is

2 \times 3^2 = 18\text{ cm}

. Dividing the dimensions by

18\text{ cm}

gives

7

tiles along the length and

10

tiles along the width. Multiplying

7 \times 10

yields a total of

70

tiles.

Adım Adım Çözüm

1

Find the prime factorization of both dimensions of the floor.

126 = 2^1 \times 3^2 \times 7^1

and

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

Prime factorization allows systematic determination of the greatest common divisor.

2

Calculate the Greatest Common Divisor (GCD) of 126 and 180 to determine the largest possible square tile side length.

\gcd(126, 180) = 2^1 \times 3^2 = 18\text{ cm}

The square tiles must fit evenly along both the length and width without cutting.

3

Determine the number of tiles along each dimension.

Along length:

126 \div 18 = 7

; along width:

180 \div 18 = 10

Dividing each total length by the tile side length gives the tile count per side.

4

Calculate the total number of tiles required.

7 \times 10 = 70\text{ tiles}

The floor forms a 7 by 10 grid of tiles.

Anahtar Kavram

Greatest Common Divisor (GCD) applications in geometric tiling

Soru 205Soru →

Three industrial machines, $X$ , $Y$ , and $Z$ , continuously produce liquid chemical mixtures composed solely of Compounds $A$ and $B$ .

- Machine $X$ produces the mixture at a constant rate of $100$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $3:2$ by volume.
- Machine $Y$ produces the mixture at a constant rate of $150$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $1:4$ by volume.
- Machine $Z$ produces the mixture at a constant rate of $R$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $7:3$ by volume.

If all three machines operate simultaneously for $4$ hours, the resulting combined mixture contains equal total volumes of Compound $A$ and Compound $B$ .

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

Geçerli olan tümünü seçin

The operating rate

R

of Machine

Z

is

175

liters per hour.

Machine

Z

accounts for more than

40\%

of the total volume of the combined mixture produced in

4

hours.

In the 4-hour output, the volume of Compound

B

produced by Machine

Y

is greater than the total volume of Compound

B

produced by Machines

X

and

Z

combined.

The ratio of the volume of Compound

A

produced by Machine

X

to the volume of Compound

A

produced by Machine

Z

is

2:3

.

If Machine

Y

were turned off and only Machines

X

and

Z

operated for

4

hours, the combined output would have Compound

A

and Compound

B

in a ratio of

7:3

.

Cevabı ve açıklamayı göster

Cevap: The operating rate $R$ of Machine $Z$ is $175$ liters per hour.; Machine $Z$ accounts for more than $40\%$ of the total volume of the combined mixture produced in $4$ hours.; In the 4-hour output, the volume of Compound $B$ produced by Machine $Y$ is greater than the total volume of Compound $B$ produced by Machines $X$ and $Z$ combined.

Cevap

The statements asserting that Machine Z's rate is 175 liters per hour, that Machine Z accounts for over 40% of the total output volume, and that Machine Y produces more Compound B than Machines X and Z combined are all true.

The correct options are those stating that Machine Z operates at 175 liters per hour, that Machine Z accounts for over 40% of the 4-hour output, and that Machine Y produces more Compound B than Machines X and Z combined. Machine X produces 240 liters of Compound A and 160 liters of Compound B in 4 hours. Machine Y produces 120 liters of Compound A and 480 liters of Compound B in 4 hours. Machine Z produces 2.8R liters of Compound A and 1.2R liters of Compound B in 4 hours. Equating total Compound A (360 + 2.8R) to total Compound B (640 + 1.2R) gives 1.6R = 280, so R = 175 liters per hour. Machine Z thus contributes 700 liters out of 1700 total liters, which is approximately 41.18% (greater than 40%). Machine Y produces 480 liters of Compound B, which exceeds the 370 liters of Compound B produced by Machines X and Z combined (160 + 210 liters).

Adım Adım Çözüm

1

Calculate the volumes of Compound A and Compound B produced by Machines X and Y in 4 hours.

Machine X produces

400

L total (

240

L of A,

160

L of B). Machine Y produces

600

L total (

120

L of A,

480

L of B). Combined from X and Y:

360

L of A and

640

L of B.

Decompose each machine's total 4-hour output into individual compound amounts using part-to-whole fraction multipliers.

2

Formulate Machine Z's 4-hour output expressions and solve for rate R.

Machine Z produces

4R

L total (

2.8R

L of A,

1.2R

L of B). Setting total A equal to total B:

360 + 2.8R = 640 + 1.2R \implies 1.6R = 280 \implies R = 175

L/hr.

The condition of equal total volumes of Compound A and Compound B forms a single linear equation in terms of R.

3

Determine Machine Z's proportion of the total volume produced by all three machines.

Machine Z volume =

4 \times 175 = 700

L. Total volume =

400 + 600 + 700 = 1700

L. Percentage =

\frac{700}{1700} \times 100\% \approx 41.18\% > 40\%

.

Evaluate whether Machine Z contributes more than 40% of the complete 1700-liter mixture.

4

Compare Compound B output from Machine Y against the combined Compound B output from X and Z.

Machine Y produces

480

L of B. Machines X and Z produce

160 + (1.2 \times 175) = 370

L of B.

480 > 370

is true.

Direct numerical comparison verifies the inequality statement.

5

Evaluate the remaining ratio claims.

Compound A ratio X to Z is

240 : 490 = 24 : 49 \neq 2 : 3

. Combined X and Z ratio of A to B is

730 : 370 = 73 : 37 \neq 7 : 3

. Both claims are false.

Check remaining distractors for mathematical accuracy.

Anahtar Kavram

Multi-rate and multi-component ratio mixture balancing

Soru 206Soru →

If $x$ and $y$ are real numbers such that $-1 < x < 0 < y < 1$ , which of the following statements MUST be true? Select all such statements.

Geçerli olan tümünü seçin

x^3 < x^5

\sqrt{x^2} = x

\left(y^{1/2}\right)^x > 1

(x + y)^2 = x^2 + y^2

x^4 > x^2

Cevabı ve açıklamayı göster

Cevap: $x^3 < x^5$ ; $\left(y^{1/2}\right)^x > 1$

Cevap

The statements

x^3 < x^5

and

\left(y^{1/2}\right)^x > 1

MUST be true.

For

-1 < x < 0 < y < 1

, odd powers of

x

satisfy

-1 < x < x^3 < x^5 < 0

, making the inequality comparing

x^3

and

x^5

true. Furthermore,

y^{1/2}

lies in

(0, 1)

, and raising a base in

(0, 1)

to a negative exponent

x

produces a result strictly greater than 1.

Adım Adım Çözüm

1

Analyze the odd power inequality

x^3 < x^5

for

-1 < x < 0

.

Since

x \in (-1, 0)

,

x^2 \in (0, 1)

. Multiplying

-1 < x < 0

by

x^2

gives

x < x^3 < x^5 < 0

. Thus,

x^3 < x^5

holds.

Odd powers preserve negative signs, and higher powers of fractions between 0 and 1 have smaller absolute values.

2

Evaluate the principal square root

\sqrt{x^2}

.

By definition,

\sqrt{x^2} = |x|

. Since

x < 0

,

|x| = -x \neq x

.

The principal square root of a real number is always non-negative.

3

Analyze the expression

\left(y^{1/2}\right)^x

.

Since

0 < y < 1

,

0 < y^{1/2} < 1

. Let

k = y^{1/2}

. Then

k^x = \left(\frac{1}{k}\right)^{-x}

. Since

k < 1

,

\frac{1}{k} > 1

, and since

x < 0

,

-x > 0

. A base greater than 1 raised to a positive power is greater than 1.

Negative exponents indicate the reciprocal of the base.

4

Evaluate

(x + y)^2 = x^2 + y^2

.

(x + y)^2 = x^2 + 2xy + y^2

. Because

x < 0

and

y > 0

,

2xy < 0

, so

(x + y)^2 < x^2 + y^2

.

Exponents do not distribute over addition, and cross-terms must be accounted for.

5

Evaluate

x^4 > x^2

.

For

-1 < x < 0

,

0 < x^2 < 1

. Squaring a number in

(0, 1)

yields a smaller number, so

x^4 < x^2

.

Higher even powers of quantities with magnitude less than 1 decrease in magnitude.

Anahtar Kavram

Properties of exponents, fractional powers, and principal square roots for bounded negative and positive real numbers

Tahmini Süre:2m 30s

Soru 207Soru →

At a clothing retailer, the wholesale cost of a leather coat was $50\%$ greater than the wholesale cost of a denim jacket. The following year, the wholesale cost of the leather coat increased by $20\%$ , while the wholesale cost of the denim jacket decreased by $10\%$ . What was the overall percentage increase in the combined wholesale cost of one leather coat and one denim jacket?

2\%

5\%

8\%

10\%

15\%

Cevabı ve açıklamayı göster

Cevap: $8\%$

Cevap

An overall increase of

8\%

in the combined wholesale cost.

Assuming an initial denim jacket cost of

100

, the leather coat costs

150

, giving an initial combined cost of

250

. After a

20\%

increase, the coat costs

180

, and after a

10\%

decrease, the jacket costs

90

. The new combined total is

270

. The total increase is

20

, which represents

\frac{20}{250} = 8\%

of the original combined cost.

Adım Adım Çözüm

1

Define variables for the initial costs of the items.

Let the initial cost of the denim jacket be

d = 100

. Then the initial cost of the leather coat is

1.50d = 150

.

Choosing a convenient base value makes calculating relative percentage changes straightforward.

2

Calculate the initial total cost and the new individual costs after price changes.

Initial total cost =

100 + 150 = 250

. New cost of leather coat =

150 \times (1 + 0.20) = 180

. New cost of denim jacket =

100 \times (1 - 0.10) = 90

.

Percentage increases and decreases must be applied to their respective starting costs.

3

Compute the new combined total cost and calculate the net percentage change.

New combined total cost =

180 + 90 = 270

. Dollar change =

270 - 250 = 20

. Net percentage increase =

\frac{20}{250} \times 100\% = 8\%

.

Percent change is calculated by dividing the absolute change by the original combined base value.

Anahtar Kavram

Combined Percent Change and Weighted Base Values

Soru 208Soru →

In Month 1, a data center's two facilities, Facility A and Facility B, consumed a combined total of $120,000\text{ kWh}$ of electricity. In Month 2, Facility A decreased its electricity consumption by $15\%$ , while Facility B increased its electricity consumption by $20\%$ . If the combined electricity consumption of the two facilities increased by $6\%$ overall in Month 2, how many kilowatt-hours ( $\text{kWh}$ ) of electricity did Facility A consume in Month 1?

Cevabı ve açıklamayı göster

Cevap: 48000

Cevap

Facility A consumed 48,000 kWh of electricity in Month 1.

Let

A

represent the electricity consumed by Facility A in Month 1. The remaining electricity consumed by Facility B in Month 1 is

120,000 - A

. In Month 2, the decrease in Facility A's consumption is

15\%

of

A

, or

0.15A

, and the increase in Facility B's consumption is

20\%

of

(120,000 - A)

, or

0.20(120,000 - A)

. The net increase overall is

6\%

of

120,000

, which is

7,200\text{ kWh}

. Setting up the change equation

-0.15A + 0.20(120,000 - A) = 7,200

leads to

-0.35A + 24,000 = 7,200

, which simplifies to

0.35A = 16,800

. Solving for

A

yields

A = 48,000\text{ kWh}

.

Adım Adım Çözüm

1

Define variables for the initial Month 1 consumption of each facility.

Let

A

= Month 1 consumption of Facility A. Month 1 consumption of Facility B =

120,000 - A

.

Expressing one unknown in terms of the total reduces the system to a single variable equation.

2

Determine the net kilowatt-hour change for Month 2.

Net change =

0.06 \times 120,000 = 7,200\text{ kWh}

.

An overall increase of 6% applies to the total starting value of 120,000 kWh.

3

Express the individual percentage changes in terms of

A

and equate to the net change.

-0.15A + 0.20(120,000 - A) = 7,200

.

Facility A's 15% decrease contributes

-0.15A

and Facility B's 20% increase contributes

+0.20(120,000 - A)

.

4

Solve the linear equation for

A

.

-0.35A + 24,000 = 7,200 \Rightarrow 0.35A = 16,800 \Rightarrow A = 48,000

.

Dividing

16,800

by

0.35

yields the original consumption of Facility A.

Anahtar Kavram

Weighted Percent Change and Systems of Percent Equations

Soru 209Soru →

An automated micro-dispenser delivers liquid reagents in precise doses. Each dose has a volume of $3.6 \times 10^{-5}$ liters. If a reservoir containing $0.0162$ liters of reagent is emptied completely by delivering these equal doses, how many doses were delivered?

Cevabı ve açıklamayı göster

Cevap: 450

Cevap

450

Dividing the total volume of

1.62 \times 10^{-2}

liters by the single dose volume of

3.6 \times 10^{-5}

liters yields

\frac{1.62}{3.6} \times 10^{-2 - (-5)} = 0.45 \times 10^3 = 450

doses.

Adım Adım Çözüm

1

Express the total reservoir volume in scientific notation

0.0162 = 1.62 \times 10^{-2}

liters

Converting decimals into standard scientific notation simplifies multiplication and division operations.

2

Set up the division for the number of doses

\text{Number of doses} = \frac{1.62 \times 10^{-2}}{3.6 \times 10^{-5}}

The total volume divided by the volume per single dose yields the total dose count.

3

Compute the division of coefficients and exponent terms independently

\frac{1.62}{3.6} = 0.45

and

\frac{10^{-2}}{10^{-5}} = 10^{-2 - (-5)} = 10^3

Applying the exponent quotient rule

10^a / 10^b = 10^{a-b}

yields

-2 - (-5) = 3

.

4

Convert from scientific notation to a standard integer

0.45 \times 10^3 = 450

Multiplying

0.45

by

1,000

shifts the decimal point 3 places to the right.

Anahtar Kavram

Division of numbers in scientific notation and place value manipulation

Tahmini Süre:1m 30s

Soru 210Soru →

The mass of a sample in a laboratory experiment is given by the expression $M = 0.000375 \times 10^n$ grams, where $n$ is an integer. When $M$ is written in standard scientific notation as $a \times 10^k$ , where $1 \le a < 10$ and $k$ is an integer, the exponent $k$ is equal to $-2$ . What is the value of $n$ ?

-6

-2

2

4

6

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The value of

n

is

2

.

Writing

0.000375

as

3.75 \times 10^{-4}

allows the mass

M

to be rewritten as

3.75 \times 10^{-4 + n}

. Matching this with the standard scientific notation form

a \times 10^k

(where

a = 3.75

and

k = -2

) gives the equation

-4 + n = -2

. Solving for

n

yields

n = 2

.

Adım Adım Çözüm

1

Convert the decimal decimal coefficient to scientific notation.

0.000375 = 3.75 \times 10^{-4}

Moving the decimal point 4 places to the right puts the leading number in the required range

1 \le a < 10

.

2

Substitute this conversion back into the expression for

M

and combine powers of 10.

M = (3.75 \times 10^{-4}) \times 10^n = 3.75 \times 10^{n - 4}

By exponent rules,

10^a \times 10^b = 10^{a+b}

.

3

Equate the exponent of 10 in the simplified expression to the given value of

k

.

n - 4 = -2 \implies n = 2

The scientific notation requires

a \times 10^k

where

k = -2

, so

n - 4

must equal

-2

.

Anahtar Kavram

Converting numbers between standard decimal form and scientific notation using place value and exponent properties

Soru 211Soru →

What is the smallest positive integer that is a multiple of 18, 24, and 30, and is also a perfect square?

Cevabı ve açıklamayı göster

Cevap: 3600

Cevap

The smallest positive integer that is a multiple of 18, 24, and 30, and is also a perfect square is 3600.

The least common multiple of 18, 24, and 30 is 360, which factors into

2^3 \cdot 3^2 \cdot 5^1

. For an integer to be a perfect square, all exponents in its prime factorization must be even. Multiplying 360 by

2 \cdot 5 = 10

completes the odd exponents to even values (

2^4 \cdot 3^2 \cdot 5^2

), giving 3600, which is the smallest perfect square divisible by 18, 24, and 30.

Adım Adım Çözüm

1

Find the prime factorization of 18, 24, and 30

18 = 2^1 \cdot 3^2

,

24 = 2^3 \cdot 3^1

,

30 = 2^1 \cdot 3^1 \cdot 5^1

Decomposing numbers into prime factors allows calculation of the LCM and analysis of perfect square conditions.

2

Calculate the LCM of 18, 24, and 30

\text{LCM}(18, 24, 30) = 2^3 \cdot 3^2 \cdot 5^1 = 360

Any common multiple must be a multiple of the LCM of these three numbers.

3

Determine the smallest factor required to make the prime exponents even

Multiply 360 by

2^1 \cdot 5^1 = 10

A perfect square requires all prime exponents to be even; 2 has power 3 and 5 has power 1 in 360, so one more factor of 2 and one more factor of 5 are required.

4

Compute the final result

360 \times 10 = 3600

3600 = 2^4 \cdot 3^2 \cdot 5^2 = 60^2

, which is a perfect square.

Anahtar Kavram

Prime Factorization, LCM, and Exponent Properties of Perfect Squares

Soru 212Soru →

An agricultural cooperative harvested crops from three distinct farming zones: Zone A, Zone B, and Zone C. In Season 1, Zone A accounted for $40\%$ of the cooperative's total crop yield, Zone B accounted for $35\%$ , and Zone C accounted for the remaining $25\%$ .

In Season 2:
- The crop yield in Zone A increased by $15\%$ compared to Season 1.
- The crop yield in Zone B decreased by $20\%$ compared to Season 1.
- The crop yield in Zone C increased by $p\%$ compared to Season 1.

If the cooperative's total crop yield in Season 2 was $6\%$ greater than its total crop yield in Season 1, what is the value of $p$ ?

7

11

24

28

32

Cevabı ve açıklamayı göster

Cevap: 28

Cevap

The value of

p

is

28

.

The correct value of

p

is

28

. Calculating the weighted percentage contributions of each zone shows that Zone A contributes a yield equal to

46\%

of the original total yield and Zone B contributes

28\%

, making their combined Season 2 contribution

74\%

of the original total yield. To reach the required total Season 2 yield of

106\%

of the original total, Zone C must contribute

106\% - 74\% = 32\%

of the original total yield. Since Zone C initially produced

25\%

of the total yield, the percentage increase for Zone C is

\frac{32 - 25}{25} \times 100\% = \frac{7}{25} \times 100\% = 28\%

.

Adım Adım Çözüm

1

Define total initial yield and express Season 1 yields for each zone in terms of total yield

Y

.

Zone A yield =

0.40Y

, Zone B yield =

0.35Y

, Zone C yield =

0.25Y

.

Establishing baseline values based on given initial percentage shares.

2

Calculate the Season 2 yields for Zones A and B after their respective percentage changes.

Zone A yield in Season 2 =

0.40Y \times 1.15 = 0.46Y

. Zone B yield in Season 2 =

0.35Y \times 0.80 = 0.28Y

.

Applying a 15% increase to Zone A and a 20% decrease to Zone B.

3

Set up the total Season 2 yield equation using the given 6% overall increase.

0.46Y + 0.28Y + 0.25Y\left(1 + \frac{p}{100}\right) = 1.06Y

.

The sum of all three zone yields in Season 2 equals 106% of the total Season 1 yield.

4

Solve for

p

.

0.74 + 0.25\left(1 + \frac{p}{100}\right) = 1.06 \implies 0.25\left(1 + \frac{p}{100}\right) = 0.32 \implies 1 + \frac{p}{100} = 1.28 \implies p = 28

.

Subtracting 0.74 from 1.06 gives 0.32, and dividing 0.32 by 0.25 yields 1.28, representing a 28% increase.

Anahtar Kavram

Weighted Percent Change and Base Values

Soru 213Soru →

An industrial laboratory prepares a batch of Alloy $Z$ weighing $260$ pounds, which consists solely of copper and zinc in a ratio of $7 : 6$ by weight. A technician then adds $40$ pounds of pure copper to this batch to create Alloy $W$ . What is the ratio of copper to zinc in Alloy $W$ ?

3 : 2

3 : 5

9 : 8

7 : 6

2 : 3

Cevabı ve açıklamayı göster

Cevap: 3 : 2

Cevap

3 : 2

In Alloy Z, the total weight of 260 pounds is divided into 13 equal parts of 20 pounds each. Thus, copper accounts for 7 parts (140 pounds) and zinc accounts for 6 parts (120 pounds). Adding 40 pounds of pure copper increases the copper content to 180 pounds, while zinc remains at 120 pounds. Comparing copper to zinc yields 180 : 120, which simplifies to 3 : 2.

Adım Adım Çözüm

1

Determine total ratio parts for Alloy Z.

The total number of ratio parts is

7 + 6 = 13

parts.

To find the weight of each component from a part-to-part ratio, divide the total weight by total ratio parts.

2

Calculate initial weights of copper and zinc in Alloy Z.

Copper weight

= 260 \times \frac{7}{13} = 140

pounds; Zinc weight

= 260 \times \frac{6}{13} = 120

pounds.

Multiply total weight by each component's fraction of the total.

3

Update component weights after adding pure copper.

New copper weight

= 140 + 40 = 180

pounds; Zinc weight remains

120

pounds.

Only copper is added to the mixture, leaving zinc weight unchanged.

4

Compute and simplify the final ratio of copper to zinc for Alloy W.

Ratio of copper to zinc

= 180 : 120 = 3 : 2

.

Divide both terms by their greatest common divisor, 60.

Anahtar Kavram

Ratios and Component Mixture Adjustment

Soru 214Soru →

Let $n = 2^a \cdot 3^b \cdot p^c$ , where $a$ , $b$ , and $c$ are positive integers and $p$ is a prime number strictly greater than $5$ . The integer $n$ has exactly $36$ positive divisors, and $\gcd(n, 180) = 36$ . Which of the following statements MUST be true? Select all such statements.

Geçerli olan tümünü seçin

The integer

n

is divisible by

36

.

The exponent

c

cannot exceed

3

.

The sum of the exponents

a + b + c

must equal

7

.

The least common multiple of

n

and

180

is equal to

5n

.

The exponent

a

must equal

2

.

Cevabı ve açıklamayı göster

Cevap: The integer $n$ is divisible by $36$ .; The exponent $c$ cannot exceed $3$ .; The least common multiple of $n$ and $180$ is equal to $5n$ .

Cevap

The statements asserting that

n

is divisible by

36

, that the exponent

c

cannot exceed

3

, and that

\text{lcm}(n, 180) = 5n

must be true.

The statement that

n

is divisible by

36

is true because

a \ge 2

and

b \ge 2

. The statement that

c \le 3

is true because

(a+1)(b+1) \ge 9

, bounding

c+1 \le 4

. The statement that

\text{lcm}(n, 180) = 5n

is true because

\max(a, 2) = a

,

\max(b, 2) = b

, and

180

contributes a factor of

5^1

.

Adım Adım Çözüm

1

Analyze the GCD condition to establish bounds on exponents

a

and

b

.

a \ge 2

and

b \ge 2

.

Since

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

and

\gcd(n, 180) = 36 = 2^2 \cdot 3^2

, the minimum of the exponents of

2

is

2

(so

a \ge 2

) and the minimum of the exponents of

3

is

2

(so

b \ge 2

).

2

Use the divisor count formula

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

to find constraints on

c

.

c \le 3

.

Since

a \ge 2 \implies a+1 \ge 3

and

b \ge 2 \implies b+1 \ge 3

, the product

(a+1)(b+1) \ge 9

. Therefore,

c+1 = \frac{36}{(a+1)(b+1)} \le \frac{36}{9} = 4

, which gives

c \le 3

.

3

Evaluate the divisibility of

n

by

36

.

n

is divisible by

36

.

Because

a \ge 2

and

b \ge 2

,

2^2 \cdot 3^2 = 36

divides

2^a \cdot 3^b \cdot p^c = n

.

4

Calculate

\text{lcm}(n, 180)

.

\text{lcm}(n, 180) = 5n

.

\text{lcm}(2^a \cdot 3^b \cdot p^c, 2^2 \cdot 3^2 \cdot 5^1) = 2^{\max(a,2)} \cdot 3^{\max(b,2)} \cdot 5^1 \cdot p^c = 2^a \cdot 3^b \cdot 5 \cdot p^c = 5n

since

a \ge 2

,

b \ge 2

, and

p > 5

.

5

Test counterexamples for statement constancy regarding

a+b+c=7

and

a=2

.

Neither

a+b+c=7

nor

a=2

is required.

The choice

(a, b, c) = (5, 2, 1)

satisfies

(5+1)(2+1)(1+1) = 36

and

\gcd(n, 180) = 36

, yielding

a+b+c=8 \neq 7

and

a=5 \neq 2

.

Anahtar Kavram

Prime Factorization, Greatest Common Divisor (GCD), and Least Common Multiple (LCM) properties

Soru 215Soru →

On the real number line, points $P$ , $Q$ , and $R$ have coordinates $x$ , $y$ , and $z$ , respectively, such that $|x - 2| = 5$ , $|y + 4| = 3$ , and $z$ is the midpoint of segment $PQ$ . If $x < y$ , what is the value of $z$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The coordinate of point

R

(the value of

z

) is

-2

.

Solving the two absolute value equations yields

x \in \{-3, 7\}

and

y \in \{-7, -1\}

. Testing the given constraint

x < y

across all four possible ordered pairs reveals that only

x = -3

and

y = -1

satisfy the condition, since

-3 < -1

is true while all other pairs fail. The midpoint of points with coordinates

-3

and

-1

is

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

.

Adım Adım Çözüm

1

Solve the absolute value equation

|x - 2| = 5

.

x = 7

or

x = -3

The equation

|x - 2| = 5

splits into

x - 2 = 5 \implies x = 7

and

x - 2 = -5 \implies x = -3

.

2

Solve the absolute value equation

|y + 4| = 3

.

y = -1

or

y = -7

The equation

|y + 4| = 3

splits into

y + 4 = 3 \implies y = -1

and

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

.

3

Evaluate all candidate pairs

(x, y)

under the inequality constraint

x < y

.

The only valid pair is

x = -3

and

y = -1

.

Comparing all four combinations:

7 < -1

(false),

7 < -7

(false),

-3 < -7

(false), and

-3 < -1

(true).

4

Calculate the midpoint

z

of the segment

PQ

.

z = -2

The midpoint of coordinates

-3

and

-1

on a number line is given by their average:

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

.

Anahtar Kavram

Absolute Value Equations and Midpoint on the Real Number Line

Soru 216Soru →

If $3^x + 3^{-x} = 4$ , what is the value of $\frac{27^x + 27^{-x} - 2}{9^x + 9^{-x} + 1}$ ?

\frac{10}{3}

\frac{50}{17}

\frac{62}{17}

\frac{74}{15}

\frac{50}{19}

Cevabı ve açıklamayı göster

Cevap: $\frac{10}{3}$

Cevap

\frac{10}{3}

Squaring

3^x + 3^{-x} = 4

yields

9^x + 2 + 9^{-x} = 16

, which simplifies to

9^x + 9^{-x} = 14

. Utilizing the sum of cubes identity gives

27^x + 27^{-x} = (3^x + 3^{-x})(9^x - 1 + 9^{-x}) = 4(14 - 1) = 52

. Substituting these values into the given fraction gives

\frac{52 - 2}{14 + 1} = \frac{50}{15} = \frac{10}{3}

.

Adım Adım Çözüm

1

Square the given expression

3^x + 3^{-x} = 4

to determine

9^x + 9^{-x}

.

(3^x + 3^{-x})^2 = 9^x + 2(3^x)(3^{-x}) + 9^{-x} = 9^x + 2 + 9^{-x} = 16

, which yields

9^x + 9^{-x} = 14

.

Expanding the square of a binomial requires accounting for the middle term

2 \cdot 3^x \cdot 3^{-x} = 2

.

2

Express

27^x + 27^{-x}

using the sum of cubes factorization identity.

27^x + 27^{-x} = (3^x)^3 + (3^{-x})^3 = (3^x + 3^{-x})(9^x - 3^x \cdot 3^{-x} + 9^{-x}) = 4(14 - 1) = 52

.

The identity

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

allows factoring cubic exponential expressions.

3

Substitute the evaluated terms into the target rational expression and simplify.

\frac{27^x + 27^{-x} - 2}{9^x + 9^{-x} + 1} = \frac{52 - 2}{14 + 1} = \frac{50}{15} = \frac{10}{3}.

Replacing component expressions with their computed values yields the simplified numerical fraction.

Anahtar Kavram

Evaluating high-power exponential expressions using polynomial identity transformations and exponent laws.

Soru 217Soru →

Two positive integers $x$ and $y$ satisfy $x < y$ , $\gcd(x, y) = 18$ , and $\text{lcm}(x, y) = 1080$ . If $x$ has exactly $6$ positive divisors, what is the value of $y$ ?

216

270

360

540

1080

Cevabı ve açıklamayı göster

Cevap: $1080$

Cevap

1080

Since

\gcd(x, y) = 18

and

\text{lcm}(x, y) = 1080

, we can express

x = 18m

and

y = 18n

with

\gcd(m, n) = 1

and

m < n

. Substituting into

\text{lcm}(x, y) = 18mn = 1080

gives

mn = 60

. The coprime pairs

(m, n)

with

m < n

are

(1, 60)

,

(3, 20)

,

(4, 15)

, and

(5, 12)

. Testing

x = 18m

for each pair: for

m = 1

,

x = 18 = 2^1 \cdot 3^2

, which has

(1+1)(2+1) = 6

positive divisors. This matches the condition, so

n = 60

and

y = 18 \times 60 = 1080

.

Adım Adım Çözüm

1

Express

x

and

y

in terms of their Greatest Common Divisor

Let

x = 18m

and

y = 18n

, where

m

and

n

are positive integers such that

m < n

and

\gcd(m, n) = 1

.

Since

\gcd(x, y) = 18

, both numbers must be multiples of

18

, and their remaining factor parts must be coprime.

2

Use the LCM formula

\text{lcm}(x, y) = 18mn

to find

mn

18mn = 1080 \implies mn = 60

.

The least common multiple of

18m

and

18n

with

\gcd(m,n)=1

is

18mn

.

3

Find all coprime factor pairs

(m, n)

of

60

with

m < n

The prime factorization of

60

is

2^2 \cdot 3^1 \cdot 5^1

. Coprime pairs

(m, n)

are

(1, 60)

,

(3, 20)

,

(4, 15)

, and

(5, 12)

.

Pairs must be coprime so that

\gcd(18m, 18n) = 18

remains true.

4

Determine the number of positive divisors for

x = 18m

for each candidate pair

For

m = 1

,

x = 18 = 2^1 \cdot 3^2

, which has

(1+1)(2+1) = 6

positive divisors.

The number of positive divisors of

p_1^{a_1} p_2^{a_2} \dots

is given by

(a_1 + 1)(a_2 + 1) \dots

. Only

m = 1

gives exactly

6

divisors.

5

Calculate

y

y = 18n = 18 \cdot 60 = 1080

.

Corresponding to

m = 1

,

n = 60

gives

y = 1080

.

Anahtar Kavram

Prime Factorization, GCD/LCM Coprime Relationships, and Divisor Counting Formula

Alternatif Yöntem

Alternatively, use

\gcd(x, y) \cdot \text{lcm}(x, y) = x \cdot y \implies x \cdot y = 18 \cdot 1080 = 19440

. Since

x

has

6

positive divisors and

x

is a multiple of

18

,

x

must equal

18

because

18 = 2^1 \cdot 3^2

has

(1+1)(2+1) = 6

divisors. Then

y = 19440 / 18 = 1080

.

Tahmini Süre:1m 30s

Soru 218Soru →

In Year 1, a technology firm allocated $40\%$ of its total operating budget to Research and Development (R&D) and the remaining $60\%$ to Business Operations. In Year 2, the amount allocated to R&D was increased by $25\%$ over its Year 1 amount, while the amount allocated to Business Operations was decreased by $15\%$ from its Year 1 amount. If the firm's total operating budget in Year 2 was $\$505,000$ , what was its total operating budget in Year 1?

\$454,500

\$480,952

\$500,000

\$499,950

\$510,050

Cevabı ve açıklamayı göster

Cevap: $\$500,000$

Cevap

The total operating budget in Year 1 was

\$500,000

.

The budget allocation in Year 1 assigns

40\%

to R&D and

60\%

to Operations. In Year 2, R&D becomes

0.40 \times 1.25 = 0.50

of the Year 1 budget, and Operations becomes

0.60 \times 0.85 = 0.51

of the Year 1 budget. Adding these yields a Year 2 total equal to

1.01

times the Year 1 budget. Setting

1.01 \times \text{Year 1 Budget} = \$505,000

and dividing gives

\$500,000

.

Adım Adım Çözüm

1

Define Year 1 components in terms of total Year 1 budget

B

.

R&D allocation =

0.40B

; Operations allocation =

0.60B

.

The problem specifies a

40\% / 60\%

split of the total budget

B

in Year 1.

2

Calculate Year 2 allocations after percentage changes.

Year 2 R&D =

0.40B \times 1.25 = 0.50B

; Year 2 Operations =

0.60B \times 0.85 = 0.51B

.

R&D increases by

25\%

(

1 + 0.25 = 1.25

) and Operations decreases by

15\%

(

1 - 0.15 = 0.85

).

3

Sum Year 2 allocations to express total Year 2 budget in terms of

B

.

Total Year 2 Budget =

0.50B + 0.51B = 1.01B

.

Combining both component allocations gives the total budget for Year 2.

4

Solve for the Year 1 budget

B

using the given Year 2 total.

1.01B = \$505,000 \implies B = \frac{505,000}{1.01} = \$500,000

.

Dividing the Year 2 budget by the total growth multiplier yields the original Year 1 budget.

Anahtar Kavram

Weighted Percent Change and Initial Base Calculation

Soru 219Soru →

A venture capital fund allocated an initial sum of money between Portfolio Alpha and Portfolio Beta, with Portfolio Alpha receiving $60\%$ of the total sum and Portfolio Beta receiving the remaining $40\%$ . During the first year, the value of Portfolio Alpha increased by $20\%$ , while the value of Portfolio Beta decreased by $15\%$ . During the second year, the value of Portfolio Alpha decreased by $10\%$ relative to its value at the end of the first year, while the value of Portfolio Beta increased by $25\%$ relative to its value at the end of the first year. By what percent did the total combined value of the two portfolios increase from the initial allocation to the end of the second year?

Cevabı ve açıklamayı göster

Cevap: 7.3

Cevap

The total combined value of the two portfolios increased by 7.3%.

To find the net percent change over the two-year period, track each portfolio's value year by year. Assuming an initial combined total of

100, Portfolio Alpha begins at

60 and Portfolio Beta at

40. By the end of Year 1, Alpha increases by 20% to

72 (

60 \times 1.20

), while Beta decreases by 15% to

34 (

40 \times 0.85

). In Year 2, Alpha decreases by 10% relative to its Year 1 ending value, becoming

64.80 (

72 \times 0.90

), while Beta increases by 25% relative to its Year 1 ending value, becoming

42.50 (

34 \times 1.25

). The total combined value at the end of Year 2 is

64.80 +

42.50 =

107.30. Relative to the initial $100 allocation, this represents a net increase of 7.3%.

Adım Adım Çözüm

1

Assign a convenient base value for the initial total investment.

Assume an initial total allocation of

100. Portfolio Alpha starts with

60 and Portfolio Beta starts with $40.

Percent changes are proportional and scale-invariant, making $100 a simple base value for calculation.

2

Compute the value of each portfolio at the end of Year 1.

Portfolio Alpha =

60 \times (1 + 0.20) =

72. Portfolio Beta =

40 \times (1 - 0.15) =

34.

Apply Year 1 growth (+20%) and loss (-15%) to their respective starting funds.

3

Compute the value of each portfolio at the end of Year 2.

Portfolio Alpha =

72 \times (1 - 0.10) =

64.80. Portfolio Beta =

34 \times (1 + 0.25) =

42.50.

Apply Year 2 changes (-10% and +25%) to the Year 1 ending values rather than the original principal.

4

Find the final total combined value and calculate the overall net percentage increase.

Combined Year 2 value =

64.80 +

42.50 = $107.30. Net percentage increase = \frac{107.30 - 100}{100} \times 100\% = 7.3\%.

Compare the final combined sum of

107.30 to the initial combined sum of

100.

Anahtar Kavram

Successive percent changes with shifting base values across multiple assets

Soru 220Soru →

If $a$ , $b$ , and $c$ are non-zero integers such that $a^3 b^2 c < 0$ , $a + b$ is even, and $b + c$ is odd, which of the following expressions MUST be a negative even integer?

a c (b^2 + 1)

a b c

(a + c) b^2

a^2 c (b + 1)

a c + b

Cevabı ve açıklamayı göster

Cevap: $a c (b^2 + 1)$

Cevap

The expression

a c (b^2 + 1)

must be a negative even integer.

The expression

a c (b^2 + 1)

is guaranteed to be negative because

a c < 0

(derived from

a^3 b^2 c < 0

) and

b^2 + 1 > 0

for all non-zero integers

b

. It is guaranteed to be even because

a

and

c

have opposite parity, meaning one of them must be even, rendering

a c

(and thus any integer multiple of

a c

) even.

Adım Adım Çözüm

1

Analyze the sign constraint

a^3 b^2 c < 0

.

a c < 0

, which means

a

and

c

have opposite signs.

Since

b \neq 0

,

b^2

is strictly positive. Dividing

a^3 b^2 c < 0

by

b^2

yields

a^3 c < 0

. Since

a^3

has the same sign as

a

,

a c < 0

.

2

Analyze the parity constraints

a + b

is even and

b + c

is odd.

a

and

c

have opposite parity (one is even, the other is odd).

If

a + b

is even,

a

and

b

share the same parity. If

b + c

is odd,

b

and

c

have opposite parity. Substituting the parity of

a

for

b

shows that

a

and

c

must have opposite parity.

3

Determine the sign and parity of

a c (b^2 + 1)

.

a c (b^2 + 1)

is strictly negative and even.

Since

a

and

c

have opposite parity, at least one of them is even, making the product

a c

an even integer. Since

a c < 0

and

b^2 + 1 \ge 1 > 0

, the product of negative

a c

and positive

(b^2 + 1)

is negative and even.

Anahtar Kavram

Combining sign rules (

a c < 0

) with even/odd parity logic across multiple variables.

Arithmetic

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Alternatif Yöntem

Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

Anahtar Kavram

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Cevap

Adım Adım Çözüm