Tüm alıştırma soruları

Soru 1Soru →

A baker has a flour mixture consisting only of wheat flour and rye flour. Currently, wheat flour accounts for $\frac{2}{5}$ of the total weight of the mixture. If the baker adds $9$ pounds of wheat flour to the mixture, wheat flour will account for $\frac{1}{2}$ of the new total weight of the mixture. What was the total weight, in pounds, of the original flour mixture?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The total weight of the original flour mixture was 45 pounds.

The initial total weight of the mixture is 45 pounds. Initially, wheat flour makes up

\frac{2}{5} \times 45 = 18

pounds. Adding 9 pounds of wheat flour increases the wheat flour to

18 + 9 = 27

pounds and the total weight to

45 + 9 = 54

pounds. The new fraction of wheat flour is

\frac{27}{54} = \frac{1}{2}

, which satisfies the given conditions.

Adım Adım Çözüm

1

Express the initial weight of wheat flour in terms of the initial total weight

W

.

Initial weight of wheat flour =

\frac{2}{5}W

.

Wheat flour represents

\frac{2}{5}

of the total mixture.

2

Formulate an equation reflecting the addition of 9 pounds of wheat flour.

\frac{\frac{2}{5}W + 9}{W + 9} = \frac{1}{2}

Adding 9 pounds of wheat flour increases both the amount of wheat flour and the total weight of the mixture by 9 pounds.

3

Solve the algebraic equation for

W

.

Cross-multiplying gives

2\left(\frac{2}{5}W + 9\right) = W + 9

, which simplifies to

\frac{4}{5}W + 18 = W + 9

. Subtracting

\frac{4}{5}W

and

9

from both sides gives

\frac{1}{5}W = 9

, so

W = 45

.

Isolating

W

gives the value of the original total weight.

Anahtar Kavram

Setting up and solving equations involving fractional parts when a quantity is added to both the part and the whole.

Soru 2Soru →

Let $n$ be a positive integer whose prime factorization consists only of the prime factors $2$ and $3$ . If $n$ has exactly $12$ positive divisors and $\gcd(n, 36) = 12$ , what is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 96

Cevap

The value of

n

is

96

.

Representing

n = 2^a \times 3^b

, the number of positive divisors is

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

. The greatest common divisor

\gcd(n, 36) = \gcd(2^a \times 3^b, 2^2 \times 3^2) = 2^{\min(a,2)} \times 3^{\min(b,2)} = 12 = 2^2 \times 3^1

. This requires

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

and

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

. Substituting

b = 1

into

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

gives

2(a+1) = 12

, so

a = 5

. Therefore,

n = 2^5 \times 3^1 = 32 \times 3 = 96

.

Adım Adım Çözüm

1

Set up the prime factorization of

n

and the divisor count equation.

n = 2^a \times 3^b

and

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

.

Since the prime factors of

n

are only

2

and

3

,

n

must take the form

2^a \times 3^b

, where the number of positive divisors is

(a+1)(b+1)

.

2

Analyze the exponent requirements using the greatest common divisor.

a \ge 2

and

b = 1

.

\gcd(2^a \times 3^b, 2^2 \times 3^2) = 2^{\min(a,2)} \times 3^{\min(b,2)} = 2^2 \times 3^1

. Matching powers gives

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

, and

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

.

3

Solve for exponent

a

and calculate

n

.

a = 5

, giving

n = 2^5 \times 3^1 = 96

.

Substituting

b = 1

into

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

gives

2(a+1) = 12 \implies a = 5

, which satisfies

a \ge 2

.

Anahtar Kavram

Prime exponent rules for GCD and divisor counting

Tahmini Süre:2m 0s

Soru 3Soru →

An equilateral triangle $ABC$ with side length $6\sqrt{3}$ is inscribed in a circle with center $O$ . What is the area of sector $AOB$ , divided by $\pi$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The area of sector

AOB

divided by

\pi

is 12.

Since triangle

ABC

is equilateral, its three vertices divide the circle into three congruent arcs of

120^\circ

each. Thus, central angle

\angle AOB = 120^\circ

. The relationship between the side length

s

of an inscribed equilateral triangle and the radius

R

of its circumscribed circle is

s = R\sqrt{3}

. Given

s = 6\sqrt{3}

, we solve for

R

to find

R = 6

. The area of sector

AOB

is

\frac{120^\circ}{360^\circ} \pi R^2 = \frac{1}{3} \pi (6^2) = 12\pi

. Dividing this area by

\pi

yields

12

.

Adım Adım Çözüm

1

Find the central angle

\angle AOB

corresponding to side

AB

of the inscribed equilateral triangle.

\angle AOB = 120^\circ

An inscribed equilateral triangle divides the

360^\circ

circle into three equal central angles.

2

Calculate the radius

R

of the circumscribed circle from the given side length

s = 6\sqrt{3}

.

R = 6

In an inscribed equilateral triangle,

s = R\sqrt{3}

. Substituting

6\sqrt{3} = R\sqrt{3}

gives

R = 6

.

3

Compute the area of sector

AOB

and divide by

\pi

.

12

\text{Sector Area} = \frac{120^\circ}{360^\circ} \times \pi \times 6^2 = 12\pi

. Dividing by

\pi

leaves

12

.

Anahtar Kavram

Relationship between inscribed shapes, circle radii, and sector area

Soru 4Soru →

A quality control engineer records the thickness, in millimeters, of 9 sample components: $11, 13, 15, 17, 19, 21, 23, 25,$ and $27$ . Each thickness measurement $x$ is then converted to a scaled rating $y$ using the linear formula $y = 1.5x + 4.8$ . What is the interquartile range (IQR) of the transformed dataset of $y$ -values?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The interquartile range of the transformed dataset is 15.

The interquartile range (IQR) measures the spread of the middle 50% of the data (

Q_3 - Q_1

). For the original dataset

11, 13, 15, 17, 19, 21, 23, 25, 27

, the median is

19

.

Q_1

is the median of the lower half

\{11, 13, 15, 17\}

, which is

\frac{13+15}{2} = 14

.

Q_3

is the median of the upper half

\{21, 23, 25, 27\}

, which is

\frac{23+25}{2} = 24

. Thus, the original

\text{IQR} = 24 - 14 = 10

. Under a linear transformation

y = ax + b

, measures of position shift by

a x + b

, so

Q_1(y) = 1.5(14) + 4.8 = 25.8

and

Q_3(y) = 1.5(24) + 4.8 = 40.8

. Subtracting these yields

\text{IQR}(y) = 40.8 - 25.8 = 15

. Notice that this is simply

1.5 \times 10

, as adding a constant shifts the location of the distribution but leaves measures of spread unchanged.

Adım Adım Çözüm

1

Find the quartiles of the original 9-element dataset.

Q_1 = 14

and

Q_3 = 24

The median of the dataset is 19 (the 5th value). The lower half of the data consists of

11, 13, 15, 17

, so

Q_1 = \frac{13 + 15}{2} = 14

. The upper half consists of

21, 23, 25, 27

, so

Q_3 = \frac{23 + 25}{2} = 24

.

2

Calculate the interquartile range of the original dataset.

\text{IQR}_x = 10

\text{IQR}_x = Q_3 - Q_1 = 24 - 14 = 10

.

3

Apply the linear transformation rules to find the transformed interquartile range.

\text{IQR}_y = 15

For a linear transformation

y = ax + b

, the interquartile range scales by

|a|

, so

\text{IQR}_y = |a| \cdot \text{IQR}_x = 1.5 \times 10 = 15

. The constant addition of

4.8

shifts all values equally and does not affect the spread/IQR.

Anahtar Kavram

Effect of linear transformations on measures of dispersion (IQR, standard deviation, range)

Alternatif Yöntem

Transform the individual quartiles directly:

Q_1(y) = 1.5(14) + 4.8 = 25.8

and

Q_3(y) = 1.5(24) + 4.8 = 40.8

. Then calculate the new IQR directly as

40.8 - 25.8 = 15

.

Tahmini Süre:1m 30s

Soru 5Soru →

A commercial coffee roaster creates a custom blend by mixing two existing bean blends: Blend X and Blend Y. Blend X consists of 60% Arabica beans and 40% Robusta beans by weight, whereas Blend Y consists of 30% Arabica beans and 70% Robusta beans by weight. The roaster mixes a quantity of Blend X with a quantity of Blend Y to produce a total of 50 pounds of a new mixture that is 42% Arabica beans by weight. How many pounds of Blend X are in the final mixture?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20 pounds

Let

x

represent the number of pounds of Blend X. The remaining weight of the mixture,

(50 - x)

pounds, comes from Blend Y. Setting up the equation for the total weight of Arabica beans gives

0.60x + 0.30(50 - x) = 0.42(50)

. Simplifying this expression yields

0.60x + 15 - 0.30x = 21

, which reduces to

0.30x = 6

. Dividing by

0.30

gives

x = 20

. Therefore, 20 pounds of Blend X were used.

Adım Adım Çözüm

1

Define variables for component weights

Let

x

be the pounds of Blend X. The weight of Blend Y used is

50 - x

pounds.

The total combined weight of the mixture is given as 50 pounds.

2

Formulate an equation for the total weight of Arabica beans

0.60x + 0.30(50 - x) = 0.42(50)

, which simplifies to

0.60x + 15 - 0.30x = 21

.

The sum of Arabica beans contributed by each blend must equal the total weight of Arabica beans in the combined mixture.

3

Solve the linear equation for

x

0.30x + 15 = 21 \implies 0.30x = 6 \implies x = 20

.

Subtract 15 from both sides to isolate the variable term, then divide by 0.30.

Anahtar Kavram

Linear Modeling and Mixture Problems

Tahmini Süre:1m 30s

Soru 6Soru →

For all real numbers

x

such that

x \neq 3

and

x \neq -5

, the algebraic expression

\frac{(x^2 - 9)^2 - 4(x - 3)^2}{(x - 3)(x + 5)}

can be simplified to the equivalent polynomial expression

x^2 + ax + b

, where

a

and

b

are constants. What is the value of

a + b

?

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

The value of

a + b

is

-5

.

Factoring the numerator yields

(x - 3)^2(x + 1)(x + 5)

. Canceling the common factors

(x - 3)

and

(x + 5)

with the denominator simplifies the expression to

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

. Comparing this to

x^2 + ax + b

identifies

a = -2

and

b = -3

, giving a sum of

a + b = -5

.

Adım Adım Çözüm

1

Factor the numerator by recognizing

(x^2 - 9) = (x - 3)(x + 3)

(x^2 - 9)^2 - 4(x - 3)^2 = [(x - 3)(x + 3)]^2 - 4(x - 3)^2 = (x - 3)^2 (x + 3)^2 - 4(x - 3)^2

Applying the difference of squares identity inside the squared term allows factoring out common factors.

2

Factor out

(x - 3)^2

from the numerator

(x - 3)^2 \left[ (x + 3)^2 - 4 \right]

Extracting the greatest common algebraic factor simplifies the remaining expression.

3

Apply difference of squares to

(x + 3)^2 - 4

(x + 3)^2 - 2^2 = ((x + 3) - 2)((x + 3) + 2) = (x + 1)(x + 5)

Recognizing

(x + 3)^2 - 2^2

as

A^2 - B^2

yields factored linear terms directly.

4

Substitute the fully factored numerator back into the rational expression and simplify

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

Canceling non-zero common factors

(x - 3)

and

(x + 5)

simplifies the rational function.

5

Expand

(x - 3)(x + 1)

and determine

a + b

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

, so

a = -2

and

b = -3

. Therefore,

a + b = -2 + (-3) = -5

.

Matching coefficients with

x^2 + ax + b

gives

a = -2

and

b = -3

.

Anahtar Kavram

Simplifying complex rational expressions through nested difference of squares factoring.

Soru 7Soru →

If $x$ is a real number such that $\sqrt[4]{16^{x+3}} = 8^{x-1}$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is

3

.

To solve

\sqrt[4]{16^{x+3}} = 8^{x-1}

, express both sides with the base 2. The left side simplifies to

\sqrt[4]{(2^4)^{x+3}} = 2^{\frac{4(x+3)}{4}} = 2^{x+3}

. The right side simplifies to

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

. Equating the exponents yields

x + 3 = 3x - 3

, which solves to

2x = 6

, giving

x = 3

.

Adım Adım Çözüm

1

Rewrite 16 and 8 using prime base 2

16 = 2^4

and

8 = 2^3

Converting terms to a common base allows direct comparison of exponents.

2

Simplify the left-hand side radical expression

\sqrt[4]{16^{x+3}} = \sqrt[4]{(2^4)^{x+3}} = 2^{\frac{4(x+3)}{4}} = 2^{x+3}

The

n

-th root

\sqrt[n]{a^m}

is equivalent to

a^{m/n}

.

3

Simplify the right-hand side exponential expression

8^{x-1} = (2^3)^{x-1} = 2^{3(x-1)} = 2^{3x-3}

Applying the exponent power rule

(a^m)^n = a^{m \cdot n}

requires multiplying

3

by

(x - 1)

.

4

Equate the exponents and solve for

x

x + 3 = 3x - 3 \implies 2x = 6 \implies x = 3

When

a^u = a^v

for

a > 0

and

a \neq 1

, it follows that

u = v

.

Anahtar Kavram

Solving exponential equations using prime base factorization and radical conversion rules

Soru 8Soru →

The quadratic equation $x^2 - kx + 36 = 0$ , where $k$ is a positive constant, has two distinct real roots $r_1$ and $r_2$ such that $r_2 - r_1 = 5$ . What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

The value of

k

is 13.

According to Vieta's formulas, for the quadratic equation

x^2 - kx + 36 = 0

, the sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 r_2 = 36

. Using the identity

(r_2 - r_1)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, we substitute the known values

r_2 - r_1 = 5

,

r_1 + r_2 = k

, and

r_1 r_2 = 36

. This gives

5^2 = k^2 - 4(36)

, which simplifies to

25 = k^2 - 144

. Solving for

k^2

gives

k^2 = 169

. Since

k

is specified as a positive constant,

k = 13

.

Adım Adım Çözüm

1

Apply Vieta's formulas to the given quadratic equation

The sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 r_2 = 36

.

For any quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and the product of roots is

c/a

.

2

Relate the difference of the roots to their sum and product

(r_2 - r_1)^2 = (r_1 + r_2)^2 - 4r_1 r_2

Expanding both sides shows that

r_2^2 - 2r_1 r_2 + r_1^2 = r_1^2 + 2r_1 r_2 + r_2^2 - 4r_1 r_2

, which is an algebraic identity.

3

Substitute the known values into the identity

5^2 = k^2 - 4(36) \implies 25 = k^2 - 144

We are given that

r_2 - r_1 = 5

,

r_1 r_2 = 36

, and

r_1 + r_2 = k

.

4

Solve for the positive constant

k

k^2 = 169 \implies k = 13

Adding 144 to both sides gives

k^2 = 169

. Taking the positive square root because

k > 0

yields

k = 13

.

Anahtar Kavram

Vieta's Formulas and Root Difference Identity

Soru 9Soru →

How many positive integers less than $100$ are divisible by both $4$ and $6$ , but are NOT divisible by $8$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct numerical answer is 4.

To be divisible by both 4 and 6, an integer must be a multiple of

\text{LCM}(4, 6) = 12

. The positive integers less than 100 that are multiples of 12 are 12, 24, 36, 48, 60, 72, 84, and 96 (8 integers). Among these, those divisible by 8 are multiples of

\text{LCM}(12, 8) = 24

, which are 24, 48, 72, and 96 (4 integers). Subtracting the excluded integers yields

8 - 4 = 4

.

Adım Adım Çözüm

1

Find the least common multiple of 4 and 6.

LCM(4, 6) = 12

An integer divisible by both 4 and 6 must be a multiple of their least common multiple.

2

Count positive integers less than 100 that are multiples of 12.

The multiples are 12, 24, 36, 48, 60, 72, 84, and 96, giving 8 integers.

The largest multiple of 12 strictly less than 100 is 96 (12 × 8).

3

Identify multiples of 12 that are also divisible by 8.

Since LCM(12, 8) = 24, these are the multiples of 24: 24, 48, 72, and 96, giving 4 integers.

Any integer divisible by both 12 and 8 must be a multiple of 24.

4

Subtract the excluded integers from the total count.

8 - 4 = 4

We exclude the multiples of 8 from the set of multiples of 12.

Anahtar Kavram

Divisibility, Least Common Multiple (LCM), and Set Exclusion

Soru 10Soru →

What is the least positive integer $n$ that leaves a remainder of $3$ when divided by $7$ , a remainder of $4$ when divided by $5$ , and is divisible by $9$ ?

Cevabı ve açıklamayı göster

Cevap: 234

Cevap

The least positive integer satisfying all three conditions is 234.

To find the least positive integer

n

that satisfies

n \equiv 3 \pmod{7}

,

n \equiv 4 \pmod{5}

, and

n \equiv 0 \pmod{9}

, we first find a general expression for integers meeting the first two conditions. Checking values of

5m + 4

modulo 7 gives

24

as the smallest positive integer matching both. The combined condition is

n \equiv 24 \pmod{35}

, or

n = 35k + 24

. Requiring

35k + 24

to be divisible by 9 gives

8k + 6 \equiv 0 \pmod{9}

, which simplifies to

k \equiv 6 \pmod{9}

. The smallest non-negative integer value for

k

is

6

, leading to

n = 35(6) + 24 = 234

.

Adım Adım Çözüm

1

Set up system of modular congruences for the remainders

n \equiv 3 \pmod{7}

,

n \equiv 4 \pmod{5}

, and

n \equiv 0 \pmod{9}

Translates the remainder and divisibility conditions into mathematical equations.

2

Combine the first two congruences using the Chinese Remainder Theorem approach

n \equiv 24 \pmod{35}

, so

n = 35k + 24

for an integer

k \ge 0

Since

\text{lcm}(5, 7) = 35

, the solutions to the combined system repeat every 35 integer values.

3

Enforce the divisibility condition by 9 on

n = 35k + 24

35k + 24 \equiv 0 \pmod{9} \implies 8k + 6 \equiv 0 \pmod{9} \implies -k + 6 \equiv 0 \pmod{9} \implies k \equiv 6 \pmod{9}

Reduces coefficients modulo 9 to find the values of

k

that make

n

a multiple of 9.

4

Calculate the smallest positive integer

n

corresponding to

k = 6

n = 35(6) + 24 = 234

Choosing

k = 6

yields the smallest non-negative integer for

k

that satisfies all conditions.

Anahtar Kavram

Simultaneous congruences and divisibility constraints

Soru 11Soru →

The table below shows the frequency distribution of daily passenger counts (in hundreds) for a city bus route recorded over a 40-day period.

Daily Passengers (in hundreds)	Frequency (Number of Days)
$10 - 14$	$5$
$15 - 19$	$12$
$20 - 24$	$15$
$25 - 29$	$8$

If the mean of the grouped data is estimated by using the midpoint of each class interval, what is the estimated mean daily passenger count (in hundreds)?

Cevabı ve açıklamayı göster

Cevap: 20.25

Cevap

The estimated mean daily passenger count is 20.25 hundred passengers.

To calculate the estimated mean of grouped data, find the midpoint of each interval, multiply each midpoint by its frequency, sum those products (

810

), and divide by the total number of observations (

40

). This gives

\frac{810}{40} = 20.25

.

Adım Adım Çözüm

1

Calculate the midpoints for each of the four class intervals.

The midpoints are

\frac{10+14}{2} = 12

,

\frac{15+19}{2} = 17

,

\frac{20+24}{2} = 22

, and

\frac{25+29}{2} = 27

.

To estimate the mean of grouped frequency data, each interval is represented by its center value (midpoint).

2

Multiply each class midpoint by its frequency and calculate the total sum of these products.

(12 \times 5) + (17 \times 12) + (22 \times 15) + (27 \times 8) = 60 + 204 + 330 + 216 = 810

.

Multiplying each midpoint by its frequency computes the total estimated value contributed by all observations in that class.

3

Divide the total estimated value by the total sample size (total frequency).

\frac{810}{40} = 20.25

.

The weighted average (grouped mean) is the sum of weighted midpoints divided by the total frequency.

Anahtar Kavram

Estimated Mean of Grouped Data

Soru 12Soru →

If $x$ and $y$ are positive integers such that $2^{x+1} + 2^x = 3^{y+2} - 3^y$ , what is the value of $x + y$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x + y

is

4

.

Factoring the left side gives

2^x(2 + 1) = 3 \cdot 2^x

, while factoring the right side gives

3^y(9 - 1) = 8 \cdot 3^y = 2^3 \cdot 3^y

. Equating the two expressions gives

3 \cdot 2^x = 2^3 \cdot 3^y

. Rearranging terms to separate bases yields

2^{x-3} = 3^{y-1}

. Because 2 and 3 share no common prime factors, this equality holds for integers if and only if both exponents are equal to 0. Solving

x - 3 = 0

gives

x = 3

, and solving

y - 1 = 0

gives

y = 1

. Both are positive integers. Thus,

x + y = 3 + 1 = 4

.

Adım Adım Çözüm

1

Factor out common terms on both sides of the equation.

2^x(2 + 1) = 3^y(9 - 1) \implies 3 \cdot 2^x = 8 \cdot 3^y

Factoring simplifies sums of powers with identical bases.

2

Rewrite integers using prime factorizations and re-group bases.

3 \cdot 2^x = 2^3 \cdot 3^y \implies 2^{x-3} = 3^{y-1}

Dividing both sides by

2^3 \cdot 3

separates the base-2 and base-3 exponential terms.

3

Set each exponent to zero using prime independence.

x - 3 = 0 \implies x = 3

and

y - 1 = 0 \implies y = 1

Powers of distinct prime numbers 2 and 3 can only be equal if both powers equal

1

(

2^0 = 3^0 = 1

).

4

Calculate the required sum

x + y

.

3 + 1 = 4

Evaluates the requested combined value of the variables.

Anahtar Kavram

Solving exponential equations involving distinct prime bases through factoring and exponent properties.

Soru 13Soru →

If

x

is a real number satisfying the equation

\frac{\left(2^{3x+2} + 2^{3x}\right)^3}{\left(4^{2x+1} - 4^{2x}\right)^2} = \frac{2000}{9}

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

Factoring out

2^{3x}

in the numerator gives

2^{3x}(2^2 + 1) = 5 \cdot 2^{3x}

. Cubing this yields

125 \cdot 2^{9x}

. In the denominator, rewriting

4^{2x}

as

2^{4x}

and factoring gives

2^{4x}(4 - 1) = 3 \cdot 2^{4x}

. Squaring this yields

9 \cdot 2^{8x}

. Taking the quotient gives

\frac{125}{9} \cdot 2^x

. Setting this equal to

\frac{2000}{9}

leads directly to

125 \cdot 2^x = 2000 \implies 2^x = 16 \implies x = 4

.

Adım Adım Çözüm

1

Factor out common exponential terms inside the parentheses

Numerator inside becomes

5 \cdot 2^{3x}

and denominator inside becomes

3 \cdot 2^{4x}

Factoring out

2^{3x}

from

2^{3x+2} + 2^{3x}

isolates the constant multiplier

(4+1)

, and expressing

4^{2x}

as

2^{4x}

allows base unification.

2

Raise the simplified terms to their respective outer powers

Numerator becomes

125 \cdot 2^{9x}

and denominator becomes

9 \cdot 2^{8x}

Using power rules

(a \cdot b)^n = a^n b^n

and

(a^m)^n = a^{m n}

.

3

Simplify the fraction by subtracting exponents of like bases

The left side simplifies to

\frac{125}{9} \cdot 2^x

\ me{2^{9x}}{2^{8x}} = 2^{9x-8x} = 2^x

using the quotient rule for exponents.

4

Solve the resulting single-variable exponential equation

2^x = 16

, which yields

x = 4

Multiplying both sides by

9

yields

125 \cdot 2^x = 2000

, so

2^x = 16 = 2^4

.

Anahtar Kavram

Factoring exponential expressions and applying power of a power and quotient rules

Soru 14Soru →

A manufacturing company produces two models of office chairs, Model P and Model Q. Producing each Model P chair requires $3$ hours of assembly and $1$ hour of finishing. Producing each Model Q chair requires $2$ hours of assembly and $2$ hours of finishing. On a given day, the assembly department was scheduled for $96$ total hours of work and the finishing department was scheduled for $48$ total hours of work. If both departments operated at full capacity and used all scheduled hours, how many Model P chairs were produced on that day?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

24

Letting

p

be the number of Model P chairs and

q

be the number of Model Q chairs, the hours used by the assembly department give

3p + 2q = 96

and the hours used by the finishing department give

p + 2q = 48

. Subtracting the finishing equation from the assembly equation eliminates

2q

, resulting in

2p = 48

, or

p = 24

.

Adım Adım Çözüm

1

Define variables for the unknown quantities.

Let

p

equal the number of Model P chairs produced and

q

equal the number of Model Q chairs produced.

Assigning variables to the unknown quantities enables the construction of algebraic equations.

2

Formulate a system of linear equations representing total department hours.

Assembly department equation:

3p + 2q = 96

; Finishing department equation:

p + 2q = 48

.

The sum of hours required for both models in each department must equal that department's total scheduled hours.

3

Solve the system of equations for

p

using the elimination method.

Subtracting

p + 2q = 48

from

3p + 2q = 96

yields

2p = 48

, which gives

p = 24

.

Eliminating

q

directly provides the value for

p

, which corresponds to the target quantity requested in the problem.

Anahtar Kavram

Modeling real-world resource allocation using systems of linear equations

Soru 15Soru →

If

x

is a positive integer such that

\sqrt{4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1}} = 30,000

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x

is 4.

Factoring the common exponential term

4^x \cdot 5^{2x}

inside the radical yields

\sqrt{(4^x \cdot 5^{2x})(4 + 5)} = \sqrt{9 \cdot 4^x \cdot (5^2)^x} = \sqrt{9 \cdot (4 \cdot 25)^x} = \sqrt{9 \cdot 100^x} = \sqrt{9 \cdot 10^{2x}} = 3 \cdot 10^x

. Setting

3 \cdot 10^x = 30,000

gives

10^x = 10,000 = 10^4

, which means

x = 4

.

Adım Adım Çözüm

1

Separate the addition in exponents using exponent rules.

4^{x+1} \cdot 5^{2x} = 4^x \cdot 4^1 \cdot 5^{2x}

and

4^x \cdot 5^{2x+1} = 4^x \cdot 5^{2x} \cdot 5^1

.

Applying the product rule of exponents

a^{m+n} = a^m \cdot a^n

prepares terms for factoring.

2

Factor out the common expression

4^x \cdot 5^{2x}

from the sum inside the radical.

4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1} = (4^x \cdot 5^{2x})(4 + 5) = 9 \cdot 4^x \cdot 5^{2x}

.

Factoring converts the sum under the square root into a single product.

3

Combine terms with powers into base 10.

4^x \cdot 5^{2x} = 4^x \cdot (5^2)^x = 4^x \cdot 25^x = (4 \cdot 25)^x = 100^x = 10^{2x}

.

Using power of a power

(a^m)^n = a^{mn}

and power of a product

a^n b^n = (ab)^n

simplifies the expression into powers of 10.

4

Take the square root of the simplified product.

\sqrt{9 \cdot 10^{2x}} = \sqrt{9} \cdot \sqrt{10^{2x}} = 3 \cdot 10^x

.

Applying the product rule for radicals

\sqrt{ab} = \sqrt{a}\sqrt{b}

and halving the exponent

(10^{2x})^{1/2} = 10^x

.

5

Equate the simplified expression to 30,000 and solve for

x

.

3 \cdot 10^x = 30,000 \implies 10^x = 10,000 \implies 10^x = 10^4 \implies x = 4

.

Dividing both sides by 3 isolates

10^x

, and matching exponential bases gives

x = 4

.

Anahtar Kavram

Exponent Rules and Radical Simplification

Tahmini Süre:2m 0s

Soru 16Soru →

A hotel renovated its guest rooms and purchased a total of 100 lighting fixtures, consisting of wall sconces costing $\$45$ each and ceiling pendants costing $\$70$ each. If the average (arithmetic mean) cost per fixture for the entire purchase was $\$52$ , how many wall sconces were purchased?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

72

Setting up the weighted total cost equation

45x + 70(100 - x) = 5200

simplifies to

-25x = -1800

, giving

x = 72

wall sconces.

Adım Adım Çözüm

1

Define the variable for the unknown quantity

Let

x

represent the number of wall sconces purchased. The number of ceiling pendants is

100 - x

.

Defining the target quantity as

x

allows the problem to be modeled using a single-variable linear equation.

2

Set up the algebraic equation for total cost

45x + 70(100 - x) = 52 \times 100

The total cost of all fixtures is the sum of the total sconce cost and total pendant cost, which equals the overall average cost multiplied by the total number of fixtures.

3

Expand and solve the linear equation

45x + 7000 - 70x = 5200 \implies -25x = -1800 \implies x = 72

Simplifying algebraic terms isolates

x

to find the exact number of wall sconces.

Anahtar Kavram

Linear Algebraic Modeling and Weighted Averages

Soru 17Soru →

The quadratic equation $x^2 + bx + 45 = 0$ , where $b$ is a constant, has two negative integer roots $p$ and $q$ such that $p < q$ . If $q - p = 4$ , what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The value of

b

is 14.

For the quadratic equation

x^2 + bx + 45 = 0

, the roots

p

and

q

must satisfy

pq = 45

and

p + q = -b

. The negative integer factor pairs of 45 with

p < q

are

(-45, -1)

,

(-15, -3)

, and

(-9, -5)

. Calculating the difference

q - p

for each pair yields 44, 12, and 4, respectively. The condition

q - p = 4

uniquely identifies the roots as

p = -9

and

q = -5

. Summing these roots gives

p + q = -14

, so

b = -(-14) = 14

.

Adım Adım Çözüm

1

Set up the relationships for the roots of the quadratic equation.

pq = 45

and

p + q = -b

.

For any quadratic equation

x^2 + bx + c = 0

, the product of the roots equals

c

and the sum of the roots equals

-b

.

2

Find all negative integer factor pairs

(p, q)

of 45 such that

p < q

.

The possible pairs are

(-45, -1)

,

(-15, -3)

, and

(-9, -5)

.

Since both roots are negative integers, their product is positive 45.

3

Determine the difference

q - p

for each pair to match the given condition

q - p = 4

.

For

(-45, -1)

,

q - p = -1 - (-45) = 44

. For

(-15, -3)

,

q - p = -3 - (-15) = 12

. For

(-9, -5)

,

q - p = -5 - (-9) = 4

.

The pair

(-9, -5)

satisfies

q - p = 4

, establishing

p = -9

and

q = -5

.

4

Calculate the coefficient

b

.

b = -(p + q) = -(-9 + (-5)) = -(-14) = 14

.

Substituting the root values into

b = -(p + q)

yields the final answer.

Anahtar Kavram

Factoring Quadratics and Relationships Between Roots and Coefficients

Soru 18Soru →

The quadratic equation $2x^2 + px + q = 0$ , where $p$ and $q$ are constants, has roots $r$ and $s$ . The quadratic equation $x^2 + (p - 2)x + 24 = 0$ has roots $r + 2$ and $s + 2$ . What is the value of $q$ ?

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

The value of

q

is 32.

Applying Vieta's formulas to

2x^2 + px + q = 0

gives

r + s = -p/2

and

rs = q/2

. For the second equation

x^2 + (p-2)x + 24 = 0

, the sum of roots is

(r+2) + (s+2) = -(p-2)

, which simplifies to

(r+s) + 4 = 2 - p

. Substituting

r+s = -p/2

yields

-p/2 + 4 = 2 - p

, solving to

p = -4

and

r+s = 2

. The product of roots for the second equation is

(r+2)(s+2) = rs + 2(r+s) + 4 = 24

. Substituting

rs = q/2

and

r+s = 2

gives

q/2 + 4 + 4 = 24

, which simplifies to

q/2 = 16

and

q = 32

.

Adım Adım Çözüm

1

Express the sum and product of roots

r

and

s

in terms of

p

and

q

using Vieta's formulas.

r + s = -\frac{p}{2}

and

rs = \frac{q}{2}

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

2

Relate the sum of the shifted roots

(r + 2)

and

(s + 2)

to the coefficients of the second quadratic equation.

(r + 2) + (s + 2) = -(p - 2) \implies (r + s) + 4 = 2 - p

.

The coefficient of

x

in

x^2 + (p - 2)x + 24 = 0

is

(p - 2)

, so the sum of its roots equals

-(p - 2)

.

3

Substitute

r + s = -\frac{p}{2}

into the sum relation to determine

p

.

-\frac{p}{2} + 4 = 2 - p \implies \frac{p}{2} = -2 \implies p = -4

.

Solving the linear equation for

p

yields

p = -4

, which means

r + s = 2

.

4

Expand the product of the shifted roots

(r + 2)(s + 2) = 24

and solve for

q

.

rs + 2(r + s) + 4 = 24 \implies \frac{q}{2} + 2(2) + 4 = 24 \implies \frac{q}{2} + 8 = 24 \implies q = 32

.

Substituting

rs = \frac{q}{2}

and

r + s = 2

isolates

q

, yielding

q = 32

.

Anahtar Kavram

Relating roots and coefficients of quadratic equations using Vieta's formulas and algebraic expansion.

Soru 19Soru →

If $(5.0 \times 10^{-4}) \times (4.0 \times 10^{7}) = 2.0 \times 10^n$ , what is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

n

is 4.

Multiplying the coefficients yields

5.0 \times 4.0 = 20.0

, and multiplying the powers of ten yields

10^{-4} \times 10^7 = 10^3

. Combining these gives

20.0 \times 10^3

. Rewriting

20.0 \times 10^3

into standard scientific notation gives

2.0 \times 10^4

. Therefore,

n = 4

.

Adım Adım Çözüm

1

Multiply the numerical coefficients

5.0 \times 4.0 = 20.0

When multiplying numbers in scientific notation, separate the coefficients from the exponential terms.

2

Add the exponents of the base 10 terms

10^{-4} \times 10^{7} = 10^{3}

By exponent rules,

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

.

3

Adjust the product to standard scientific notation form

20.0 \times 10^3 = 2.0 \times 10^4

Shift the decimal point one place to the left to obtain a coefficient

2.0

(

1 \le a < 10

), which increases the power of 10 by 1.

4

Determine the exponent value

n

n = 4

Comparing

2.0 \times 10^4

to

2.0 \times 10^n

gives

n = 4

.

Anahtar Kavram

Scientific notation multiplication and place value rules

Soru 20Soru →

In a department of $90$ employees, $55$ speak French, $45$ speak Spanish, and $10$ speak neither French nor Spanish. How many employees speak both French and Spanish?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

Subtracting the 10 employees who speak neither language from the total of 90 leaves 80 employees who speak at least one language. By the principle of inclusion-exclusion, Total(At least one) = French + Spanish - Both. Substituting the known values gives 80 = 55 + 45 - Both, which simplifies to 80 = 100 - Both, so Both = 20.

Adım Adım Çözüm

1

Find the number of employees who speak at least one of the languages

80 employees

Subtract the 10 employees who speak neither language from the total department size of 90.

2

Sum the total counts for each language group

100

Add the number of French speakers (55) to Spanish speakers (45).

3

Calculate the overlap (intersection) using inclusion-exclusion

20 employees

Subtract the count of employees speaking at least one language (80) from the sum of the individual language groups (100).

Anahtar Kavram

Principle of Inclusion-Exclusion for Two Sets

Tahmini Süre:45s

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

Sık Yapılan Hatalar

Alternatif Yöntem

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

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

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

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

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