Arithmetic Alıştırma Soruları — GRE General Test — Sayfa 10

Soru 181Soru →

Let $m$ and $n$ be positive integers such that $m$ is divisible by $18$ and $n$ is divisible by $15$ . Which of the following integers MUST be a divisor of the product $m \cdot n$ ? Select all such values.

Geçerli olan tümünü seçin

54

90

135

36

60

Cevabı ve açıklamayı göster

Cevap: $54$ ; $90$ ; $135$

Cevap

The integers

54

,

90

, and

135

must be divisors of the product

m \cdot n

.

Since

m

is a multiple of

18

(

2^1 \times 3^2

) and

n

is a multiple of

15

(

3^1 \times 5^1

), their product

m \cdot n

must be a multiple of

18 \times 15 = 270

. The prime factorization of

270

is

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

. Any integer that divides

270

is guaranteed to divide

m \cdot n

for all valid values of

m

and

n

. The numbers

54

(

2^1 \times 3^3

),

90

(

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

), and

135

(

3^3 \times 5^1

) are all divisors of

270

.

Adım Adım Çözüm

1

Express

m

and

n

in terms of their minimal prime factorizations.

m = 18a = 2^1 \times 3^2 \times a

and

n = 15b = 3^1 \times 5^1 \times b

for positive integers

a

and

b

.

Divisibility conditions specify the minimum prime factors that

m

and

n

must contain.

2

Find the minimal guaranteed prime factorization of the product

m \cdot n

.

m \cdot n = (2^1 \times 3^2 \times a) \times (3^1 \times 5^1 \times b) = 2^1 \times 3^3 \times 5^1 \times (ab) = 270 \times ab

.

Multiplying

m

and

n

combines their guaranteed prime factor powers.

3

Determine which choices divide

270 = 2^1 \times 3^3 \times 5^1

without requiring additional factors of

a

or

b

.

54 = 2^1 \times 3^3

,

90 = 2^1 \times 3^2 \times 5^1

, and

135 = 3^3 \times 5^1

all divide

270

. Numbers requiring

2^2

(

36

and

60

) do not necessarily divide

270

.

A number MUST divide

m \cdot n

if its prime factor powers do not exceed the minimum guaranteed powers in

m \cdot n

.

Anahtar Kavram

Divisibility of Integer Products via Prime Factorization

Soru 182Soru →

A sequence $a_1, a_2, a_3, \dots, a_n$ is defined by $a_1 = 1$ and $a_n = 2a_{n-1} + 3$ for all integers $n \geq 2$ . What is the value of $a_4$ ?

11

13

25

29

61

Cevabı ve açıklamayı göster

Cevap: $29$

Cevap

The value of

a_4

is

29

.

Using the recursive rule

a_n = 2a_{n-1} + 3

starting with

a_1 = 1

, we compute

a_2 = 2(1) + 3 = 5

,

a_3 = 2(5) + 3 = 13

, and finally

a_4 = 2(13) + 3 = 29

.

Adım Adım Çözüm

1

Calculate the second term

a_2

using the recursive formula with

a_1 = 1

.

a_2 = 2(1) + 3 = 5

The recursive rule requires the previous term to compute the next term.

2

Calculate the third term

a_3

using

a_2 = 5

.

a_3 = 2(5) + 3 = 13

Apply the recursive definition

a_n = 2a_{n-1} + 3

for

n = 3

.

3

Calculate the fourth term

a_4

using

a_3 = 13

.

a_4 = 2(13) + 3 = 29

Apply the recursive definition

a_n = 2a_{n-1} + 3

for

n = 4

.

Anahtar Kavram

Recursive Sequences

Soru 183Soru →

Let $A = 0.000032 \times 10^{n}$ and $B = 8.0 \times 10^{n-3}$ , where $n$ is an integer. If $\frac{A^2}{B} = 1.28 \times 10^{-4}$ , what is the value of $n$ ?

-2

1

3

4

9

Cevabı ve açıklamayı göster

Cevap: $3$

Cevap

The value of

n

is

3

.

Rewriting

0.000032

as

3.2 \times 10^{-5}

allows

A

to be expressed as

3.2 \times 10^{n-5}

. Squaring

A

yields

10.24 \times 10^{2n-10}

. Dividing by

B = 8.0 \times 10^{n-3}

gives

1.28 \times 10^{n-7}

. Equating

n - 7 = -4

directly leads to

n = 3

.

Adım Adım Çözüm

1

Convert

A

to standard scientific notation in terms of

n

A = 3.2 \times 10^{-5} \times 10^n = 3.2 \times 10^{n-5}

The decimal

0.000032

equals

3.2 \times 10^{-5}

because the decimal point is shifted

5

places to the right.

2

Calculate

A^2

A^2 = (3.2 \times 10^{n-5})^2 = (3.2)^2 \times 10^{2(n-5)} = 10.24 \times 10^{2n-10}

Apply the power of a product rule

(ab)^k = a^k b^k

and exponent power rule

(10^p)^q = 10^{pq}

.

3

Divide

A^2

by

B

\frac{A^2}{B} = \frac{10.24 \times 10^{2n-10}}{8.0 \times 10^{n-3}} = \left(\frac{10.24}{8.0}\right) \times 10^{(2n-10) - (n-3)} = 1.28 \times 10^{n-7}

Divide coefficients

(10.24 / 8.0 = 1.28)

and subtract powers of ten exponents

((2n - 10) - (n - 3) = n - 7)

.

4

Equate the simplified expression to the given value and solve for

n

1.28 \times 10^{n-7} = 1.28 \times 10^{-4} \implies n - 7 = -4 \implies n = 3

Since the coefficients match (

1.28

), equate the exponents of

10

to solve for

n

.

Anahtar Kavram

Scientific Notation Operations and Exponent Laws

Tahmini Süre:2m 0s

Soru 184Soru →

At the beginning of 2024, a university endowment fund was reallocated across various asset classes. Over the course of 2024, the total value of the endowment fund increased by $30\%$ . In 2025, market downturns caused the total value of the fund to decrease by $20\%$ relative to its value at the end of 2024. If the total value of the endowment fund at the end of 2025 was $\$520,000$ , what was the total value of the fund at the beginning of 2024?

\$455,000

\$468,000

\$480,000

\$499,200

\$500,000

Cevabı ve açıklamayı göster

Cevap: $\$500,000$

Cevap

\$500,000

The option stating

\$500,000

is correct because a

30\%

increase followed by a

20\%

decrease corresponds to multiplying the initial value

V

by

1.30 \times 0.80 = 1.04

. Solving

1.04V = 520,000

yields

V = 500,000

.

Adım Adım Çözüm

1

Define the initial value variable and express successive percent changes as growth/decay multipliers.

Let

V

be the initial value at the start of 2024. A

30\%

increase corresponds to a multiplier of

1 + 0.30 = 1.30

. A

20\%

decrease corresponds to a multiplier of

1 - 0.20 = 0.80

.

Successive percentage changes compound multiplicatively on the original base value.

2

Formulate the net equation for the final value.

V \times 1.30 \times 0.80 = 520,000 \implies V \times 1.04 = 520,000

Multiplying

1.30

by

0.80

yields a net factor of

1.04

, representing a net

4\%

increase over the two-year period.

3

Solve for the initial value

V

.

V = \frac{520,000}{1.04} = 500,000

Dividing the final amount by the combined multiplier isolates the original starting value.

Anahtar Kavram

Successive Percent Changes and Base Value Calculation

Tahmini Süre:1m 30s

Soru 185Soru →

An investment fund allocated its total initial capital equally into two separate accounts, Account X and Account Y. Account X earns simple annual interest at a rate of $r\%$ , while Account Y earns compound annual interest at the same rate of $r\%$ , compounded annually. At the end of 2 years, the total interest earned by Account Y exceeded the total interest earned by Account X by an amount equal to $4\%$ of the initial amount placed in Account X. What was the net percentage increase in the fund's total combined capital over the 2-year period?

40%

42%

44%

60%

84%

Cevabı ve açıklamayı göster

Cevap: 42%

Cevap

42%

The correct answer of 42% is found by recognizing that the 2-year difference between compound interest and simple interest at rate

r\%

equals

P(r/100)^2

. Setting this difference to

0.04P

yields

r = 20\%

. At

r = 20\%

, Account X produces

0.40P

in simple interest, while Account Y produces

0.44P

in compound interest. Together, they generate

0.84P

in interest on a total initial investment of

2P

, resulting in a net fund increase of

(0.84P / 2P) \times 100\% = 42\%

.

Adım Adım Çözüm

1

Set up expressions for interest earned by each account after 2 years

Let

P

be the initial capital in Account X (so Account Y also starts with

P

, making total initial capital

2P

). Simple interest for Account X is

I_X = P \cdot \frac{2r}{100} = 0.02r P

. Compound interest for Account Y is

I_Y = P\left[\left(1 + \frac{r}{100}\right)^2 - 1\right] = P\left(\frac{2r}{100} + \frac{r^2}{10000}\right)

.

Establishes the interest expressions based on simple vs. compound interest formulas over a 2-year period.

2

Calculate the annual interest rate

r

Difference in interest:

I_Y - I_X = P\left(\frac{r^2}{10000}\right)

. Given this equals

4\%

of

P

(

0.04P

), we have

\frac{r^2}{10000} = 0.04 \implies r^2 = 400 \implies r = 20\%

.

Isolates the interest rate

r

using the difference between compound and simple interest.

3

Determine total interest earned across both accounts

For Account X:

I_X = 0.02(20)P = 0.40P

. For Account Y:

I_Y = 0.40P + 0.04P = 0.44P

. Total interest earned =

0.40P + 0.44P = 0.84P

.

Sums the returns of both portfolio components.

4

Compute the net percentage increase of the total fund

Total initial capital =

2P

. Net percentage increase =

\frac{\text{Total Interest}}{\text{Total Initial Capital}} \times 100\% = \frac{0.84P}{2P} \times 100\% = 42\%

.

Calculates the overall percent change using the correct part-to-whole base (total initial investment

2P

).

Anahtar Kavram

Simple vs. Compound Interest Base Shift and Net Percent Change

Soru 186Soru →

Let $x$ , $y$ , and $z$ be non-zero integers that satisfy the following three conditions:

I. $(-1)^{x y + z} = -1$
II. $(-1)^{x z + y} = 1$
III. $x^3 y z^2 < 0$

Which of the following statements MUST be true?

x

and

y

are both even integers, and

x y < 0

x

is an odd integer, and

z

is an even integer

y

and

z

are both odd integers

x + z

is an even integer

x y z > 0

Cevabı ve açıklamayı göster

Cevap: $x$ and $y$ are both even integers, and $x y < 0$

Cevap

x

and

y

are both even integers, and

x y < 0

From Condition I,

(-1)^{xy+z} = -1

implies

xy + z

is odd, so

xy

and

z

have opposite parity. From Condition II,

(-1)^{xz+y} = 1

implies

xz + y

is even, so

xz

and

y

have the same parity. If

x

were odd,

xy

would share the parity of

y

, and

xz

would share the parity of

z

. This would mean

y

and

z

must have opposite parity (from Condition I) and the same parity (from Condition II), which is impossible. Thus,

x

must be even. With

x

even,

xy

is even, so

z

must be odd for

xy + z

to be odd. Likewise,

xz

is even, so

y

must be even for

xz + y

to be even. Finally, from Condition III, since

z^2 > 0

for non-zero

z

,

x^3 y z^2 < 0

reduces to

x^3 y < 0

, which means

xy < 0

. Thus,

x

and

y

are both even integers, and

xy < 0

.

Adım Adım Çözüm

1

Analyze Condition I for parity requirements

x y + z

must be an odd integer

Since

(-1)^k = -1

if and only if

k

is an odd integer,

x y + z

is odd. Thus,

x y

and

z

must have opposite parity.

2

Analyze Condition II for parity requirements

x z + y

must be an even integer

Since

(-1)^k = 1

if and only if

k

is an even integer,

x z + y

is even. Thus,

x z

and

y

must have the same parity.

3

Determine the parity of

x

,

y

, and

z

using proof by contradiction

x

is even,

y

is even, and

z

is odd

If

x

were odd, then

x y

would have the same parity as

y

, and

x z

would have the same parity as

z

. Condition I would require

y

and

z

to have opposite parity, while Condition II would require

y

and

z

to have the same parity, a contradiction. Therefore,

x

must be even. Since

x

is even,

x y

is even, making

z

odd (from Condition I). Similarly,

x z

is even, making

y

even (from Condition II).

4

Analyze Condition III for sign requirements

x y < 0

Since

z

is a non-zero integer,

z^2 > 0

. The inequality

x^3 y z^2 < 0

simplifies to

x^3 y < 0

. Since

x^3

has the same sign as

x

,

x^3 y < 0

implies

x y < 0

.

Anahtar Kavram

Parity rules for exponentiation and basic arithmetic operations, combined with sign rules for products

Soru 187Soru →

A positive integer $n$ has the prime factorization $2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are positive integers. Given that $\gcd(n, 7{}200) = 360$ and $\text{lcm}(n, 1{}080) = 3{}240$ , what is the value of $a + b + c$ ?

6

7

8

9

10

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

By prime factorizing each given term,

7{}200 = 2^5 \cdot 3^2 \cdot 5^2

,

360 = 2^3 \cdot 3^2 \cdot 5^1

,

1{}080 = 2^3 \cdot 3^3 \cdot 5^1

, and

3{}240 = 2^3 \cdot 3^4 \cdot 5^1

. Applying the definition of GCD as taking the minimum exponent gives

\min(a, 5) = 3 \implies a = 3

and

\min(c, 2) = 1 \implies c = 1

. Applying the definition of LCM as taking the maximum exponent gives

\max(b, 3) = 4 \implies b = 4

. Thus,

a + b + c = 3 + 4 + 1 = 8

.

Adım Adım Çözüm

1

Find the prime factorizations of the given integers and the GCD/LCM values.

7{}200 = 2^5 \cdot 3^2 \cdot 5^2

,

360 = 2^3 \cdot 3^2 \cdot 5^1

,

1{}080 = 2^3 \cdot 3^3 \cdot 5^1

, and

3{}240 = 2^3 \cdot 3^4 \cdot 5^1

.

Decomposing into prime factors allows direct comparison of exponents using GCD (minimum exponent) and LCM (maximum exponent) rules.

2

Analyze the GCD condition

\gcd(n, 7{}200) = 360

.

\min(a, 5) = 3 \implies a = 3

,

\min(b, 2) = 2 \implies b \ge 2

, and

\min(c, 2) = 1 \implies c = 1

.

The exponent of each prime factor in

\gcd(x, y)

is the minimum of their respective exponents in

x

and

y

.

3

Analyze the LCM condition

\text{lcm}(n, 1{}080) = 3{}240

.

\max(a, 3) = 3 \implies a \le 3

,

\max(b, 3) = 4 \implies b = 4

, and

\max(c, 1) = 1 \implies c \le 1

.

The exponent of each prime factor in

\text{lcm}(x, y)

is the maximum of their respective exponents in

x

and

y

.

4

Combine the exponent constraints to determine

a

,

b

, and

c

, then calculate their sum.

a = 3

,

b = 4

, and

c = 1

, so

a + b + c = 3 + 4 + 1 = 8

.

Combining

a=3

,

b \ge 2

with

b=4

, and

c=1

with

c \le 1

uniquely specifies

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

.

Anahtar Kavram

Prime Exponent Analysis of Greatest Common Divisor and Least Common Multiple

Tahmini Süre:2m 0s

Soru 188Soru →

A technology firm monitors its active user base and average monthly server cost per user across four consecutive quarters:

- In Quarter 1, the total server cost was $C$ .
- In Quarter 2, the active user base increased by $25\%$ relative to Quarter 1, while the average monthly server cost per user decreased by $20\%$ relative to Quarter 1.
- In Quarter 3, the active user base decreased by $x\%$ relative to Quarter 2, while the average monthly server cost per user increased by $x\%$ relative to Quarter 2, where $x > 0$ .
- In Quarter 4, the active user base increased by $20\%$ relative to Quarter 3, while the average monthly server cost per user decreased by $10\%$ relative to Quarter 3.

If the total server cost in Quarter 4 was $3.68\%$ greater than the total server cost in Quarter 1, what is the value of $x$ ?

4

15

20

25

40

Cevabı ve açıklamayı göster

Cevap: $20$

Cevap

The value of

x

is

20

.

The correct response is 20. Total cost in Quarter 1 is

C_1 = N_1 P_1 = C

. In Quarter 2,

C_2 = (1.25 N_1)(0.80 P_1) = 1.00 C

. In Quarter 3, the user base and cost per user change by

-x\%

and

+x\%

, giving

C_3 = C \left(1 - \frac{x}{100}\right)\left(1 + \frac{x}{100}\right) = C \left(1 - \frac{x^2}{10,000}\right)

. In Quarter 4, the overall factor becomes

1.20 \times 0.90 = 1.08

, so

C_4 = 1.08 C \left(1 - \frac{x^2}{10,000}\right)

. Setting

1.08 \left(1 - \frac{x^2}{10,000}\right) = 1.0368

yields

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

, from which

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

, giving

x^2 = 400

and

x = 20

.

Adım Adım Çözüm

1

Define total cost variables for Quarter 1 and Quarter 2.

Quarter 1 cost

C_1 = N_1 P_1 = C

. Quarter 2 user base

N_2 = 1.25 N_1

and cost per user

P_2 = 0.80 P_1

, so

C_2 = (1.25 N_1)(0.80 P_1) = 1.00 N_1 P_1 = C

.

Total cost is the product of user count and cost per user; successive changes in Quarter 2 offset each other completely.

2

Calculate total cost in Quarter 3 in terms of

x

.

Quarter 3 user base

N_3 = N_2\left(1 - \frac{x}{100}\right)

and cost per user

P_3 = P_2\left(1 + \frac{x}{100}\right)

. Total cost

C_3 = N_3 P_3 = C_2 \left(1 - \frac{x}{100}\right)\left(1 + \frac{x}{100}\right) = C \left(1 - \frac{x^2}{10,000}\right)

.

Applying the difference of squares identity

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

for compound percentage change.

3

Calculate total cost in Quarter 4 and set up the equation with Quarter 1 cost.

Quarter 4 user base

N_4 = 1.20 N_3

and cost per user

P_4 = 0.90 P_3

, giving

C_4 = (1.20)(0.90) C_3 = 1.08 C_3 = 1.08 C \left(1 - \frac{x^2}{10,000}\right)

. Since

C_4 = 1.0368 C

, we get

1.08 \left(1 - \frac{x^2}{10,000}\right) = 1.0368

.

Quarter 4 expenditure compounds the Quarter 3 cost by a factor of

1.20 \times 0.90 = 1.08

.

4

Solve for

x

.

1 - \frac{x^2}{10,000} = \frac{1.0368}{1.08} = 0.96 \implies \frac{x^2}{10,000} = 0.04 \implies x^2 = 400 \implies x = 20

.

Dividing

1.0368

by

1.08

yields

0.96

, giving

x^2 = 400

and

x = 20

since

x > 0

.

Anahtar Kavram

Compounding successive percent changes across multiple factors and using algebraic identities for net percent change.

Tahmini Süre:2m 30s

Soru 189Soru →

A car travels a total distance of $120$ miles. For the first $60$ miles, the car travels at an average speed of $30$ miles per hour, and for the remaining $60$ miles, the car travels at an average speed of $60$ miles per hour. What is the average speed of the car, in miles per hour, for the entire $120$ -mile trip?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The average speed of the car for the entire trip is

40

miles per hour.

The overall average speed is determined by dividing total distance (

120

miles) by total time (

3

hours), yielding

40

miles per hour.

Adım Adım Çözüm

1

Calculate time taken for the first segment

Time =

2

hours

Using

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

,

\frac{60}{30} = 2

hours.

2

Calculate time taken for the second segment

Time =

1

hour

Using

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

,

\frac{60}{60} = 1

hour.

3

Determine total distance and total time

Total distance =

120

miles, Total time =

3

hours

Add distances (

60 + 60 = 120

) and times (

2 + 1 = 3

).

4

Calculate overall average speed

Average speed =

40

miles per hour

Divide total distance by total time:

\frac{120}{3} = 40

.

Anahtar Kavram

Average Speed and Rates

Tahmini Süre:1m 0s

Soru 190Soru →

A sequence of numbers begins with $t_1 = 4$ . For all integers $n \geq 2$ , each term is defined by $t_n = 3t_{n-1} - 2$ . What is the value of $t_5$ ?

Cevabı ve açıklamayı göster

Cevap: 244

Cevap

The 5th term of the sequence,

t_5

, is equal to 244.

Applying the recursive relation

t_n = 3t_{n-1} - 2

step-by-step starting from

t_1 = 4

gives

t_2 = 10

,

t_3 = 28

,

t_4 = 82

, and finally

t_5 = 244

.

Adım Adım Çözüm

1

Find the second term,

t_2

, using

t_1 = 4

.

t_2 = 3(4) - 2 = 12 - 2 = 10

Substitute

n = 2

into the recursive rule

t_n = 3t_{n-1} - 2

.

2

Find the third term,

t_3

, using

t_2 = 10

.

t_3 = 3(10) - 2 = 30 - 2 = 28

Substitute

n = 3

into the recursive rule.

3

Find the fourth term,

t_4

, using

t_3 = 28

.

t_4 = 3(28) - 2 = 84 - 2 = 82

Substitute

n = 4

into the recursive rule.

4

Find the fifth term,

t_5

, using

t_4 = 82

.

t_5 = 3(82) - 2 = 246 - 2 = 244

Substitute

n = 5

into the recursive rule.

Anahtar Kavram

Evaluating terms in a recursively defined sequence

Soru 191Soru →

At the beginning of a quarter, the cost of raw material for a manufacturing plant was $x$ dollars per ton. During the first month, the cost per ton increased by $15\%$ . During the second month, the cost per ton decreased by $20\%$ relative to the first month's price. If the cost per ton at the end of the second month was $\$552$ , what was the initial cost $x$ , in dollars?

Cevabı ve açıklamayı göster

Cevap: 600

Cevap

600

A

15\%

increase followed by a

20\%

decrease yields a combined multiplier of

(1 + 0.15)(1 - 0.20) = 1.15 \times 0.80 = 0.92

. Setting

0.92x = 552

gives

x = \frac{552}{0.92} = 600

.

Adım Adım Çözüm

1

Calculate the price multiplier for the first month

Price after month 1 is

1.15x

A

15\%

increase scales the initial value by

(1 + 0.15) = 1.15

.

2

Calculate the price multiplier for the second month

Price after month 2 is

1.15x \times 0.80 = 0.92x

A

20\%

decrease applies to the new base price, scaling it by

(1 - 0.20) = 0.80

.

3

Solve for the initial price

x

x = 600

Divide the given final price of

\$552

by the overall multiplier

0.92

.

Anahtar Kavram

Successive Percent Changes and Base Shifts

Tahmini Süre:1m 30s

Soru 192Soru →

A contractor prepares a concrete mix using gravel and sand in a ratio of $4 : 7$ by weight. If the contractor uses $28$ tons of sand for a project, what is the total weight, in tons, of the concrete mix?

16

21

44

49

77

Cevabı ve açıklamayı göster

Cevap: $44$

Cevap

The total weight of the concrete mix is

44

tons.

The ratio of gravel to sand is

4 : 7

. Sand corresponds to

7

parts of the total mixture and weighs

28

tons, meaning

1

part represents

4

tons. The total mixture comprises

4 + 7 = 11

parts. Multiplying

11

parts by

4

tons per part gives a total weight of

44

tons.

Adım Adım Çözüm

1

Determine the weight per ratio part using the known quantity of sand.

Since

7

parts equal

28

tons of sand,

1

part

= \frac{28}{7} = 4

tons.

The ratio specifies that sand accounts for

7

equal parts of the total mixture.

2

Calculate the weight of the gravel component.

Gravel weight

= 4 \text{ parts} \times 4 \text{ tons/part} = 16

tons.

The ratio specifies that gravel accounts for

4

parts of the mixture.

3

Compute the total weight of the concrete mixture.

Total weight

= 16 \text{ tons (gravel)} + 28 \text{ tons (sand)} = 44

tons.

The total weight of the mixture is the sum of the weights of all individual ingredients.

Anahtar Kavram

Solving Proportions from Given Component Values

Soru 193Soru →

On a real number line, point $A$ has coordinate $-5$ and point $B$ has coordinate $11$ . Point $C$ has coordinate $x$ such that the distance between $C$ and the midpoint of line segment $AB$ is equal to $\frac{1}{3}$ of the distance between $C$ and point $B$ . What is the maximum possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The maximum possible value of

x

is

5

.

The midpoint of

A(-5)

and

B(11)

is

M = 3

. Setting up the distance equation

|x - 3| = \frac{1}{3}|x - 11|

leads to

3|x - 3| = |x - 11|

. Evaluating the two cases

3(x - 3) = x - 11

and

3(x - 3) = -(x - 11)

yields solutions

x = -1

and

x = 5

. The maximum possible value among these is 5.

Adım Adım Çözüm

1

Calculate the coordinate of the midpoint of line segment

AB

.

Midpoint coordinate

M = 3

.

The midpoint of two coordinates

a

and

b

on a number line is given by

\frac{a + b}{2} = \frac{-5 + 11}{2} = 3

.

2

Formulate the distance equation using absolute values.

3|x - 3| = |x - 11|

.

The distance between

x

and

3

is

|x - 3|

and the distance between

x

and

11

is

|x - 11|

. The given condition is

|x - 3| = \frac{1}{3}|x - 11|

.

3

Solve the absolute value equation for all possible values of

x

.

x = -1

and

x = 5

.

Splitting

3(x - 3) = x - 11

yields

x = -1

, and splitting

3(x - 3) = -(x - 11)

yields

x = 5

.

4

Select the maximum value among all valid solutions.

5

Comparing

x = -1

and

x = 5

, the maximum value is

5

.

Anahtar Kavram

Distance on a number line using absolute value and midpoint formula

Tahmini Süre:2m 0s

Soru 194Soru →

For how many integer values of $x$ does the inequality $||2x - 7| - 5| \le 4$ hold true?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

To solve

||2x - 7| - 5| \le 4

, break the outer absolute value into

-4 \le |2x - 7| - 5 \le 4

. Adding 5 to all parts yields

1 \le |2x - 7| \le 9

. This produces two simultaneous conditions:

|2x - 7| \le 9

, which gives

-1 \le x \le 8

, and

|2x - 7| \ge 1

, which gives

x \le 3

or

x \ge 4

. Intersecting these solution sets yields the real intervals

[-1, 3]

and

[4, 8]

. The integers contained within these intervals are

-1, 0, 1, 2, 3

and

4, 5, 6, 7, 8

, giving a total of 10 valid integer values.

Adım Adım Çözüm

1

Unpack the outer absolute value inequality

-4 \le |2x - 7| - 5 \le 4

An inequality of the form

|u| \le k

for

k \ge 0

is equivalent to

-k \le u \le k

.

2

Isolate the inner absolute value term

1 \le |2x - 7| \le 9

Adding 5 across the compound inequality isolates the term

|2x - 7|

.

3

Solve the upper bound inequality

|2x - 7| \le 9

-1 \le x \le 8

-9 \le 2x - 7 \le 9 \implies -2 \le 2x \le 16 \implies -1 \le x \le 8

.

4

Solve the lower bound inequality

|2x - 7| \ge 1

x \le 3

or

x \ge 4

The inequality

|u| \ge 1

splits into

u \ge 1

or

u \le -1

, yielding

2x - 7 \ge 1 \implies x \ge 4

or

2x - 7 \le -1 \implies x \le 3

.

5

Determine the overlapping interval and count integer solutions

10 integer solutions

The intersection of

[-1, 8]

with

((-\infty, 3] \cup [4, \infty))

is

[-1, 3] \cup [4, 8]

. The integers in this domain are

-1, 0, 1, 2, 3, 4, 5, 6, 7, 8

, which equals 10 integer values.

Anahtar Kavram

Solving nested absolute value inequalities on the real number line

Soru 195Soru →

At a public university, the ratio of the number of undergraduate students to the number of graduate students at the beginning of an academic year was $4$ to $1$ . By the end of the academic year, the number of undergraduate students had increased by $15\%$ , and the total student population (undergraduate and graduate combined) had increased by $18\%$ . By what percent did the number of graduate students increase?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30%

Representing the initial populations by their ratio components (

4

parts undergraduate,

1

part graduate,

5

parts total), an overall growth of

18\%

across

5

total parts yields

5 \times 0.18 = 0.90

total parts of growth. The undergraduate population grew by

15\%

, contributing

4 \times 0.15 = 0.60

parts of growth. The remaining

0.90 - 0.60 = 0.30

parts of growth must come from the graduate student component. Since the initial graduate population was

1

part, a growth of

0.30

parts corresponds to a

30\%

increase.

Adım Adım Çözüm

1

Set up initial component values using the ratio

4:1

Undergraduates =

400

, Graduates =

100

, Total =

500

Choosing convenient numbers compatible with the

4:1

ratio simplifies percentage calculations.

2

Calculate the updated undergraduate count after a

15\%

increase

New Undergraduates =

460

An increase of

15\%

means the new value is

115\%

of the original

400

.

3

Calculate the updated total count after an

18\%

increase

New Total =

590

An overall increase of

18\%

means the new total is

118\%

of the original

500

.

4

Subtract the new undergraduate count from the new total count to find the new graduate count

New Graduates =

130

The total population is the sum of undergraduate and graduate populations.

5

Calculate the percent change for the graduate population

30\%

The increase of

30

graduate students over the initial

100

graduate students represents a

30\%

increase.

Anahtar Kavram

Weighted percentage change across subgroup components

Tahmini Süre:2m 0s

Soru 196Soru →

At a logistics company, fuel expenditure accounted for $40\%$ of total operational costs in Year 1, while maintenance expenditure accounted for the remaining $60\%$ . In Year 2, fuel expenditure increased by $25\%$ , and maintenance expenditure decreased by $15\%$ . If there were no other operational costs in either year, by what percentage did the total operational cost change from Year 1 to Year 2?

An increase of

1\%

An increase of

5\%

An increase of

9\%

An increase of

10\%

A decrease of

1\%

Cevabı ve açıklamayı göster

Cevap: An increase of $1\%$

Cevap

An increase of

1\%

The total operational cost increases by

1\%

because fuel cost increases by

0.40 \times 25\% = +10\%

of the total cost, while maintenance cost decreases by

0.60 \times 15\% = -9\%

of the total cost, yielding a net change of

+10\% - 9\% = +1\%

.

Adım Adım Çözüm

1

Assign an arbitrary base value or variable for Year 1 total cost

Let the total operational cost in Year 1 be

C = 100

. Thus, fuel expenditure is

40

and maintenance expenditure is

60

.

Choosing a concrete baseline value makes multi-step percentage change calculations clear and straightforward.

2

Calculate Year 2 cost for each component

Year 2 fuel cost

= 40 \times (1 + 0.25) = 50

. Year 2 maintenance cost

= 60 \times (1 - 0.15) = 51

.

Apply the individual percentage increase (

+25\%

) and decrease (

-15\%

) to the respective Year 1 component values.

3

Sum the Year 2 component costs and compute total percentage change

Total Year 2 cost

= 50 + 51 = 101

. Percent change

= \frac{101 - 100}{100} \times 100\% = +1\%

.

Compare the new total operational cost to the original baseline cost to determine net percentage change.

Anahtar Kavram

Weighted Percentage Change

Tahmini Süre:1m 30s

Soru 197Soru →

An astronomical digital sensor captures $4.5 \times 10^7$ raw data packets during an observation run, where each packet has a size of $1.6 \times 10^{-4}$ megabytes. A compression algorithm reduces the total volume of these combined packets by $75\%$ . What is the total volume, in megabytes, of the compressed data expressed in scientific notation?

1.8 \times 10^3

5.4 \times 10^3

7.2 \times 10^3

1.8 \times 10^4

7.2 \times 10^{-3}

Cevabı ve açıklamayı göster

Cevap: $1.8 \times 10^3$

Cevap

1.8 \times 10^3

megabytes

Multiplying

4.5 \times 10^7

by

1.6 \times 10^{-4}

gives

7.2 \times 10^3

megabytes. Decreasing this total by

75\%

retains

25\%

of the volume, which equals

0.25 \times 7.2 \times 10^3 = 1.8 \times 10^3

megabytes.

Adım Adım Çözüm

1

Calculate the total uncompressed size by multiplying the packet count by individual packet size.

(4.5 \times 10^7) \times (1.6 \times 10^{-4}) = (4.5 \times 1.6) \times 10^{7 + (-4)} = 7.2 \times 10^3

megabytes.

Total volume is the product of quantity and unit size.

2

Determine the remaining fraction of data after a

75\%

size reduction.

100\% - 75\% = 25\% = 0.25

A

75\%

reduction leaves

25\%

of the original data.

3

Multiply the uncompressed volume by

0.25

and write the result in standard scientific notation.

0.25 \times (7.2 \times 10^3) = 1.8 \times 10^3

megabytes.

The coefficient

1.8

meets the condition

1 \leq a < 10

for scientific notation

a \times 10^n

.

Anahtar Kavram

Decimals and Scientific Notation

Soru 198Soru →

At the beginning of Year 1, a commercial real estate portfolio consisted of two properties, Property A and Property B, where the value of Property A was $50\%$ greater than the value of Property B. During Year 1, the value of Property A increased by $20\%$ while the value of Property B decreased by $20\%$ . During Year 2, the value of Property A decreased by $x\%$ while the value of Property B increased by $x\%$ . If the combined value of the two properties at the end of Year 2 was equal to the combined value of the two properties at the beginning of Year 1, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

Tracking base value changes across both properties yields a Year 1 ending total of

1.8B + 0.8B = 2.6B

. Applying Year 2 percentage adjustments gives a total value of

1.8B(1 - 0.01x) + 0.8B(1 + 0.01x) = 2.6B - 0.01Bx

. Setting this equal to the initial portfolio value of

2.5B

gives

0.1B = 0.01Bx

, which solves to

x = 10

.

Adım Adım Çözüm

1

Define initial values for both properties relative to a single variable at the beginning of Year 1

Property B value =

B

, Property A value =

1.5B

, and total portfolio value =

2.5B

Setting Property B as

B

allows all subsequent values and combined sums to be expressed cleanly in terms of

B

.

2

Calculate individual property values at the end of Year 1 after applying respective

+20\%

and

-20\%

changes

Property A value =

1.5B \times 1.20 = 1.8B

; Property B value =

B \times 0.80 = 0.8B

Year 1 percentage changes must be calculated using each property's initial starting base.

3

Formulate algebraic expressions for the property values at the end of Year 2

Total Year 2 end value =

1.8B \left(1 - \frac{x}{100}\right) + 0.8B \left(1 + \frac{x}{100}\right) = 2.6B - 0.01Bx

Year 2 percentage shifts apply to the new updated base values (

1.8B

and

0.8B

) obtained at the end of Year 1.

4

Set the Year 2 end total value equal to the Year 1 start total value and solve for

x

2.6B - 0.01Bx = 2.5B \implies 0.1B = 0.01Bx \implies x = 10

Equating the two expressions eliminates

B

from both sides, leaving a linear equation in

x

.

Anahtar Kavram

Successive Percentage Changes and Base Shifts across Multiple Portfolio Assets

Soru 199Soru →

Three automated production lines—Line A, Line B, and Line C—run continuous maintenance cycles every $45$ minutes, $60$ minutes, and $75$ minutes, respectively. All three lines completed a maintenance cycle simultaneously at 8:00 a.m. on Monday. A supervisory inspection is triggered whenever at least two of the three lines complete a maintenance cycle at the exact same time. Between 8:01 a.m. on Monday and 8:00 a.m. on Tuesday, inclusive, how many supervisory inspections will be triggered?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

To find the number of times at least two lines finish maintenance simultaneously, we determine the LCM for each pair of intervals (180, 225, and 300 minutes) and calculate how many multiples of these values occur within 1,440 minutes. Within this duration, there are 8 multiples of 180, 6 multiples of 225, and 4 multiples of 300. The single time all three coincide (at 900 minutes) is included in all three pairwise counts. Subtracting 2 instances of this overlap yields 8 + 6 + 4 - 2 = 16 total supervisory inspections.

Adım Adım Çözüm

1

Prime factorize each interval length

45 = 3^2 × 5; 60 = 2^2 × 3 × 5; 75 = 3 × 5^2

Prime factorization allows systematic calculation of pairwise and triple LCMs.

2

Compute pairwise and triple LCMs

lcm(45, 60) = 180; lcm(45, 75) = 225; lcm(60, 75) = 300; lcm(45, 60, 75) = 900

The LCM determines the interval between simultaneous completions.

3

Count occurrences within the 1,440-minute time window

A & B: 8 times; A & C: 6 times; B & C: 4 times; All three: 1 time

Dividing the total time window by each LCM yields the frequency of simultaneous maintenance events.

4

Apply Principle of Inclusion-Exclusion for sets

8 + 6 + 4 - 2(1) = 16 distinct inspection events

Events where all three lines coincide (at t = 900) are counted in all three pair counts and must be adjusted so that the time point is counted exactly once.

Anahtar Kavram

Least Common Multiple (LCM) and Inclusion-Exclusion Principle

Soru 200Soru →

On the real number line, $x$ is a real number such that its distance from $1$ is at most $6$ , and its distance from $-2$ is at least $4$ . Which of the following inequalities represents all possible values of $x$ ?

2 \le x \le 7

-5 \le x \le 7

-6 \le x \le 7

-5 \le x \le 2

-5 \le x \le -2 \text{ or } 2 \le x \le 7

Cevabı ve açıklamayı göster

Cevap: $2 \le x \le 7$

Cevap

The inequality

2 \le x \le 7

represents all possible values of

x

.

The distance condition 'at most 6 units from 1' translates to

|x - 1| \le 6

, yielding the interval

[-5, 7]

. The condition 'at least 4 units from -2' translates to

|x + 2| \ge 4

, yielding

x \le -6

or

x \ge 2

. Finding the values that belong to both conditions requires taking the intersection of

[-5, 7]

and

(-\infty, -6] \cup [2, \infty)

. Since

[-5, 7]

does not overlap with

(-\infty, -6]

, the only overlapping region is

[2, 7]

, which corresponds to

2 \le x \le 7

.

Adım Adım Çözüm

1

Express the geometric distance conditions as absolute value inequalities

Condition 1:

|x - 1| \le 6

. Condition 2:

|x - (-2)| = |x + 2| \ge 4

.

The distance between two numbers

a

and

b

on the number line is given by

|a - b|

.

2

Solve the first inequality

|x - 1| \le 6

-6 \le x - 1 \le 6 \implies -5 \le x \le 7

.

An absolute value inequality of the form

|u| \le k

(for

k \ge 0

) unwraps to

-k \le u \le k

.

3

Solve the second inequality

|x + 2| \ge 4

x + 2 \ge 4

or

x + 2 \le -4 \implies x \ge 2

or

x \le -6

.

An absolute value inequality of the form

|u| \ge k

(for

k \ge 0

) unwraps to

u \ge k

or

u \le -k

.

4

Find the intersection of the two solution sets

Combine

[-5, 7]

with

(-\infty, -6] \cup [2, \infty)

. The intersection of

[-5, 7]

and

(-\infty, -6]

is empty since

-5 > -6

. The intersection of

[-5, 7]

and

[2, \infty)

is

[2, 7]

, which means

2 \le x \le 7

.

The variable

x

must satisfy both conditions simultaneously.

Anahtar Kavram

Distance on a number line expressed as absolute value inequalities and finding overlapping intervals

Tahmini Süre:2m 0s

Arithmetic

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