Arithmetic Alıştırma Soruları — GRE General Test — Sayfa 7

Soru 121Soru →

Two positive integers $a$ and $b$ have prime factorizations of the form $a = 2^x \times 3^y \times 5^z$ and $b = 2^u \times 3^v \times 5^w$ , where $x, y, z, u, v, w$ are non-negative integers. The greatest common divisor (GCD) of $a$ and $b$ is $180$ , and the least common multiple (LCM) of $a$ and $b$ is $54,000$ . If $a$ has exactly $24$ positive divisors, what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 18000

Cevap

18000

Prime factorizing

\text{GCD}(a,b) = 180 = 2^2 \times 3^2 \times 5^1

and

\text{LCM}(a,b) = 54,000 = 2^4 \times 3^3 \times 5^3

reveals that the pair of exponents for bases

2, 3, 5

are

\{2, 4\}

,

\{2, 3\}

, and

\{1, 3\}

respectively. Applying the total divisor formula

(x+1)(y+1)(z+1) = 24

for

a

,

x+1

must divide

24

. Since

x \in \{2, 4\}

,

x

must be

2

(since

5

does not divide

24

). This leaves

(y+1)(z+1) = 8

. With

y \in \{2, 3\}

,

y+1

must be

4

(since

3

does not divide

8

), giving

y = 3

and

z = 1

. Thus,

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

. The remaining exponent choices for

b

are

u = 4

,

v = 2

, and

w = 3

, giving

b = 2^4 \times 3^2 \times 5^3 = 18,000

.

Adım Adım Çözüm

1

Factorize GCD and LCM into prime bases.

GCD = 2^2 * 3^2 * 5^1; LCM = 2^4 * 3^3 * 5^3.

GCD represents the minimum prime exponents and LCM represents the maximum prime exponents of a and b.

2

Set up candidate exponent pairs for primes 2, 3, and 5.

{x, u} = {2, 4}, {y, v} = {2, 3}, and {z, w} = {1, 3}.

For any prime p, min(exp_p(a), exp_p(b)) = exp_p(GCD) and max(exp_p(a), exp_p(b)) = exp_p(LCM).

3

Solve for x, y, and z using the divisor count formula d(a) = (x+1)(y+1)(z+1) = 24.

x = 2, y = 3, z = 1.

If x = 4, then x+1 = 5, which is not a factor of 24. Thus x = 2. Then (y+1)(z+1) = 8. Since y in {2, 3}, y+1 = 4 implies y = 3 and z = 1.

4

Determine the exponents of b and calculate b.

u = 4, v = 2, w = 3, so b = 2^4 * 3^2 * 5^3 = 18,000.

Complementary values from the sets {x,u}, {y,v}, and {z,w} give the exponents for b.

Anahtar Kavram

Relationship between prime factor exponents in GCD, LCM, and divisor counting functions.

Soru 122Soru →

An employee's annual salary was increased by $20\%$ at the beginning of 2024. At the end of 2024, the salary was decreased by $20\%$ from its updated value. At the beginning of 2025, the salary was increased by $10\%$ from its end-of-2024 value. If the initial salary before any of these changes was $S$ , which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

The salary at the end of 2024 was

4\%

less than the original salary

S

.

The overall percentage change from the original salary

S

to the salary at the beginning of 2025 is a

5.6\%

increase.

The salary at the end of 2024 was equal to the original salary

S

because the

20\%

increase and

20\%

decrease offset each other.

The salary at the beginning of 2025 was

10\%

greater than the original salary

S

.

The net change over the entire period is equivalent to a single decrease of

10\%

followed by an increase of

15.6\%

.

Cevabı ve açıklamayı göster

Cevap: The salary at the end of 2024 was $4\%$ less than the original salary $S$ .; The overall percentage change from the original salary $S$ to the salary at the beginning of 2025 is a $5.6\%$ increase.

Cevap

The statement that the salary at the end of 2024 was 4% less than the original salary S, and the statement that the overall percentage change from the original salary S to the salary at the beginning of 2025 is a 5.6% increase, are both correct.

The initial salary

S

experiences successive changes: first multiplied by

1.20

, then by

0.80

, and finally by

1.10

. At the end of 2024, the value is

1.20 \times 0.80 S = 0.96S

, which is

4\%

less than

S

. At the beginning of 2025, the value is

0.96S \times 1.10 = 1.056S

, which reflects a net increase of

5.6\%

over the initial amount

S

.

Adım Adım Çözüm

1

Calculate the salary after the first 20% increase at the beginning of 2024.

Salary =

S \times (1 + 0.20) = 1.20S

An increase of 20% scales the initial value

S

by a multiplier of 1.20.

2

Calculate the salary after the 20% decrease at the end of 2024.

Salary =

1.20S \times (1 - 0.20) = 1.20S \times 0.80 = 0.96S

The 20% decrease is calculated using the updated base of

1.20S

, yielding

0.96S

, which represents a 4% overall decrease from

S

.

3

Calculate the salary after the 10% increase at the beginning of 2025.

Final Salary =

0.96S \times (1 + 0.10) = 0.96S \times 1.10 = 1.056S

The 10% increase is calculated using the base value of

0.96S

.

4

Determine the net percent change from the original salary

S

to the beginning of 2025.

Net Change =

\frac{1.056S - S}{S} \times 100\% = 0.056 \times 100\% = 5.6\%

increase.

Comparing the final multiplier

1.056

to the original base multiplier

1.000

shows a 5.6% overall growth.

Anahtar Kavram

Successive Percentage Changes and Base Value Shifts

Soru 123Soru →

Let $N = 2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are positive integers. The integer $N$ is divisible by $24$ , has exactly $36$ positive integer divisors, and $\frac{N}{5}$ is not divisible by $25$ . What is the least possible value of $a + b + c$ ?

6

7

8

9

10

Cevabı ve açıklamayı göster

Cevap: $7$

Cevap

7

The correct answer is

7

. Since

24 = 2^3 \cdot 3^1

divides

N

,

a \ge 3

and

b \ge 1

. The condition that

\frac{N}{5}

is not divisible by

25

restricts

c

to either

1

or

2

. For

c=2

, the divisor equation becomes

(a+1)(b+1)(3) = 36

, so

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

. Taking

a+1=4

(

a=3

) and

b+1=3

(

b=2

) satisfies

a \ge 3

and gives a minimal sum of

a+b+c = 3+2+2 = 7

.

Adım Adım Çözüm

1

Analyze divisibility conditions on exponents

Since

N

is divisible by

24 = 2^3 \cdot 3^1

, we must have

a \ge 3

and

b \ge 1

. Since

\frac{N}{5} = 2^a \cdot 3^b \cdot 5^{c-1}

is not divisible by

25 = 5^2

,

c-1 < 2 \implies c \le 2

. Because

c

is a positive integer,

c

can be

1

or

2

.

Establish structural constraints on prime exponents from the divisibility statements.

2

Set up the total divisor formula

The total number of positive integer divisors of

N

is

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

.

Apply the standard formula for counting divisors using prime factorization.

3

Evaluate candidate values for

c = 1

and

c = 2

If

c = 1

, then

c+1 = 2

, so

(a+1)(b+1) = 18

. Since

a \ge 3 \implies a+1 \ge 4

, possible integer pairs for

(a+1, b+1)

are

(6, 3)

giving

a=5, b=2 \implies a+b+c = 8

, or

(9, 2)

giving

a=8, b=1 \implies a+b+c = 10

.
If

c = 2

, then

c+1 = 3

, so

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

. With

a+1 \ge 4

, possible pairs for

(a+1, b+1)

are

(4, 3)

giving

a=3, b=2 \implies a+b+c = 3+2+2 = 7

, or

(6, 2)

giving

a=5, b=1 \implies a+b+c = 8

.

Test allowable values of

c

to find all valid exponent triples

(a, b, c)

and minimize their sum.

4

Identify the minimum sum

The minimum sum of

a + b + c

is

7

, achieved when

a = 3

,

b = 2

, and

c = 2

.

Compare sums across all valid exponent combinations.

Anahtar Kavram

Prime Factorization, Divisibility Rules, and Number of Divisors Formula

Tahmini Süre:2m 0s

Soru 124Soru →

On the real number line, $x$ is a real number such that $|x + 2| \le 5$ . Which of the following could be the value of $x$ ? Select all such values.

Geçerli olan tümünü seçin

-8

-7

0

3

4

Cevabı ve açıklamayı göster

Cevap: $-7$ ; $0$ ; $3$

Cevap

The values that satisfy the inequality are

-7

,

0

, and

3

.

The inequality

|x + 2| \le 5

represents all real numbers

x

whose distance from

-2

on the number line is at most

5

. This gives the closed interval

[-7, 3]

. The values

-7

,

0

, and

3

all fall within this range.

Adım Adım Çözüm

1

Set up the compound inequality for the absolute value expression.

-5 \le x + 2 \le 5

For any real number expression

A

and non-negative constant

k

,

|A| \le k

is equivalent to

-k \le A \le k

.

2

Isolate

x

by subtracting

2

from all parts of the inequality.

-5 - 2 \le x \le 5 - 2 \implies -7 \le x \le 3

Subtracting a constant from all parts preserves the inequality signs.

3

Evaluate which given choices lie within the interval

[-7, 3]

.

-7

,

0

, and

3

lie within

[-7, 3]

, while

-8

and

4

lie outside.

Only values inside or on the boundaries of

[-7, 3]

satisfy the inequality.

Anahtar Kavram

Absolute Value Inequalities and Real Number Line Distance

Soru 125Soru →

Let $a$ and $b$ be positive integers such that $15a = 28b$ . Which of the following statements MUST be true? Select all that apply.

Geçerli olan tümünü seçin

a

is divisible by

7

b

is divisible by

15

a

must be an odd integer

b

must be a prime number

a + b

must be divisible by

30

Cevabı ve açıklamayı göster

Cevap: $a$ is divisible by $7$ ; $b$ is divisible by $15$

Cevap

The statements asserting that

a

is divisible by

7

and that

b

is divisible by

15

must be true.

Equating prime factorizations gives

(3 \cdot 5) \cdot a = (2^2 \cdot 7) \cdot b

. Because

15

and

28

share no common prime factors,

a

must contain

2^2 \cdot 7 = 28

as a factor, and

b

must contain

3 \cdot 5 = 15

as a factor. Therefore,

a

is divisible by

7

(since

7

divides

28

) and

b

is divisible by

15

.

Adım Adım Çözüm

1

Analyze the prime factorization of both sides of the equation.

The given equation

15a = 28b

can be rewritten as

(3 \cdot 5) \cdot a = (2^2 \cdot 7) \cdot b

.

Prime factorizations allow us to identify fundamental divisibility relationships between variables.

2

Determine the prime factors required for

a

and

b

.

a

must be divisible by

28

(i.e.,

a = 28k

for some integer

k \ge 1

), and

b

must be divisible by

15

(i.e.,

b = 15k

for the same integer

k

).

Since

\gcd(15, 28) = 1

, all prime factors of

28

(

2^2

and

7

) must come from

a

, and all prime factors of

15

(

3

and

5

) must come from

b

.

3

Evaluate each choice against these general forms.

Since

a = 28k

,

a

is a multiple of

7

, making the first statement true. Since

b = 15k

,

b

is a multiple of

15

, making the second statement true.

Any factor of

28

is automatically a factor of

a

, and any factor of

15

is automatically a factor of

b

.

Anahtar Kavram

Fundamental Theorem of Arithmetic and Coprime Divisibility Rules

Soru 126Soru →

An agricultural research station divides an experimental plot of land among three crops: Wheat, Rye, and Barley. Initially, Wheat occupies $\frac{4}{9}$ of the total plot area, Rye occupies $\frac{1}{3}$ of the total area, and Barley occupies the remaining area. First, $\frac{1}{3}$ of the Wheat area is converted to Rye. Next, $\frac{2}{5}$ of the new total Rye area is converted to Barley. Finally, $\frac{1}{4}$ of the new total Barley area is converted back to Wheat. If the final area occupied by Wheat is $126$ hectares, what was the total area, in hectares, of the experimental plot?

210

315

270

378

540

Cevabı ve açıklamayı göster

Cevap: 315

Cevap

315 hectares

By tracking the dynamic changes to each crop's area step-by-step, the final Wheat area is

\frac{54}{135} = \frac{2}{5}

of the total land area. Setting

\frac{2}{5} T = 126

yields

T = 315

hectares.

Adım Adım Çözüm

1

Determine initial fraction allocations for each crop

Wheat:

\frac{4}{9}

, Rye:

\frac{1}{3} = \frac{3}{9}

, Barley:

1 - \left(\frac{4}{9} + \frac{3}{9}\right) = \frac{2}{9}

The sum of all three crop areas must equal the total area (

1

).

2

Apply the first conversion (

\frac{1}{3}

of Wheat area converted to Rye)

Wheat transferred:

\frac{1}{3} \times \frac{4}{9} = \frac{4}{27}

. New Wheat:

\frac{4}{9} - \frac{4}{27} = \frac{8}{27}

. New Rye:

\frac{1}{3} + \frac{4}{27} = \frac{13}{27}

. Barley:

\frac{2}{9} = \frac{6}{27}

.

Wheat loses

\frac{1}{3}

of its current area, and Rye gains that exact amount.

3

Apply the second conversion (

\frac{2}{5}

of new Rye area converted to Barley)

Rye transferred:

\frac{2}{5} \times \frac{13}{27} = \frac{26}{135}

. New Rye:

\frac{13}{27} - \frac{26}{135} = \frac{39}{135}

. New Barley:

\frac{6}{27} + \frac{26}{135} = \frac{56}{135}

. Wheat remains

\frac{8}{27} = \frac{40}{135}

.

Rye loses

\frac{2}{5}

of its updated area, and Barley gains that exact amount.

4

Apply the third conversion (

\frac{1}{4}

of new Barley area converted to Wheat)

Barley transferred:

\frac{1}{4} \times \frac{56}{135} = \frac{14}{135}

. New Wheat:

\frac{40}{135} + \frac{14}{135} = \frac{54}{135} = \frac{2}{5}

.

Barley loses

\frac{1}{4}

of its updated area, and Wheat gains that exact amount.

5

Solve for the total plot area

T

\frac{2}{5} T = 126 \implies T = \frac{126 \times 5}{2} = 315

hectares.

Set the final calculated fraction of Wheat equal to the given area of

126

hectares.

Anahtar Kavram

Sequential fractional updates of non-disjoint sub-amounts

Tahmini Süre:2m 30s

Soru 127Soru →

An investor divides a total initial investment of $\$10,000$ between Account $X$ and Account $Y$ . Account $X$ earns simple annual interest at a rate of $6\%$ , while Account $Y$ earns annual interest at a rate of $10\%$ compounded annually. If no additional deposits or withdrawals are made and the total interest earned from both accounts combined at the end of $2$ years is $\$1,560$ , how much money was originally invested in Account $Y$ ?

\$3,500

\$4,000

\$4,500

\$6,000

\$6,400

Cevabı ve açıklamayı göster

Cevap: $\$4,000$

Cevap

\$4,000

The correct answer of

\$4,000

is determined by calculating the interest earned by each account over two years. Account X earns simple interest of

6\% \times 2 = 12\%

, producing interest equal to

0.12(10,000 - P_Y)

. Account Y earns compound interest at

10\%

annually, yielding an interest factor of

1.10^2 - 1 = 0.21

, or

0.21 P_Y

. Adding the interest expressions yields

1,200 + 0.09 P_Y = 1,560

, which simplifies to

0.09 P_Y = 360

and gives

P_Y = \$4,000

.

Adım Adım Çözüm

1

Define variables for the initial principal in each account

Let

P_Y

be the principal in Account

Y

. Then the principal in Account

X

is

10,000 - P_Y

.

The total investment across both accounts is

\$10,000

.

2

Calculate the interest earned by Account

X

over

2

years

\text{Interest}_X = (10,000 - P_Y) \times 0.06 \times 2 = 0.12(10,000 - P_Y) = 1,200 - 0.12 P_Y.

Account

X

earns simple annual interest of

6\%

per year for

2

years.

3

Calculate the interest earned by Account

Y

over

2

years

\text{Interest}_Y = P_Y \times ((1 + 0.10)^2 - 1) = P_Y \times (1.21 - 1) = 0.21 P_Y.

Account

Y

earns interest compounded annually at

10\%

per year for

2

years.

4

Formulate and solve the equation for total interest

(1,200 - 0.12 P_Y) + 0.21 P_Y = 1,560 \implies 1,200 + 0.09 P_Y = 1,560 \implies 0.09 P_Y = 360 \implies P_Y = 4,000.

The combined interest from both accounts equals

\$1,560

.

Anahtar Kavram

Simple vs Compound Interest over multiple time periods

Soru 128Soru →

Let $m$ and $n$ be positive integers whose prime factorizations consist only of the prime factors $2$ , $3$ , and $5$ . The greatest common divisor of $m$ and $n$ is $360$ , and their least common multiple is $108,000$ . If the power of $2$ in the prime factorization of $m$ is strictly greater than the power of $2$ in the prime factorization of $n$ , and $m$ has exactly $36$ positive divisors, what is the total number of positive divisors of $n$ ?

32

48

64

72

96

Cevabı ve açıklamayı göster

Cevap: $64$

Cevap

The total number of positive divisors of

n

is

64

.

By writing the GCD and LCM in prime factorized form,

\text{GCD}(m,n) = 2^3 \times 3^2 \times 5^1

and

\text{LCM}(m,n) = 2^5 \times 3^3 \times 5^3

. Given

a > x

for the exponent of

2

, we obtain

a=5

and

x=3

. Using the divisor count formula for

m

,

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

, which requires

(b+1)(c+1) = 6

. Testing possible exponent values

b \in \{2,3\}

reveals that only

b=2

yields an integer exponent

c=1

. This uniquely determines

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

. Calculating

(3+1)(3+1)(3+1)

gives

64

positive divisors.

Adım Adım Çözüm

1

Express the greatest common divisor (GCD) and least common multiple (LCM) in prime factorization form.

\text{GCD}(m, n) = 360 = 2^3 \times 3^2 \times 5^1

and

\text{LCM}(m, n) = 108,000 = 2^5 \times 3^3 \times 5^3

.

Prime factorizations allow direct comparison of the minimum and maximum exponents for each prime factor.

2

Determine the exponent of

2

for both integers.

For prime factor

2

,

\min(a, x) = 3

and

\max(a, x) = 5

. Since the exponent of

2

in

m

is strictly greater than in

n

(

a > x

),

a = 5

and

x = 3

.

The GCD gives the lower exponent bound while the LCM gives the upper exponent bound.

3

Use the number of divisors of

m

to solve for its remaining exponents.

Let

m = 2^5 \times 3^b \times 5^c

. The number of positive divisors of

m

is

(5+1)(b+1)(c+1) = 6(b+1)(c+1) = 36

, so

(b+1)(c+1) = 6

. Since

b \in \{2, 3\}

, if

b = 3

then

b+1=4

which does not divide

6

. Thus,

b = 2

and

c = 1

.

The divisor count formula requires integer exponents, uniquely identifying

b = 2

and

c = 1

for

m

.

4

Determine the exponents for

n

and calculate its divisor count.

Since

b = 2

and

\max(b, y) = 3

,

y = 3

. Since

c = 1

and

\max(c, z) = 3

,

z = 3

. Thus

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

, and its divisor count is

(3+1)(3+1)(3+1) = 64

.

Combining the max/min constraints from LCM/GCD yields all exponents of

n

.

Anahtar Kavram

GCD and LCM Prime Exponent Relations & Divisor Counting Formula

Tahmini Süre:3m 0s

Soru 129Soru →

The market value of a rare piece of artwork increased by $x\%$ during its first decade in a gallery and then decreased by $x\%$ during its second decade. If the artwork's market value at the end of the second decade was $\$19,200$ , which represented a net decrease of $\$800$ from its initial market value prior to the two decades, what is the value of $x$ ?

4

10

16

20

25

Cevabı ve açıklamayı göster

Cevap: $20$

Cevap

The value of

x

is

20

.

The initial value of the artwork is

\$19,200 + \$800 = \$20,000

. Applying an increase of

x\%

followed by a decrease of

x\%

gives a net value of

\$20,000 \times (1 + \frac{x}{100})(1 - \frac{x}{100}) = \$20,000 \left(1 - \frac{x^2}{10,000}\right)

. Setting this equal to

\$19,200

yields

1 - \frac{x^2}{10,000} = 0.96

, which simplifies to

\frac{x^2}{10,000} = 0.04

, so

x^2 = 400

and

x = 20

.

Adım Adım Çözüm

1

Determine the initial market value before the two decades.

Initial Value

= \$19,200 + \$800 = \$20,000

.

The final value of

\$19,200

reflects a net decrease of

\$800

from the original starting value.

2

Express the successive percentage changes in terms of

x

.

Final Value

= \text{Initial Value} \times \left(1 + \frac{x}{100}\right) \times \left(1 - \frac{x}{100}\right) = \text{Initial Value} \times \left(1 - \frac{x^2}{10,000}\right)

.

An increase of

x\%

followed by a decrease of

x\%

on the new base yields the difference of squares formula for percent multipliers.

3

Set up the equation with known numerical values and solve for

x^2

.

20,000 \left(1 - \frac{x^2}{10,000}\right) = 19,200 \implies 1 - \frac{x^2}{10,000} = \frac{19,200}{20,000} = 0.96 \implies \frac{x^2}{10,000} = 0.04 \implies x^2 = 400

.

Dividing the final value by the initial value gives the net multiplier of

0.96

, meaning a

4\%

overall decrease.

4

Take the positive square root to find

x

.

x = \sqrt{400} = 20

.

Since

x

represents a positive percentage rate,

x = 20

.

Anahtar Kavram

Successive percentage changes and base shifts using the difference of squares

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

.

Tahmini Süre:2m 0s

Soru 130Soru →

At the beginning of 2025, Company X's allocation for Marketing was $40\%$ greater than its allocation for Research. Over the course of the year, the Marketing allocation was decreased by $25\%$ , while the Research allocation was increased by $p\%$ . At the end of 2025, the revised Research allocation was $20\%$ greater than the revised Marketing allocation. What is the value of $p$ ?

20

25

26

38

50

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

The value of

p

is 26.

The correct option correctly applies successive percentage changes to each variable. Starting with Research as

R

, Marketing is

1.40R

. Reducing Marketing by

25\%

yields

1.40R \times 0.75 = 1.05R

. Increasing this result by

20\%

gives the final Research amount of

1.20 \times 1.05R = 1.26R

, corresponding to a

26\%

increase over the initial Research allocation

R

.

Adım Adım Çözüm

1

Define variables for the initial allocations of both departments.

Let

R

represent the initial Research allocation. Then the initial Marketing allocation is

1.40R

.

Marketing was

40\%

greater than Research at the start of the year.

2

Calculate the revised Marketing allocation at the end of the year.

Revised Marketing allocation

= 1.40R \times (1 - 0.25) = 1.40R \times 0.75 = 1.05R

.

The Marketing allocation decreased by

25\%

from its initial value.

3

Determine the revised Research allocation using the end-of-year relationship.

Revised Research allocation

= 1.20 \times 1.05R = 1.26R

.

The revised Research allocation was

20\%

greater than the revised Marketing allocation.

4

Find the percentage increase

p

for the Research allocation.

R \left(1 + \frac{p}{100}\right) = 1.26R \implies 1 + \frac{p}{100} = 1.26 \implies p = 26

.

The change from

R

to

1.26R

represents a

26\%

increase.

Anahtar Kavram

Successive percent changes and base shifts in multi-variable algebraic scenarios

Tahmini Süre:2m 30s

Soru 131Soru →

On a number line, point $P$ corresponds to $-3$ and point $Q$ corresponds to $5$ . If point $R$ is the midpoint of line segment $PQ$ , what is the value of $|R - P|$ ?

2

4

8

1

15

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The correct distance between point P and midpoint R is 4.

The total distance between

P (-3)

and

Q (5)

on the real number line is

|5 - (-3)| = 8

. Since

R

is the midpoint of segment

PQ

, it splits the total length into two equal halves of length

4

. Therefore, the distance from

P

to

R

, represented by

|R - P|

, is

4

.

Adım Adım Çözüm

1

Calculate the total distance between points P and Q on the number line.

Distance

PQ = |5 - (-3)| = |5 + 3| = 8

.

The distance between any two points

a

and

b

on a number line is given by

|b - a|

.

2

Find the coordinate of the midpoint R.

Coordinate of

R = \frac{-3 + 5}{2} = \frac{2}{2} = 1

.

The midpoint coordinate is the average of the coordinates of the two endpoints.

3

Calculate the absolute distance between R and P.

|R - P| = |1 - (-3)| = |1 + 3| = 4

.

The absolute value gives the non-negative distance between the midpoint and point P.

Anahtar Kavram

Distance on a Number Line and Midpoint Formula

Soru 132Soru →

Let $M = 3.2 \times 10^{-5}$ and $N = 8.0 \times 10^{-4}$ . Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

N - M = 7.68 \times 10^{-4}

The ratio

\frac{N}{M}

is equal to

25

\sqrt{M \cdot N} = 1.6 \times 10^{-5}

M + N = 1.12 \times 10^{-3}

\frac{1}{M} = 3.125 \times 10^4

Cevabı ve açıklamayı göster

Cevap: $N - M = 7.68 \times 10^{-4}$ ; The ratio $\frac{N}{M}$ is equal to $25$ ; $\frac{1}{M} = 3.125 \times 10^4$

Cevap

The correct statements are

N - M = 7.68 \times 10^{-4}

, the ratio

\frac{N}{M}

is equal to

25

, and

\frac{1}{M} = 3.125 \times 10^4

.

The statement

N - M = 7.68 \times 10^{-4}

is true because expressing both numbers with the exponent

10^{-4}

gives

(8.0 - 0.32) \times 10^{-4} = 7.68 \times 10^{-4}

. The statement that the ratio

\frac{N}{M}

is equal to

25

is true because

\frac{8.0}{3.2} \times 10^{-4 - (-5)} = 2.5 \times 10 = 25

. The statement

\frac{1}{M} = 3.125 \times 10^4

is true because

\frac{1}{3.2} \times 10^5 = 0.3125 \times 10^5 = 3.125 \times 10^4

.

Adım Adım Çözüm

1

Evaluate

N - M

N - M = 7.68 \times 10^{-4}

Convert

M

to the exponent

10^{-4}

:

M = 0.32 \times 10^{-4}

. Then

N - M = (8.0 - 0.32) \times 10^{-4} = 7.68 \times 10^{-4}

.

2

Evaluate

\frac{N}{M}

\frac{N}{M} = 25

Divide coefficients and subtract exponents:

\frac{8.0}{3.2} \times 10^{-4 - (-5)} = 2.5 \times 10^1 = 25

.

3

Evaluate

\sqrt{M \cdot N}

\sqrt{M \cdot N} = 1.6 \times 10^{-4}

Compute

M \cdot N = 25.6 \times 10^{-9} = 2.56 \times 10^{-8}

. Take the square root:

\sqrt{2.56} \times \sqrt{10^{-8}} = 1.6 \times 10^{-4}

.

4

Evaluate

M + N

M + N = 8.32 \times 10^{-4}

Express with common powers of ten:

0.32 \times 10^{-4} + 8.0 \times 10^{-4} = 8.32 \times 10^{-4}

.

5

Evaluate

\frac{1}{M}

\frac{1}{M} = 3.125 \times 10^4

Compute

\frac{1}{3.2 \times 10^{-5}} = 0.3125 \times 10^5 = 3.125 \times 10^4

.

Anahtar Kavram

Operations with numbers in scientific notation and place value adjustments

Soru 133Soru →

At a financial trading firm, incoming trade orders pass through a three-stage automated verification pipeline:

1. In Stage 1, $\frac{1}{6}$ of all incoming orders are rejected as invalid, and $\frac{1}{5}$ of the remaining valid orders are flagged for manual review.
2. In Stage 2, of the orders that were cleared directly without manual review, $\frac{3}{8}$ are executed immediately, while the rest are placed in a waiting queue.
3. In Stage 3, $\frac{1}{3}$ of the queued orders are processed, leaving the remaining queued orders unprocessed.

If exactly $250$ queued orders remain unprocessed at the end of Stage 3, how many incoming orders were flagged for manual review in Stage 1?

100

150

180

200

225

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

150 orders were flagged for manual review in Stage 1.

Let

N

represent the total number of incoming trade orders. In Stage 1, after

\frac{1}{6}N

orders are rejected,

\frac{5}{6}N

remain. Flagging

\frac{1}{5}

of these remaining valid orders means

\frac{1}{5} \times \frac{5}{6}N = \frac{1}{6}N

orders are flagged for manual review, leaving

\frac{5}{6}N - \frac{1}{6}N = \frac{2}{3}N

orders cleared directly. In Stage 2,

\frac{5}{8}

of the cleared orders enter the queue, which equals

\frac{5}{8} \times \frac{2}{3}N = \frac{5}{12}N

. In Stage 3,

\frac{2}{3}

of the queued orders remain unprocessed, giving

\frac{2}{3} \times \frac{5}{12}N = \frac{5}{18}N

. Setting

\frac{5}{18}N = 250

yields

N = 900

. The number of orders flagged for manual review is

\frac{1}{6} \times 900 = 150

.

Adım Adım Çözüm

1

Determine the fraction of orders cleared directly in Stage 1.

The fraction of valid orders remaining after rejection is

1 - \frac{1}{6} = \frac{5}{6}

. The fraction flagged for manual review is

\frac{1}{5} \times \frac{5}{6} = \frac{1}{6}

of the total. Thus, the fraction cleared directly is

\frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}

of total incoming orders.

Sequential fractional deductions must be applied to the remaining amounts at each sub-stage.

2

Determine the fraction of orders placed in the queue during Stage 2.

Of the

\frac{2}{3}

directly cleared orders,

\frac{3}{8}

are executed immediately, so

1 - \frac{3}{8} = \frac{5}{8}

are placed in the queue. The fraction of total incoming orders queued is

\frac{5}{8} \times \frac{2}{3} = \frac{5}{12}

.

Fraction of a fraction requires multiplying the operational fraction by the current subtotal.

3

Determine the fraction of total incoming orders that remain unprocessed at the end of Stage 3.

Since

\frac{1}{3}

of queued orders are processed,

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

of the queued orders remain unprocessed. The fraction of total orders unprocessed is

\frac{2}{3} \times \frac{5}{12} = \frac{5}{18}

.

Multiply the remaining fraction in Stage 3 by the total queued fraction from Stage 2.

4

Calculate the total number of incoming orders (

N

) and the target flagged orders.

Setting

\frac{5}{18}N = 250

gives

N = 250 \times \frac{18}{5} = 900

total incoming orders. The number of orders flagged for manual review is

\frac{1}{6} \times 900 = 150

.

Equate the given numerical quantity to its fractional equivalent to find the total, then compute the required part.

Anahtar Kavram

Multi-step fraction calculations involving fractions of remaining quantities and working backward from a known part to the whole.

Soru 134Soru →

A boutique owner increases the original price of a handbag by $25\%$ . One month later, the owner reduces the updated price by $20\%$ . Which of the following statements about the price of the handbag must be true? Select all such statements.

Geçerli olan tümünü seçin

The final price of the handbag is equal to its original price.

The overall net percent change in the price of the handbag is

0\%

.

The dollar amount of the

25\%

increase is equal to the dollar amount of the

20\%

discount.

The final price of the handbag is

5\%

greater than its original price.

The dollar amount of the

20\%

discount is less than the dollar amount of the

25\%

increase.

Cevabı ve açıklamayı göster

Cevap: The final price of the handbag is equal to its original price.; The overall net percent change in the price of the handbag is $0\%$ .; The dollar amount of the $25\%$ increase is equal to the dollar amount of the $20\%$ discount.

Cevap

The correct statements are: the final price is equal to the original price, the overall net percent change is 0%, and the dollar amount of the 25% increase equals the dollar amount of the 20% discount.

Applying a 25% increase to an initial amount

P

yields

1.25P

, representing a

0.25P

increase. Taking 20% off of

1.25P

calculates to

0.20 \times 1.25P = 0.25P

. Subtracting

0.25P

from

1.25P

returns the price to

1.00P

. Therefore, the final price is equal to the original price, the overall percent change is 0%, and the two dollar changes are equal.

Adım Adım Çözüm

1

Define a variable for the original price and calculate the price after the 25% increase.

Let the original price be

P

. The price after a

25\%

increase is

P + 0.25P = 1.25P

. The dollar increase is

0.25P

.

Establishing the intermediate price is necessary to apply the subsequent discount correctly.

2

Calculate the price after the 20% discount applied to the updated price.

The discount amount is

20\%

of

1.25P

, which is

0.20 \times 1.25P = 0.25P

. The final price is

1.25P - 0.25P = 1.00P

.

Percentage discounts must be evaluated using the price immediately preceding the discount as the base value.

3

Evaluate the net change and compare dollar amounts.

Final price = Original price (

P

). Net percent change =

\frac{1.00P - P}{P} \times 100\% = 0\%

. Dollar increase (

0.25P

) = Dollar decrease (

0.25P

).

Comparing final values to initial values confirms which statements are true.

Anahtar Kavram

Successive Percentage Changes and Base Value Shifts

Soru 135Soru →

On Monday, a community kitchen starts with a bulk container of rice. First, $\frac{1}{3}$ of the initial supply of rice is used for lunch. Next, $\frac{1}{4}$ of the remaining rice in the container is used for dinner. After preparing both meals, exactly $60$ kilograms of rice remain in the container. How many kilograms of rice were in the container initially?

84 kg

90 kg

120 kg

144 kg

180 kg

Cevabı ve açıklamayı göster

Cevap: 120 kg

Cevap

120 kg

The correct option is 120 kg. After using

\frac{1}{3}

of the initial supply for lunch,

\frac{2}{3}

of the supply remains. Using

\frac{1}{4}

of this remaining amount for dinner leaves

\frac{3}{4}

of the

\frac{2}{3}

remaining. Multiplying these fractions gives

\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}

of the original supply. Since

\frac{1}{2}

of the original supply equals

60

kg, the initial total supply was

60 \times 2 = 120

kg.

Adım Adım Çözüm

1

Determine the fraction of rice remaining after lunch.

Since

\frac{1}{3}

of the initial supply is used for lunch,

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

of the initial supply remains.

The remaining portion after lunch is the base for the next fractional reduction.

2

Determine the fraction of rice used for dinner and the fraction remaining overall.

Dinner uses

\frac{1}{4}

of the remaining

\frac{2}{3}

, which is

\frac{1}{4} \times \frac{2}{3} = \frac{1}{6}

of the initial total supply. The total fraction used is

\frac{1}{3} + \frac{1}{6} = \frac{1}{2}

. Thus, the fraction remaining after dinner is

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

of the initial total.

Sequential fractional reductions require multiplying the second fraction by the remaining fraction from the first step.

3

Solve for the initial total mass of rice.

Let

R

be the initial mass in kilograms. Then

\frac{1}{2} R = 60 \implies R = 120

kg.

Setting the calculated remaining fraction equal to the given numerical remaining quantity yields the starting total.

Anahtar Kavram

Sequential Fraction of Remaining Quantities

Tahmini Süre:1m 30s

Soru 136Soru →

Let $N$ be a positive integer with exactly $12$ positive divisors. If the sum of the distinct prime factors of $N$ is $12$ and $N$ is not divisible by $4$ , what is the least possible value of $N$ ?

Cevabı ve açıklamayı göster

Cevap: 126

Cevap

126

The least possible value of

N

is 126. The only sets of distinct prime factors summing to 12 are

\{5, 7\}

and

\{2, 3, 7\}

. Using

\{2, 3, 7\}

produces smaller candidates. Because

N

is not divisible by 4, the exponent of 2 must be 1. The total divisor condition

(1+1)(b+1)(c+1) = 12

requires

(b+1)(c+1) = 6

, giving exponent pairs

(1, 2)

or

(2, 1)

for bases 3 and 7. Assigning the exponent 2 to 3 and 1 to 7 minimizes

N

, giving

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

.

Adım Adım Çözüm

1

Find all sets of distinct prime factors that sum to 12.

The possible sets of distinct prime factors are

\{5, 7\}

and

\{2, 3, 7\}

.

Testing combinations of prime numbers (

2, 3, 5, 7, 11, \dots

):

5 + 7 = 12

and

2 + 3 + 7 = 12

are the only valid sets of distinct primes summing to 12.

2

Analyze the set of prime factors

\{2, 3, 7\}

under the condition that

N

is not divisible by 4.

The prime factorization is

N = 2^a \cdot 3^b \cdot 7^c

, where

a = 1

.

Since 2 is a prime factor of

N

,

a \ge 1

. Because

N

is not divisible by 4 (

2^2

),

a

must be strictly less than 2. Thus,

a = 1

.

3

Determine the exponents

b

and

c

using the total number of positive divisors.

The number of divisors is

(1+1)(b+1)(c+1) = 12

, which simplifies to

(b+1)(c+1) = 6

. The possible integer pairs

(b, c)

for

b, c \ge 1

are

(1, 2)

and

(2, 1)

.

The divisor count formula for

N = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}

is

(e_1+1)(e_2+1)\dots(e_k+1) = 12

.

4

Calculate values of

N

for these exponent pairs.

For

(b, c) = (1, 2)

,

N = 2^1 \cdot 3^1 \cdot 7^2 = 294

. For

(b, c) = (2, 1)

,

N = 2^1 \cdot 3^2 \cdot 7^1 = 126

.

To make

N

as small as possible, assign the larger exponent to the smaller prime base (

3^2 \cdot 7^1 < 3^1 \cdot 7^2

).

5

Compare with candidate values from the alternative set of prime factors

\{5, 7\}

.

For

\{5, 7\}

,

N = 5^3 \cdot 7^2 = 6125

or

5^5 \cdot 7^1 = 21875

, both of which are much larger than 126.

The combination

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

yields exponents of 3 and 2 (or 5 and 1), resulting in much higher prime powers.

Anahtar Kavram

Divisor count formula combined with prime factorization constraints

Soru 137Soru →

A residential solar power system generated $4{}000$ kilowatt-hours ( $\text{kWh}$ ) of electricity during its first year of operation. In the second year, electricity generation decreased by $10\%$ . In the third year, electricity generation increased by $10\%$ relative to the second year's generation. How many kilowatt-hours of electricity did the solar system generate in the third year?

3{}240

3{}600

3{}960

4{}000

4{}400

Cevabı ve açıklamayı göster

Cevap: $3{}960$

Cevap

The solar power system generated

3{}960\text{ kWh}

of electricity in the third year.

In the first year, generation was

4{}000\text{ kWh}

. A

10\%

decrease yields

4{}000 \times (1 - 0.10) = 3{}600\text{ kWh}

for the second year. In the third year, a

10\%

increase over the second year's generation yields

3{}600 \times (1 + 0.10) = 3{}960\text{ kWh}

.

Adım Adım Çözüm

1

Calculate the electricity generated in the second year after a

10\%

decrease from the first year.

4{}000 - (0.10 \times 4{}000) = 4{}000 - 400 = 3{}600\text{ kWh}

A

10\%

decrease means retaining

90\%

of the original

4{}000\text{ kWh}

base.

2

Calculate the electricity generated in the third year after a

10\%

increase over the second year's total.

3{}600 + (0.10 \times 3{}600) = 3{}600 + 360 = 3{}960\text{ kWh}

The

10\%

increase applies to the new base value of

3{}600\text{ kWh}

, not the initial

4{}000\text{ kWh}

.

Anahtar Kavram

Successive Percent Changes and Base Shifts

Tahmini Süre:45s

Soru 138Soru →

If $x$ is a positive integer such that $120$ is a factor of $x^2$ , what is the least possible value of $x$ ?

60

30

120

15

360

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60

The prime factorization of 120 is

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

. For

x^2

to be divisible by 120,

x^2

must contain at least three 2s, one 3, and one 5. Because

x^2

is a perfect square, all exponents in its prime factorization must be even numbers. Thus, the minimum prime factorization for

x^2

is

2^4 \times 3^2 \times 5^2

, which means

x

must be at least

2^2 \times 3^1 \times 5^1 = 60

.

Adım Adım Çözüm

1

Find the prime factorization of 120.

120 = 2^3 \times 3^1 \times 5^1

Decomposing 120 into prime factors reveals the minimal prime factor exponents required for

x^2

.

2

Determine the prime factors required for

x^2

to be divisible by 120.

The prime factorization of

x^2

must contain even exponents for all its prime factors and must be divisible by

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

. Therefore,

x^2

must contain at least

2^4 \times 3^2 \times 5^2

.

Because

x^2

is a perfect square, every prime factor in its prime factorization must have an even exponent.

3

Calculate the minimum value of

x

.

x = \sqrt{2^4 \times 3^2 \times 5^2} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60

Taking the square root of

2^4 \times 3^2 \times 5^2

gives the least positive integer

x

satisfying the condition.

Anahtar Kavram

Prime Factorization and Perfect Squares

Soru 139Soru →

At the start of a fiscal year, a venture capital fund divided its total investment capital between Asset Class $X$ and Asset Class $Y$ . Over the course of the year, the value of Asset Class $X$ increased by $25\%$ , while the value of Asset Class $Y$ decreased by $40\%$ . At the end of the year, the total value of the fund was exactly equal to its total value at the start of the year. Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

At the start of the year, the value of Asset Class

X

was

60\%

greater than the value of Asset Class

Y

.

At the start of the year, Asset Class

X

accounted for exactly

60\%

of the total capital of the fund.

At the end of the year, Asset Class

X

represented more than

75\%

of the total value of the fund.

The ratio of the final value of Asset Class

X

to the final value of Asset Class

Y

was

5

to

2

.

The dollar amount by which Asset Class

X

increased was equal to the dollar amount by which Asset Class

Y

decreased.

Cevabı ve açıklamayı göster

Cevap: At the start of the year, the value of Asset Class $X$ was $60\%$ greater than the value of Asset Class $Y$ .; At the end of the year, Asset Class $X$ represented more than $75\%$ of the total value of the fund.; The dollar amount by which Asset Class $X$ increased was equal to the dollar amount by which Asset Class $Y$ decreased.

Cevap

The statements that must be true are: at the start of the year, the value of Asset Class X was 60% greater than the value of Asset Class Y; at the end of the year, Asset Class X represented more than 75% of the total value of the fund; and the dollar amount by which Asset Class X increased was equal to the dollar amount by which Asset Class Y decreased.

The problem establishes that

0.25X = 0.40Y

, which yields

X = 1.60Y

. This confirms that at the start of the year, the value of Asset Class

X

was

60\%

greater than Asset Class

Y

. Furthermore, the final value of Asset Class

X

becomes

2.00Y

out of a total final fund value of

2.60Y

, giving it a share of

\frac{10}{13} \approx 76.92\%

, which is strictly greater than

75\%

. Finally, because the total portfolio value is unchanged, the exact dollar increase in Asset Class

X

(

0.25X

) must balance the dollar decrease in Asset Class

Y

(

0.40Y

).

Adım Adım Çözüm

1

Set up an equation using initial variables

X

and

Y

for the two asset classes.

Initial total value =

X + Y

. Final value of

X = 1.25X

, final value of

Y = 0.60Y

. Final total value =

1.25X + 0.60Y

.

Percentage increases and decreases apply directly as multiplier factors to their respective initial base values.

2

Equate the initial and final total fund values to find the relationship between

X

and

Y

.

1.25X + 0.60Y = X + Y \implies 0.25X = 0.40Y \implies X = 1.60Y

.

The problem states the net total value of the fund remained unchanged over the year.

3

Evaluate the initial relationship between Asset Class

X

and Asset Class

Y

.

Since

X = 1.60Y = (1 + 0.60)Y

, Asset Class

X

was

60\%

greater than Asset Class

Y

at the start of the year.

A multiplier of

1.60

corresponds to a

60\%

increase over the base value

Y

.

4

Determine the initial and final percentage allocations of Asset Class

X

relative to the total fund.

Initial share =

\frac{1.60Y}{2.60Y} = \frac{8}{13} \approx 61.54\%

. Final share =

\frac{1.25(1.60Y)}{1.25(1.60Y) + 0.60Y} = \frac{2.00Y}{2.60Y} = \frac{10}{13} \approx 76.92\%

.

Evaluating part-to-whole fractions demonstrates that the final share of Asset Class

X

(

76.92\%

) exceeds

75\%

.

5

Analyze dollar change equality between the assets.

Dollar gain in

X = 0.25X

. Dollar loss in

Y = 0.40Y

. Since

0.25X = 0.40Y

, the dollar gain in

X

equals the dollar loss in

Y

.

Zero net change in total fund value guarantees that positive dollar changes cancel negative dollar changes exactly.

Anahtar Kavram

Weighted Percentage Changes and Base Value Shifting

Tahmini Süre:3m 0s

Soru 140Soru →

A retail store tracks the price of a commercial oven across three consecutive quarters.

- In Quarter 1, the original price $P$ was increased by $x\%$ .
- In Quarter 2, the price was decreased by $x\%$ from its Quarter 1 price.
- In Quarter 3, the price was increased by $25\%$ from its Quarter 2 price.

If the price at the end of Quarter 3 is $20\%$ greater than the original price $P$ , which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

The value of

x

is

20

.

The price at the end of Quarter 2 was

4\%

less than the original price

P

.

The dollar amount of the price decrease in Quarter 2 was equal to the dollar amount of the price increase in Quarter 1.

The overall net percent change from the original price

P

to the price at the end of Quarter 2 was

0\%

.

The price at the end of Quarter 1 was

20\%

greater than the price at the end of Quarter 2.

Cevabı ve açıklamayı göster

Cevap: The value of $x$ is $20$ .; The price at the end of Quarter 2 was $4\%$ less than the original price $P$ .

Cevap

The statements holding that the value of

x

is

20

and that the price at the end of Quarter 2 was

4\%

less than the original price

P

are both true.

The statement showing

x = 20

is correct because solving the multi-stage percent equation

1.25 P (1 - \frac{x^2}{10,000}) = 1.20P

yields

x = 20

. The statement showing that the Quarter 2 price was

4\%

less than

P

is correct because

P_2 = P(1 - \frac{400}{10,000}) = 0.96P

, which represents a

4\%

reduction relative to

P

.

Adım Adım Çözüm

1

Express the price after each quarter algebraically in terms of initial price

P

and rate

x

.

Quarter 1 price:

P_1 = P\left(1 + \frac{x}{100}\right)

. Quarter 2 price:

P_2 = P_1\left(1 - \frac{x}{100}\right) = P\left(1 - \frac{x^2}{10,000}\right)

. Quarter 3 price:

P_3 = 1.25 P_2 = 1.25 P\left(1 - \frac{x^2}{10,000}\right)

.

Successive percentage changes compound sequentially on updated base values.

2

Set the Quarter 3 price expression equal to

1.20P

and solve for

x

.

1.25 P\left(1 - \frac{x^2}{10,000}\right) = 1.20 P \implies 1 - \frac{x^2}{10,000} = \frac{1.20}{1.25} = 0.96 \implies \frac{x^2}{10,000} = 0.04 \implies x^2 = 400 \implies x = 20

.

A price

20\%

greater than

P

equals

1.20P

.

3

Evaluate intermediate values to test each statement.

P_1 = 1.20P

,

P_2 = 0.96P

. The Quarter 1 increase is

0.20P

, while the Quarter 2 decrease is

0.20 \times 1.20P = 0.24P

.

Comparing exact calculated dollar and percentage values allows for direct validation of each option.

Anahtar Kavram

Successive Percentage Changes and Base Shift Dependencies

Arithmetic

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