All practice questions

Question 2021Question →

On a number line, point $A$ is located at $10$ and point $B$ is located at $2$ . If point $P$ , with coordinate $x > 0$ , is three times as far from point $A$ as it is from point $B$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

4

The distance from

P(x)

to

A(10)

is

|x - 10|

and to

B(2)

is

|x - 2|

. Setting

|x - 10| = 3|x - 2|

gives two equations:

x - 10 = 3x - 6

, which yields

x = -2

, and

x - 10 = -3x + 6

, which yields

x = 4

. Since

x > 0

, the correct value is 4.

Step-by-Step Solution

1

Formulate the distance relationship using absolute value notation.

|x - 10| = 3|x - 2|

The distance between two points

u

and

v

on the real number line is given by

|u - v|

.

2

Split the absolute value equation into two linear equations representing possible cases.

x - 10 = 3(x - 2)

or

x - 10 = -3(x - 2)

The equality

|a| = |b|

implies

a = b

or

a = -b

.

3

Solve each case algebraically.

Case 1 gives

x - 10 = 3x - 6 \implies 2x = -4 \implies x = -2

. Case 2 gives

x - 10 = -3x + 6 \implies 4x = 16 \implies x = 4

.

Standard linear equation solving.

4

Apply the given domain condition

x > 0

.

x = 4

The solution

x = -2

is negative and therefore violates the condition

x > 0

.

Key Concept

Distance on a number line represented by absolute value equations

Question 2022Question →

A research laboratory conducted $6$ experimental trials to measure the duration, in milliseconds, of a specific chemical reaction. The durations recorded for $5$ of the trials were $240$ , $215$ , $260$ , $225$ , and $245$ . If the median duration of all $6$ trials was $235$ milliseconds, what was the duration, in milliseconds, of the 6th trial?

Show answer & explanation

Answer: 230

Answer

230

For a dataset with 6 numbers, the median is the arithmetic mean of the 3rd and 4th numbers in ascending order. Arranging the 5 given numbers gives

215, 225, 240, 245, 260

. Since the target median is

235

, the sum of the two middle numbers must be

235 \times 2 = 470

. Placing

x = 230

into the dataset yields the ordered set

215, 225, 230, 240, 245, 260

, where the 3rd and 4th numbers are

230

and

240

. Their mean is

(230 + 240) / 2 = 235

, which matches the given median.

Step-by-Step Solution

1

Order the 5 given numbers from least to greatest

The sorted list of known values is

215, 225, 240, 245, 260

.

Calculating or using median requires ordering the data points.

2

Express the median condition for an even number of data points (

n = 6

)

\text{Median} = \frac{\text{3rd value} + \text{4th value}}{2} = 235

, so

\text{3rd value} + \text{4th value} = 470

.

For an even number of values, the median is the average of the two middle numbers.

3

Determine the position and value of the unknown 6th trial

x

If

x \le 225

, the 3rd and 4th values would be

225

and

240

(median

232.5

). If

x \ge 245

, the 3rd and 4th values would be

240

and

245

(median

242.5

). Thus,

x

must lie between

225

and

240

.

Analyzing boundary conditions places

x

as the 3rd value and

240

as the 4th value.

4

Solve for

x

x + 240 = 470 \implies x = 230

.

The sum of the two middle values must equal

470

to yield a median of

235

.

Key Concept

Finding a missing value in a dataset given the median of an even number of observations.

Question 2023Question →

A reliability engineering test evaluates two independent components, Component X and Component Y, in a machine. The probability that Component X fails during operation is $0.20$ , and the probability that Component Y fails during operation is $0.30$ . What is the probability that at least one of the two components operates successfully during operation?

0.44

0.50

0.56

0.70

0.94

Show answer & explanation

Answer: $0.94$

Answer

The probability that at least one of the two components operates successfully is

0.94

.

The correct answer is

0.94

. The complement of the event 'at least one component operates successfully' is the event 'both components fail'. Because Component X and Component Y fail independently,

P(\text{both fail}) = P(\text{X fails}) \times P(\text{Y fails}) = 0.20 \times 0.30 = 0.06

. Subtracting this complementary probability from

1

yields

1 - 0.06 = 0.94

.

Step-by-Step Solution

1

Determine the probability that each component fails.

P(\text{X fails}) = 0.20

and

P(\text{Y fails}) = 0.30

.

These probabilities are explicitly given in the problem statement.

2

Calculate the joint probability that BOTH components fail simultaneously using the multiplication rule for independent events.

P(\text{both fail}) = P(\text{X fails}) \times P(\text{Y fails}) = 0.20 \times 0.30 = 0.06

.

Since the components fail independently, their joint failure probability is the product of their individual failure probabilities.

3

Apply the complement rule to find the probability that at least one component operates successfully.

P(\text{at least one succeeds}) = 1 - P(\text{both fail}) = 1 - 0.06 = 0.94

.

The event 'at least one component succeeds' is the exact complement of 'both components fail'.

Key Concept

Independent Events and Complement Probability Rule

Question 2024Question →

A commercial coffee roasting facility operates three roasters: $X$ , $Y$ , and $Z$ .

- Roaster $X$ processes coffee beans at a constant rate of $60\text{ kg/hr}$ , producing a blend with an Arabica-to-Robusta ratio of $3:2$ by weight.
- Roaster $Y$ processes coffee beans at a constant rate of $90\text{ kg/hr}$ , producing a blend with an Arabica-to-Robusta ratio of $2:1$ by weight.
- Roaster $Z$ processes coffee beans at a constant rate of $150\text{ kg/hr}$ , producing a blend with an Arabica-to-Robusta ratio of $1:4$ by weight.

All three roasters operate simultaneously for $4$ hours to complete a production order. Which of the following statements regarding the production output over this $4$ -hour period must be true? Select all such statements.

Select all that apply

The total weight of Arabica beans processed across all three roasters is

504\text{ kg}

.

Arabica beans account for exactly

42\%

of the total weight of coffee beans processed.

The overall ratio of total Arabica weight to total Robusta weight produced is

21:29

.

The combined hourly production rate of Robusta beans across the three roasters is

150\text{ kg/hr}

.

Roaster

Z

produces a greater total weight of Arabica beans than Roaster

X

.

Show answer & explanation

Answer: The total weight of Arabica beans processed across all three roasters is $504\text{ kg}$ .; Arabica beans account for exactly $42\%$ of the total weight of coffee beans processed.; The overall ratio of total Arabica weight to total Robusta weight produced is $21:29$ .

Answer

The statements confirming that total Arabica weight is

504\text{ kg}

, Arabica accounts for

42\%

of total weight, and the overall ratio of Arabica to Robusta is

21:29

are all correct.

The total production from Roaster

X

is

240\text{ kg}

(

144\text{ kg}

Arabica,

96\text{ kg}

Robusta), from Roaster

Y

is

360\text{ kg}

(

240\text{ kg}

Arabica,

120\text{ kg}

Robusta), and from Roaster

Z

is

600\text{ kg}

(

120\text{ kg}

Arabica,

480\text{ kg}

Robusta). The total Arabica weight is

144 + 240 + 120 = 504\text{ kg}

, which constitutes

\frac{504}{1,200} = 42\%

of the

1,200\text{ kg}

total output. The remaining

696\text{ kg}

is Robusta, giving an Arabica to Robusta ratio of

504:696 = 21:29

.

Step-by-Step Solution

1

Calculate total output and Arabica quantity for each roaster over 4 hours

Roaster

X

:

240\text{ kg}

total,

\frac{3}{5} \times 240 = 144\text{ kg}

Arabica. Roaster

Y

:

360\text{ kg}

total,

\frac{2}{3} \times 360 = 240\text{ kg}

Arabica. Roaster

Z

:

600\text{ kg}

total,

\frac{1}{5} \times 600 = 120\text{ kg}

Arabica.

Converting hourly rates to total batch weights over

4

hours and applying part-to-whole fractions determined by each ratio.

2

Aggregate total Arabica and Robusta weights across all roasters

Total Arabica =

144 + 240 + 120 = 504\text{ kg}

. Total coffee =

240 + 360 + 600 = 1,200\text{ kg}

. Total Robusta =

1,200 - 504 = 696\text{ kg}

.

Adding individual component weights to get total aggregate weights.

3

Evaluate percentage and ratio statements

Arabica percentage =

\frac{504}{1,200} \times 100\% = 42\%

. Arabica to Robusta ratio =

\frac{504}{696} = \frac{21}{29}

.

Verifying overall proportions and simplifying the fraction by dividing by

24

.

Key Concept

Multi-stage weighted component ratios and rate conversion

Question 2025Question →

In his reevaluation of early twentieth-century monetary policy, economist Arthur Pendelton argues that central bankers during the Interwar period were not merely conservative functionaries adhering to outdated doctrine; rather, they were 'trimming their sails to the prevailing trade winds' of domestic political pressure while publicly professing an unyielding commitment to the gold standard. Far from operating as disinterested technocrats governed strictly by abstract theory, these policy makers constantly calibrated their interest rate adjustments to appease vocal domestic electoral blocs. Consequently, their public declarations of monetary orthodoxy functioned less as an operational blueprint than as an ideological veil, masking a series of opportunistic compromises that ultimately undermined international financial stability.

Based on the passage, the figurative expression 'trimming their sails to the prevailing trade winds' serves to convey which of the following regarding the central bankers? Select all that apply.

Select all that apply

Adapting strategic policy choices to accommodate external political forces

Maintaining a pragmatically flexible posture despite public assertions of unyielding principle

Reducing maritime shipping subsidies to balance domestic trade deficits

Surrendering all decision-making authority due to complete helplessness against economic crises

Systematically modernizing commercial regulations to promote international trade expansion

Show answer & explanation

Answer: Adapting strategic policy choices to accommodate external political forces; Maintaining a pragmatically flexible posture despite public assertions of unyielding principle

Answer

The correct selections are the options stating that the central bankers adapted strategic policy choices to accommodate external political forces and that they maintained a pragmatically flexible posture despite public assertions of unyielding principle.

The phrase 'trimming their sails to the prevailing trade winds' uses nautical imagery to describe adjusting one's actions to accommodate shifting external forces. In context, the author uses this expression to show that central bankers calibrated interest rates in response to domestic political pressure while maintaining a public façade of adherence to the gold standard. Thus, the selections indicating political adaptation and pragmatic flexibility behind a rigid public stance correctly interpret the metaphor's contextual meaning.

Step-by-Step Solution

1

Analyze the figurative expression within its immediate sentence context.

The phrase 'trimming their sails to the prevailing trade winds' is contrasted with 'publicly professing an unyielding commitment' and directly linked to responding to 'domestic political pressure'.

Understanding how the metaphor operates rhetorically requires identifying the contrast signals and defining context clues.

2

Determine the metaphorical meaning of the phrase.

'Trimming sails' means adjusting nautical sails to handle wind direction, metaphorically indicating modifying behavior or policy to suit changing political conditions.

Idioms in academic text translate physical or functional analogies into abstract behavioral descriptions.

3

Evaluate the option choices against the derived contextual meaning.

Options reflecting political adaptation and flexible pragmatism beneath an unyielding façade accurately capture the dual implications of the metaphor.

Multiple-choice select-all questions require identifying all valid contextual nuances while eliminating literal, extreme, or out-of-scope distractors.

Key Concept

Interpreting Idiomatic Usage and Figurative Language in Context

Question 2026Question →

If $a$ and $b$ are nonzero real numbers such that $a^2 b^3 < 0$ and $\sqrt{a^2 b^4} = -a b^2$ , which of the following statements must be true? Select all such statements.

Select all that apply

a + b < 0

\frac{a^3}{b} > 0

\sqrt{a^4 b^2} = a^2 b

(-a)^3 b^2 < 0

\sqrt{(a + b)^2} = a + b

Show answer & explanation

Answer: $a + b < 0$ ; $\frac{a^3}{b} > 0$

Answer

The statements that must be true are the inequality asserting that the sum of the variables is negative (

a + b < 0

) and the inequality asserting that the quotient of the cubed variable and the second variable is positive (

\frac{a^3}{b} > 0

).

Analyzing the given constraints reveals that both variables are negative. From

a^2 b^3 < 0

, since

a^2 > 0

, we must have

b^3 < 0

, so

b < 0

. Next, from

\sqrt{a^2 b^4} = |a| b^2 = -a b^2

, dividing by

b^2 > 0

gives

|a| = -a

, which implies

a < 0

. Thus,

a < 0

and

b < 0

. The statement asserting

a + b < 0

is true because the sum of two negative numbers is negative. The statement asserting

\frac{a^3}{b} > 0

is true because

a^3 < 0

and

b < 0

, and dividing two negative numbers yields a positive quotient.

Step-by-Step Solution

1

Determine the sign of

b

using the given inequality

a^2 b^3 < 0

.

b < 0

Since

a

is a nonzero real number,

a^2 > 0

. For the product

a^2 b^3

to be negative,

b^3

must be negative, which implies

b < 0

.

2

Determine the sign of

a

using the identity

\sqrt{a^2 b^4} = -a b^2

.

a < 0

Simplify the radical:

\sqrt{a^2 b^4} = \sqrt{a^2} \cdot \sqrt{(b^2)^2} = |a| b^2

. Equating this to

-a b^2

gives

|a| b^2 = -a b^2

. Since

b \neq 0

,

b^2 > 0

, so dividing by

b^2

yields

|a| = -a

. For a nonzero real number,

|a| = -a

implies

a < 0

.

3

Evaluate statement

a + b < 0

.

True

The sum of two negative numbers (

a < 0

and

b < 0

) is always negative.

4

Evaluate statement

\frac{a^3}{b} > 0

.

True

Since

a < 0

,

a^3 < 0

. Dividing the negative quantity

a^3

by the negative quantity

b

yields a positive result.

5

Evaluate statement

\sqrt{a^4 b^2} = a^2 b

.

False

\sqrt{a^4 b^2} = \sqrt{a^4}\sqrt{b^2} = a^2 |b|

. Since

b < 0

,

|b| = -b

, so

\sqrt{a^4 b^2} = -a^2 b

.

6

Evaluate statement

(-a)^3 b^2 < 0

.

False

Since

a < 0

,

-a > 0

, making

(-a)^3 > 0

. Since

b \neq 0

,

b^2 > 0

. The product of two positive numbers is positive, so

(-a)^3 b^2 > 0

.

7

Evaluate statement

\sqrt{(a + b)^2} = a + b

.

False

\sqrt{x^2} = |x|

for any real

x

. Since

a + b < 0

,

\sqrt{(a + b)^2} = |a + b| = -(a + b) \neq a + b

.

Key Concept

Properties of even exponents, odd exponents, and principal square roots of negative variable terms.

Question 2027Question →

A manufacturing plant operates two assembly lines, Line X and Line Y. Line X produces 15 units per hour, and Line Y produces 22 units per hour. On a certain day, Line X operated for 3 hours longer than Line Y did, and the two lines produced a total of 415 units. How many hours did Line Y operate?

Show answer & explanation

Answer: 10

Answer

10

Let

h

represent the number of hours Line Y operated. Because Line X operated for 3 hours longer than Line Y, Line X operated for

h + 3

hours. The total number of units produced by both lines is the sum of their individual outputs:

15(h + 3) + 22h = 415

. Distributing 15 yields

15h + 45 + 22h = 415

. Combining like terms gives

37h + 45 = 415

. Subtracting 45 from both sides yields

37h = 370

, and dividing by 37 gives

h = 10

. Therefore, Line Y operated for 10 hours.

Step-by-Step Solution

1

Define the variable representing Line Y's operating time in hours.

Let

h

represent the number of hours Line Y operated. Line X's operating time is

h + 3

hours.

Line X operated for 3 hours longer than Line Y.

2

Formulate a linear equation in one variable for total units produced.

15(h + 3) + 22h = 415

Total production is the sum of production from Line X (

15 \times (h + 3)

) and Line Y (

22 \times h

).

3

Distribute and combine like terms on the left side of the equation.

15h + 45 + 22h = 415 \implies 37h + 45 = 415

Apply the distributive property and combine variable terms.

4

Isolate the variable

h

.

37h = 370 \implies h = 10

Subtract 45 from both sides of the equation and divide by 37.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 2028Question →

A convex hexagon has five interior angles measuring $115^\circ$ , $125^\circ$ , $130^\circ$ , $140^\circ$ , and $150^\circ$ . What is the measure, in degrees, of the exterior angle adjacent to the sixth interior angle?

60^\circ

80^\circ

120^\circ

140^\circ

150^\circ

Show answer & explanation

Answer: $120^\circ$

Answer

The measure of the exterior angle adjacent to the sixth interior angle is

120^\circ

.

The sum of the interior angles of a 6-sided polygon (hexagon) is given by

(6 - 2) \times 180^\circ = 720^\circ

. The sum of the five given interior angles is

115^\circ + 125^\circ + 130^\circ + 140^\circ + 150^\circ = 660^\circ

, which leaves

720^\circ - 660^\circ = 60^\circ

for the sixth interior angle. Since an interior angle and its adjacent exterior angle are supplementary, the exterior angle is

180^\circ - 60^\circ = 120^\circ

.

Step-by-Step Solution

1

Calculate the sum of the interior angles for a convex hexagon.

Using

(n - 2) \times 180^\circ

with

n = 6

, the total interior angle sum is

(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

.

The sum of interior angles of any convex polygon with

n

sides is

(n - 2) \times 180^\circ

.

2

Find the sum of the five given interior angles.

115^\circ + 125^\circ + 130^\circ + 140^\circ + 150^\circ = 660^\circ

.

Summing the five known angle values is necessary to find the remaining sixth angle.

3

Determine the measure of the sixth interior angle.

720^\circ - 660^\circ = 60^\circ

.

Subtracting the sum of the five interior angles from the total interior angle sum yields the sixth interior angle.

4

Calculate the supplementary exterior angle.

180^\circ - 60^\circ = 120^\circ

.

An interior angle and its adjacent exterior angle form a straight line and are supplementary (

180^\circ

).

Key Concept

Interior and Exterior Angles of Convex Polygons

Question 2029Question →

The prime factorizations of two positive integers $A$ and $B$ are given by $A = 2^a \times 3^5 \times 5^b$ and $B = 2^4 \times 3^c \times 7^2$ , where $a$ , $b$ , and $c$ are positive integers. If the greatest common divisor of $A$ and $B$ is $\gcd(A, B) = 2^2 \times 3^3$ , and their least common multiple is \text{lcm} $(A, B) = 2^4 \times 3^5 \times 5^3 \times 7^2$ , what is the value of $a + b + c$ ?

Show answer & explanation

Answer: 8

Answer

The correct answer is 8.

For any two positive integers expressed in prime factorized form, the greatest common divisor contains each prime factor raised to the minimum of its exponents in the two numbers, while the least common multiple contains each prime factor raised to the maximum of its exponents. For prime factor 2, the GCD has exponent 2, so min(a, 4) = 2, giving a = 2. For prime factor 3, the GCD has exponent 3, so min(5, c) = 3, giving c = 3. For prime factor 5, the LCM has exponent 3, so max(b, 0) = 3, giving b = 3. Adding these values together yields a + b + c = 2 + 3 + 3 = 8.

Step-by-Step Solution

1

Analyze the prime factor

2

min(a, 4) = 2, so a = 2

The greatest common divisor takes the minimum exponent for each prime factor shared between A and B.

2

Analyze the prime factor

3

min(5, c) = 3, so c = 3

The exponent of 3 in the GCD is 3, which must equal the smaller of the two exponents 5 and c.

3

Analyze the prime factor

5

max(b, 0) = 3, so b = 3

The least common multiple takes the maximum exponent for each prime factor present in either number.

4

Calculate the sum

a + b + c

2 + 3 + 3 = 8

Add the solved values of the three unknown prime exponents.

Key Concept

Relating prime factor exponents to GCD (minimum exponent) and LCM (maximum exponent)

Question 2030Question →

If

x

is a positive integer such that

\frac{4^x + 4^x + 4^x + 4^x}{2^x + 2^x} = 512

what is the value of

x

?

Show answer & explanation

Answer: 8

Answer

The value of

x

is 8.

Combining four terms of

4^x

yields

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

. Combining two terms of

2^x

yields

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

. Dividing the numerator by the denominator gives

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

. Since

512 = 2^9

, setting

2^{x+1} = 2^9

gives

x + 1 = 9

, which leads directly to

x = 8

.

Step-by-Step Solution

1

Simplify the numerator by combining identical added terms.

The numerator

4^x + 4^x + 4^x + 4^x

equals

4 \cdot 4^x

, which simplifies to

4^{x+1}

.

Adding four identical quantities is equivalent to multiplying that quantity by 4.

2

Simplify the denominator by combining identical added terms.

The denominator

2^x + 2^x

equals

2 \cdot 2^x

, which simplifies to

2^{x+1}

.

Adding two identical quantities is equivalent to multiplying that quantity by 2.

3

Convert the numerator to base 2 and simplify the fraction.

Since

4^{x+1} = (2^2)^{x+1} = 2^{2x+2}

, the fraction becomes

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

.

Converting all terms to a common base allows using the exponent quotient rule

a^m / a^n = a^{m-n}

.

4

Solve for

x

by equating the simplified power to 512.

Setting

2^{x+1} = 512 = 2^9

yields

x + 1 = 9

, so

x = 8

.

When exponential expressions with the same positive base (other than 1) are equal, their exponents must be equal.

Key Concept

Combining repeated addition of exponential terms and converting powers to a common base using exponent laws (

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

and

\frac{a^m}{a^n} = a^{m-n}

).

Question 2031Question →

Given non-zero real numbers $x$ and $y$ where $|x| \neq |y|$ , simplify the complex rational expression:

\frac{\frac{x^3 + y^3}{x^2 - y^2} - \frac{x^2y - xy^2}{(x - y)^2}}{\frac{x^4 - y^4}{x^3 + x^2y + xy^2 + y^3}}

Which of the following represents the completely simplified expression?

1

x - y

x + y

\frac{x - y}{x + y}

\frac{x^2 + y^2}{x - y}

Show answer & explanation

Answer: 1

Answer

1

Both the entire complex numerator and the entire complex denominator independently simplify to

x - y

. Consequently, dividing the numerator

x - y

by the denominator

x - y

gives

1

.

Step-by-Step Solution

1

Simplify the first term of the main numerator

\frac{x^3 + y^3}{x^2 - y^2} = \frac{(x + y)(x^2 - xy + y^2)}{(x - y)(x + y)} = \frac{x^2 - xy + y^2}{x - y}

Factor the sum of cubes in the numerator and the difference of squares in the denominator, then cancel the common factor

(x + y)

.

2

Simplify the second term of the main numerator

\frac{x^2y - xy^2}{(x - y)^2} = \frac{xy(x - y)}{(x - y)^2} = \frac{xy}{x - y}

Factor out the greatest common factor

xy

from the numerator and cancel one factor of

(x - y)

.

3

Subtract the simplified terms in the main numerator

\frac{x^2 - xy + y^2}{x - y} - \frac{xy}{x - y} = \frac{x^2 - 2xy + y^2}{x - y} = \frac{(x - y)^2}{x - y} = x - y

Combine the numerators over the common denominator

(x - y)

, factor the perfect square trinomial

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

, and simplify.

4

Simplify the main denominator

\frac{x^4 - y^4}{x^3 + x^2y + xy^2 + y^3} = \frac{(x - y)(x + y)(x^2 + y^2)}{x^2(x + y) + y^2(x + y)} = \frac{(x - y)(x + y)(x^2 + y^2)}{(x + y)(x^2 + y^2)} = x - y

Factor the numerator using difference of squares twice, factor the denominator by grouping, and cancel common factors

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

.

5

Divide the main numerator by the main denominator

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

Divide the simplified main numerator (

x - y

) by the simplified main denominator (

x - y

).

Key Concept

Multi-step algebraic expression simplification using special factoring identities (sum/difference of cubes, difference of squares, quadratic trinomials, and factoring by grouping).

Estimated Time:2m 0s

Question 2032Question →

A textile mill uses two weaving looms, Loom $P$ and Loom $Q$ , to produce a fabric blend composed of silk, wool, and cotton.

- Loom $P$ produces silk, wool, and cotton in the ratio $2 : 3 : 5$ by weight, operating at a constant output rate of $120\text{ kg per hour}$ .
- Loom $Q$ produces silk, wool, and cotton in the ratio $1 : 4 : 3$ by weight, operating at a constant output rate of $160\text{ kg per hour}$ .

If both looms operate simultaneously for 5 hours, what is the ratio of the total weight of wool produced to the total weight of cotton produced in the combined output?

29 : 30

7 : 8

32 : 35

29 : 70

30 : 29

Show answer & explanation

Answer: $29 : 30$

Answer

The ratio of the total weight of wool produced to the total weight of cotton produced is

29 : 30

.

To find the overall ratio of wool to cotton, determine the actual mass of each material produced per hour by each machine. Loom

P

outputs

\frac{3}{10} \times 120 = 36\text{ kg}

of wool and

\frac{5}{10} \times 120 = 60\text{ kg}

of cotton per hour. Loom

Q

outputs

\frac{4}{8} \times 160 = 80\text{ kg}

of wool and

\frac{3}{8} \times 160 = 60\text{ kg}

of cotton per hour. Combining both looms yields

116\text{ kg}

of wool and

120\text{ kg}

of cotton per hour. Over 5 hours, the combined output is

580\text{ kg}

of wool and

600\text{ kg}

of cotton. The ratio of total wool to total cotton is

580 : 600

, which simplifies to

29 : 30

.

Step-by-Step Solution

1

Calculate the hourly production rates of wool and cotton for Loom

P

.

Loom

P

produces fabric at

120\text{ kg/hr}

with ratio parts

2 + 3 + 5 = 10

.
- Wool rate from

P = \frac{3}{10} \times 120 = 36\text{ kg/hr}

- Cotton rate from

P = \frac{5}{10} \times 120 = 60\text{ kg/hr}

Converting the ratio into component rates by multiplying each component's fraction by Loom

P

's total hourly output.

2

Calculate the hourly production rates of wool and cotton for Loom

Q

.

Loom

Q

produces fabric at

160\text{ kg/hr}

with ratio parts

1 + 4 + 3 = 8

.
- Wool rate from

Q = \frac{4}{8} \times 160 = 80\text{ kg/hr}

- Cotton rate from

Q = \frac{3}{8} \times 160 = 60\text{ kg/hr}

Converting the ratio into component rates by multiplying each component's fraction by Loom

Q

's total hourly output.

3

Determine total quantities produced over the 5-hour period.

Total wool

= (36 + 80) \times 5 = 116 \times 5 = 580\text{ kg}

.
Total cotton

= (60 + 60) \times 5 = 120 \times 5 = 600\text{ kg}

.

Combining the hourly outputs from both machines and multiplying by the total duration of 5 hours.

4

Compute and simplify the ratio of total wool to total cotton.

\text{Ratio} = \frac{580}{600} = \frac{29}{30}

, which is expressed as

29 : 30$.

Dividing both terms of the ratio by their greatest common divisor, 20.

Key Concept

Combining weighted rates across multiple ratio-based sub-components

Estimated Time:2m 0s

Question 2033Question →

In his 1890 treatise on comparative philology, historian Julian Thorne reexamined the formalization of sound laws in historical linguistics. Mid-nineteenth-century scholars often viewed language shifts as arbitrary historical accidents, whereas later Neogrammarians insisted that phonetic change operated with absolute, exceptionless necessity. Thorne contends that in their eagerness to establish linguistics as a rigorous science, these theorists put the cart before the horse by treating abstract, retrospective regularities as the primary engine driving speech transformation. In Thorne’s estimation, the codified sound laws were merely descriptive summaries compiled from thousands of individual spoken interactions, rather than pre-existing structural imperatives that dictated how speakers altered their pronunciation over time. By mistaking the observed outcome of linguistic evolution for its initiating cause, the Neogrammarians inverted the proper relationship between living language and theoretical abstraction.

Which of the following best captures the conceptual error described by the author's use of the phrase "put the cart before the horse"?

Reversing the logical relationship between an observed outcome and its underlying cause

Misinterpreting historical developments in agricultural transport technology and animal usage

Rejecting the overall scientific validity and utility of historical linguistics as a discipline

Assuming that living spoken dialects will cease evolving once formal grammatical rules are codified

Demonstrating severe scholarly hostility toward earlier philologists who studied dialectal variation

Show answer & explanation

Answer: Reversing the logical relationship between an observed outcome and its underlying cause

Answer

Reversing the logical relationship between an observed outcome and its underlying cause

The correct answer accurately identifies the metaphorical meaning of 'put the cart before the horse' in context. The passage explicitly states that the Neogrammarians treated retrospective summaries of language (the effect/outcome) as the driving force behind speech transformation (the cause), thereby inverting the natural causal sequence.

Step-by-Step Solution

1

Locate the target phrase within the passage text

The phrase appears when describing how Neogrammarians treated 'abstract, retrospective regularities as the primary engine driving speech transformation.'

Contextual analysis is required to determine how the figurative phrase functions in this specific argument.

2

Analyze the surrounding elaboration provided by the author

The author explicitly clarifies the phrase in the subsequent sentences: the theorists mistook 'the observed outcome of linguistic evolution for its initiating cause.'

The author provides a direct contextual definition of the figurative expression immediately following its use.

3

Match the contextual meaning to the correct option while eliminating distractors

The choice describing the reversal of the logical relationship between an observed outcome and its underlying cause accurately reflects the inverted causal order described by the idiom.

The idiom 'put the cart before the horse' metaphorically signifies placing an effect or secondary item before its primary cause.

Key Concept

Interpreting Idiomatic Usage and Figurative Language in Context

Estimated Time:1m 30s

Question 2034Question →

Let $n = 2^a \cdot 3^b \cdot 5^c$ be a positive integer, where $a$ , $b$ , and $c$ are non-negative integers. If the greatest common divisor of $n$ and $360$ is $45$ , and the least common multiple of $n$ and $90$ is $450$ , what is the value of $a + b + c$ ?

2

3

4

5

6

Show answer & explanation

Answer: 4

Answer

The value of

a + b + c

is

4

.

Prime factorizing the given values yields

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

,

45 = 2^0 \cdot 3^2 \cdot 5^1

,

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

, and

450 = 2^1 \cdot 3^2 \cdot 5^2

. Because

\gcd(n, 360) = 45

, taking the minimum exponent of 2 implies

\min(a, 3) = 0

, so

a = 0

. Taking the minimum exponent of 3 implies

b \ge 2

, and for 5 implies

c \ge 1

. Next, using

\text{lcm}(n, 90) = 450

, taking the maximum exponent of 3 gives

\max(b, 2) = 2

, which forces

b = 2

. Taking the maximum exponent of 5 gives

\max(c, 1) = 2

, which forces

c = 2

. Therefore,

a + b + c = 0 + 2 + 2 = 4

.

Step-by-Step Solution

1

Express all given integers in their prime factorizations.

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

,

45 = 3^2 \cdot 5^1 = 2^0 \cdot 3^2 \cdot 5^1

,

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

, and

450 = 2^1 \cdot 3^2 \cdot 5^2

.

Prime factorizations allow determination of exponents using exponent rules for GCD and LCM.

2

Apply the GCD condition

\gcd(n, 360) = 45

.

\min(a, 3) = 0 \implies a = 0

,

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

, and

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

.

The greatest common divisor takes the minimum exponent for each prime factor shared between the numbers.

3

Apply the LCM condition

\text{lcm}(n, 90) = 450

using

a = 0

.

\max(b, 2) = 2 \implies b \le 2

(so

b = 2

), and

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

.

The least common multiple takes the maximum exponent for each prime factor.

4

Calculate the sum

a + b + c

.

a + b + c = 0 + 2 + 2 = 4

.

Adding the individual prime factor exponents yields the requested sum.

Key Concept

Relating prime factor exponents to GCD (minimum powers) and LCM (maximum powers)

Estimated Time:1m 30s

Question 2035Question →

If $x$ is a positive real number such that $x^{\frac{3}{4}} = 27$ , what is the value of $x^{-\frac{1}{2}}$ ?

\frac{1}{9}

9

-\frac{1}{9}

\frac{1}{3}

\frac{1}{27}

Show answer & explanation

Answer: $\frac{1}{9}$

Answer

\frac{1}{9}

Raising both sides of

x^{\frac{3}{4}} = 27

to the power of

\frac{4}{3}

gives

x = (3^3)^{\frac{4}{3}} = 3^4 = 81

. Substituting

x = 81

into

x^{-\frac{1}{2}}

gives

81^{-\frac{1}{2}} = \frac{1}{\sqrt{81}} = \frac{1}{9}

.

Step-by-Step Solution

1

Solve for

x

in the equation

x^{\frac{3}{4}} = 27

.

x = 27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^4 = 81

Raise both sides to the power of

\frac{4}{3}

to isolate

x

.

2

Evaluate

x^{-\frac{1}{2}}

for

x = 81

.

81^{-\frac{1}{2}} = \frac{1}{81^{\frac{1}{2}}} = \frac{1}{\sqrt{81}} = \frac{1}{9}

Apply the negative exponent rule

a^{-n} = \frac{1}{a^n}

and the fractional exponent rule

a^{\frac{1}{2}} = \sqrt{a}

.

Key Concept

Fractional and Negative Exponents

Estimated Time:1m 30s

Question 2036Question →

An executive is monitoring two independent corporate projects, Project Alpha and Project Beta. Based on historical performance, the probability that Project Alpha meets its deadline is $\frac{4}{5}$ , and the probability that Project Beta meets its deadline is $\frac{3}{4}$ . What is the probability that exactly one of the two projects meets its deadline?

\frac{7}{20}

\frac{3}{5}

\frac{19}{20}

\frac{7}{9}

\frac{1}{20}

Show answer & explanation

Answer: $\frac{7}{20}$

Answer

The probability that exactly one project meets its deadline is

\frac{7}{20}

.

The event 'exactly one project meets its deadline' consists of two mutually exclusive scenarios: (1) Alpha meets its deadline and Beta does not, or (2) Alpha misses its deadline and Beta meets it. Using independence, the probability of Scenario 1 is

\frac{4}{5} \times \left(1 - \frac{3}{4}\right) = \frac{4}{5} \times \frac{1}{4} = \frac{4}{20}

. The probability of Scenario 2 is

\left(1 - \frac{4}{5}\right) \times \frac{3}{4} = \frac{1}{5} \times \frac{3}{4} = \frac{3}{20}

. Summing these mutually exclusive probabilities gives

\frac{4}{20} + \frac{3}{20} = \frac{7}{20}

.

Step-by-Step Solution

1

Determine the probabilities of individual events and their complements.

P(\text{Alpha meets}) = \frac{4}{5}

,

P(\text{Alpha misses}) = 1 - \frac{4}{5} = \frac{1}{5}

.

P(\text{Beta meets}) = \frac{3}{4}

,

P(\text{Beta misses}) = 1 - \frac{3}{4} = \frac{1}{4}

.

To find the probability of specific outcome combinations, the complementary probabilities for each independent event are required.

2

Identify the mutually exclusive cases that satisfy the condition 'exactly one project meets its deadline'.

Case 1: Alpha meets and Beta misses. Case 2: Alpha misses and Beta meets.

The event 'exactly one' consists of two distinct, non-overlapping scenarios.

3

Calculate the joint probability for each case using independence.

Case 1 probability:

\frac{4}{5} \times \frac{1}{4} = \frac{4}{20}

. Case 2 probability:

\frac{1}{5} \times \frac{3}{4} = \frac{3}{20}

.

Since the projects operate independently, joint probabilities are found by multiplying individual event probabilities.

4

Add the probabilities of the mutually exclusive cases.

\frac{4}{20} + \frac{3}{20} = \frac{7}{20}

.

For mutually exclusive events, the total probability of either case occurring is the sum of their individual probabilities.

Key Concept

Probability of Independent Events and Mutually Exclusive Cases

Question 2037Question →

A digital archive charges an annual subscription fee of $\$120$ , which includes $50$ free document downloads per year. For each additional document downloaded beyond the first $50$ , the archive charges a flat rate of $\$1.50$ . If a research group paid a total of $\$231$ to the archive last year, how many total documents did the research group download?

24

74

114

124

204

Show answer & explanation

Answer: 124

Answer

The total number of documents downloaded by the research group last year was 124.

Subtracting the base subscription fee of

120 from the total payment of

231 leaves

111 spent strictly on extra downloads. Dividing

111 by the $1.50 per-document rate yields 74 extra downloads. Adding the 50 included free downloads gives a total of 124 documents downloaded.

Step-by-Step Solution

1

Define the linear equation representing total cost.

120 + 1.50(x - 50) = 231

, where

x

represents the total number of documents downloaded.

The base subscription fee is

120, and the per-document rate of

1.50 applies only to documents in excess of the initial 50 free documents.

2

Isolate the variable term by subtracting the base subscription fee from both sides.

1.50(x - 50) = 231 - 120 = 111

This determines the portion of the total cost that resulted strictly from additional downloads.

3

Solve for the number of additional documents

(x - 50)

.

x - 50 = \frac{111}{1.50} = 74

Dividing the extra cost of

111 by the per-document rate of

1.50 gives the exact count of extra downloads.

4

Add the initial 50 free documents to find the total document count

x

.

x = 74 + 50 = 124

The question asks for the total documents downloaded, which combines the 50 free downloads and the 74 additional paid downloads.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 2038Question →

A chemical processing tank receives two liquid solutions, Solution $X$ and Solution $Y$ , from separate inlet pipes.

- Solution $X$ contains chemical $A$ and water in a volume ratio of $3:2$ and enters the tank at a constant rate of $150$ liters per hour.
- Solution $Y$ contains chemical $A$ and water in a volume ratio of $1:4$ and enters the tank at a constant rate of $250$ liters per hour.

Both inlet pipes run simultaneously into an initially empty tank for $4$ hours. After $4$ hours, the inlet pipes are shut off. To adjust the mixture, pure chemical $A$ is added to the tank at a constant rate of $50$ liters per hour, while water is continuously drained from the tank at a constant rate of $30$ liters per hour.

How many hours must this adjustment process run until the volume of chemical $A$ in the tank is equal to the volume of water in the tank?

Show answer & explanation

Answer: 6

Answer

6

To find the time

t

when the volumes of chemical A and water in the tank are equal, first determine the initial quantities contributed by both solutions during the 4-hour filling period. Solution X provides

150 \times 4 = 600

liters total, containing

\frac{3}{5} \times 600 = 360

liters of chemical A and

\frac{2}{5} \times 600 = 240

liters of water. Solution Y provides

250 \times 4 = 1000

liters total, containing

\frac{1}{5} \times 1000 = 200

liters of chemical A and

\frac{4}{5} \times 1000 = 800

liters of water. Adding these amounts yields

360 + 200 = 560

liters of chemical A and

240 + 800 = 1040

liters of water. In the adjustment phase of

t

hours, chemical A increases at

50

L/hr to

560 + 50t

, while water decreases at

30

L/hr to

1040 - 30t

. Equating the two expressions gives

560 + 50t = 1040 - 30t

, which simplifies to

80t = 480

, resulting in

t = 6

hours.

Step-by-Step Solution

1

Determine the volumes of chemical A and water supplied by Solution X during the first 4 hours.

Solution X delivers 600 liters in total, consisting of 360 liters of chemical A and 240 liters of water.

Solution X flows at 150 L/hr for 4 hours (150 * 4 = 600 L) with a 3:2 chemical A to water ratio, meaning chemical A represents 3/5 of the total volume and water represents 2/5.

2

Determine the volumes of chemical A and water supplied by Solution Y during the first 4 hours.

Solution Y delivers 1000 liters in total, consisting of 200 liters of chemical A and 800 liters of water.

Solution Y flows at 250 L/hr for 4 hours (250 * 4 = 1000 L) with a 1:4 chemical A to water ratio, meaning chemical A represents 1/5 of the total volume and water represents 4/5.

3

Calculate the total initial quantities of chemical A and water present in the tank prior to the adjustment phase.

Total chemical A = 560 liters; Total water = 1040 liters.

Sum the quantities from both solutions: Chemical A = 360 + 200 = 560 L; Water = 240 + 800 = 1040 L.

4

Formulate linear expressions representing the total volume of chemical A and water after t hours of adjustment.

Chemical A volume = 560 + 50t; Water volume = 1040 - 30t.

Pure chemical A is added at 50 L/hr, increasing its total volume, while water is drained at 30 L/hr, reducing its total volume.

5

Set the two component volume expressions equal to each other and solve for t.

t = 6 hours.

Solving 560 + 50t = 1040 - 30t leads to 80t = 480, which yields t = 6.

Key Concept

Multi-stream mixture rate integration and ratio equality modeling

Question 2039Question →

For all real numbers $x$ and $y$ such that $x^2 + xy + y^2 \neq 0$ , which of the following expressions are equivalent to $\frac{x^6 - y^6}{x^2 + xy + y^2}$ ? Select all such expressions.

Select all that apply

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

(x - y)(x^3 + y^3)

x^4 - x^3 y + x y^3 - y^4

(x + y)(x^3 - y^3)

x^4 - y^4

Show answer & explanation

Answer: $(x^2 - y^2)(x^2 - xy + y^2)$ ; $(x - y)(x^3 + y^3)$ ; $x^4 - x^3 y + x y^3 - y^4$

Answer

The equivalent expressions are

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

,

(x - y)(x^3 + y^3)

, and

x^4 - x^3y + xy^3 - y^4

.

Factoring the numerator

x^6 - y^6

as a difference of squares yields

(x^3 - y^3)(x^3 + y^3)

. Applying the difference of cubes identity

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

enables cancellation of the non-zero denominator

x^2 + xy + y^2

, leaving

(x - y)(x^3 + y^3)

. Expanding this product gives

x^4 - x^3y + xy^3 - y^4

. Furthermore, factoring

x^3 + y^3

as

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

and regrouping

(x - y)(x + y)

gives

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

. Consequently, the three valid equivalent forms are

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

,

(x - y)(x^3 + y^3)

, and

x^4 - x^3y + xy^3 - y^4

.

Step-by-Step Solution

1

Factor the numerator

x^6 - y^6

as a difference of squares.

x^6 - y^6 = (x^3)^2 - (y^3)^2 = (x^3 - y^3)(x^3 + y^3)

Applying the difference of squares identity

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

with

a = x^3

and

b = y^3

.

2

Apply the difference of cubes identity to

x^3 - y^3

.

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

This reveals the non-zero quadratic factor present in the denominator.

3

Substitute into the original fraction and cancel the common factor

(x^2 + xy + y^2)

.

\frac{(x - y)(x^2 + xy + y^2)(x^3 + y^3)}{x^2 + xy + y^2} = (x - y)(x^3 + y^3)

Since

x^2 + xy + y^2 \neq 0

, dividing numerator and denominator by

(x^2 + xy + y^2)

simplifies the expression to

(x - y)(x^3 + y^3)

.

4

Expand

(x - y)(x^3 + y^3)

to check for equivalent expanded polynomial forms.

(x - y)(x^3 + y^3) = x^4 + xy^3 - x^3y - y^4 = x^4 - x^3y + xy^3 - y^4

Distributing terms verifies polynomial equivalence.

5

Factor

x^3 + y^3

and regroup to find another equivalent factored representation.

(x - y)(x^3 + y^3) = (x - y)(x + y)(x^2 - xy + y^2) = (x^2 - y^2)(x^2 - xy + y^2)

Applying the sum of cubes identity

x^3 + y^3 = (x + y)(x^2 - xy + y^2)

and combining

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

.

Key Concept

Factoring higher-degree algebraic expressions using difference of squares and sum/difference of cubes identities

Question 2040Question →

A positive integer $N$ has exactly $12$ positive divisors. If the greatest common divisor of $N$ and $35$ is $7$ , and $N$ is a multiple of $6$ , which of the following values could be equal to $N$ ? Indicate all such values.

Select all that apply

84

126

210

294

441

Show answer & explanation

Answer: $84$ ; $126$ ; $294$

Answer

The positive integer

N

could be equal to

84

,

126

, or

294

.

The integer

N

must contain the prime factors

2

,

3

, and

7

, but not

5

. For

N

to have exactly

12

divisors, its prime factorization exponent set

\{a,b,c\}

must satisfy

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

, which restricts the exponents to a permutation of

\{1, 1, 2\}

. Evaluating the three permutations yields

84

,

126

, and

294

.

Step-by-Step Solution

1

Analyze the conditions given for

N

.

Since

N

is a multiple of

6

,

2 \mid N

and

3 \mid N

. Since

\gcd(N, 35) = 7

,

7 \mid N

and

5 \nmid N

. Thus,

N

must have prime factors

2, 3, 7

and no prime factor of

5

.

Establishing the prime factors of

N

based on divisibility and GCD conditions.

2

Determine the prime factorization form and divisor count.

Let

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

, where

a \ge 1, b \ge 1, c \ge 1

. The number of divisors is given by

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

.

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

\prod p_i^{e_i}

is

\prod (e_i+1)

.

3

Find all valid exponent combinations

(a, b, c)

.

The factors of

12

into three integers each

\ge 2

are

2 \times 2 \times 3

. Therefore, the set of exponents

\{a, b, c\}

must be a permutation of

\{1, 1, 2\}

.

Each exponent increment

(e+1)

must be at least

2

since every prime

2, 3, 7

is present.

4

Calculate the possible numerical values of

N

.

Case 1:

2^2 \cdot 3^1 \cdot 7^1 = 84

.
Case 2:

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

.
Case 3:

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

.

Evaluating all three possible permutations of exponents for prime bases

2, 3,

and

7

.

Key Concept

Prime Factorization, Divisor Count Formula, and GCD Constraints

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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