All practice questions

Question 361Question →

A quartermaster is organizing deployment cargo for four different platoons. Match each platoon's cargo inventory to the corresponding simplified ratio of the first listed item to the second listed item.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Platoon Alpha matches to

3:1

, Platoon Bravo matches to

2:3

, Platoon Charlie matches to

3:5

, and Platoon Delta matches to

4:1

.

Each platoon's inventory forms a direct part-to-part ratio that can be simplified by dividing both quantities by their greatest common factor (GCF). Platoon Alpha's

45:15

reduces to

3:1

(GCF is

15

). Platoon Bravo's

24:36

reduces to

2:3

(GCF is

12

). Platoon Charlie's

60:100

reduces to

3:5

(GCF is

20

). Platoon Delta's

120:30

reduces to

4:1

(GCF is

30

).

Step-by-Step Solution

1

Determine the simplified ratio for Platoon Alpha (

45

rations to

15

water).

The greatest common factor is

15

. Dividing both numbers by

15

yields

3:1

.

Simplifying the ratio reveals the baseline relationship between the two quantities.

2

Determine the simplified ratio for Platoon Bravo (

24

batteries to

36

flares).

The greatest common factor is

12

. Dividing both numbers by

12

yields

2:3

.

Both terms must be reduced completely to find the correct matching equivalent.

3

Determine the simplified ratio for Platoon Charlie (

60

sandbags to

100

stakes).

The greatest common factor is

20

. Dividing both numbers by

20

yields

3:5

.

Identifying the highest common divisor ensures the ratio is in its simplest form.

4

Determine the simplified ratio for Platoon Delta (

120

standard rounds to

30

tracer rounds).

The greatest common factor is

30

. Dividing both numbers by

30

yields

4:1

.

Applying the same simplification principle resolves the final pairing.

Key Concept

Simplifying basic part-to-part ratios by identifying and dividing by the greatest common factor.

Estimated Time:1m 30s

Question 362Question →

Read the scenario and fill in the missing values based on the given ratio.

Fill in the blanks below

A medical supply depot is distributing

2,400

trauma kits among three field hospitals—Alpha, Bravo, and Charlie—in a ratio of

3:4:5

, respectively. Based on this distribution, Hospital Charlie will receive trauma kits, which is more kits than Hospital Alpha receives.

Show answer & explanation

Answer

Hospital Charlie receives 1,000 kits, which is 400 more than Hospital Alpha's 600 kits.

By adding the ratio parts (

3+4+5=12

) and dividing the total kits by this sum (

2,400 \div 12=200

), we find the value of one part. Multiplying by Charlie's ratio (

5

) gives 1,000 kits. Multiplying by Alpha's ratio (

3

) gives 600 kits. The difference is 400 kits.

Step-by-Step Solution

1

Calculate the total number of parts in the ratio.

3 + 4 + 5 = 12

total parts.

To find out how much one part of the ratio is worth, we first need to know the sum of all the parts.

2

Determine the value of a single part.

2,400 \div 12 = 200

kits per part.

Dividing the total number of kits by the total number of parts gives the value of one part.

3

Calculate the number of kits Hospital Charlie and Hospital Alpha receive.

Charlie receives

5 \times 200 = 1,000

kits. Alpha receives

3 \times 200 = 600

kits.

Multiply the number of parts each hospital has in the ratio by the value of a single part.

4

Calculate the difference between Charlie's and Alpha's shares.

1,000 - 600 = 400

kits.

The question asks for how many more kits Charlie receives compared to Alpha.

Key Concept

To divide a quantity in a given ratio, add the parts of the ratio to find the total parts, divide the total quantity by this sum to find the value of one part, and then multiply by the specific parts for each share.

Question 363Question →

Read the scenario below and determine the technician's total gross earnings for the week.

Fill in the blanks below

An industrial equipment technician is paid an hourly rate of

\$28.00

for a standard

40

-hour workweek. Any additional hours are compensated at time-and-a-half. Last week, the technician worked a total of

47

hours and also received a flat

\$75.00

emergency call-out fee.

The technician's total gross earnings for that week were $\$$.

Show answer & explanation

Answer

The technician's total gross earnings for the week are $1,489.

The correct total is calculated by evaluating each component of the technician's compensation separately. The standard pay for the first 40 hours is

1,120. The 7 overtime hours are paid at a time-and-a-half rate of

42 per hour, contributing

294. Adding the standard pay, the overtime pay, and the

75 flat fee results in a total gross earnings of $1,489.

Step-by-Step Solution

1

Calculate the pay for the standard 40-hour workweek.

40 \text{ hours} \times \$28.00/\text{hour} = \$1,120.00

The standard hourly wage applies only to the first 40 hours worked.

2

Determine the overtime pay rate and the number of overtime hours.

Overtime rate =

\$28.00 \times 1.5 = \$42.00

per hour. Overtime hours =

47 - 40 = 7

hours.

Any hours worked beyond the standard 40 hours are paid at the time-and-a-half rate.

3

Calculate the total overtime pay.

7 \text{ hours} \times \$42.00/\text{hour} = \$294.00

This represents the earnings from the additional 7 hours worked.

4

Sum the standard pay, overtime pay, and the flat emergency fee to find the total gross earnings.

\$1,120.00 + \$294.00 + \$75.00 = \$1,489.00

Total gross earnings include all forms of compensation before taxes and deductions.

Key Concept

Calculating total gross pay involving a standard hourly wage, time-and-a-half overtime pay, and additional flat fees.

Question 364Question →

Consider the polynomial $8x^2 - 10x - 3$ . When this expression is factored completely into the form $(4x + a)(2x - b)$ , where $a$ and $b$ are positive integers, what are the corresponding values for $a$ and $b$ ?

Fill in the blanks below

The value of

a

is and the value of

b

is .

Show answer & explanation

Answer

The value of a is 1 and the value of b is 3.

By applying the AC method, the quadratic

8x^2 - 10x - 3

can be rewritten as

8x^2 - 12x + 2x - 3

and factored by grouping into

4x(2x - 3) + 1(2x - 3) = (4x + 1)(2x - 3)

. Aligning this with

(4x + a)(2x - b)

directly shows that

a = 1

and

b = 3

.

Step-by-Step Solution

1

Identify the type of polynomial and the appropriate factoring method.

The polynomial

8x^2 - 10x - 3

is a non-monic quadratic trinomial (where the leading coefficient is not 1). We can use the ac-method.

The ac-method involves finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

2

Calculate the ac product and find the target sum.

The product is

a \times c = 8 \times (-3) = -24

. The target sum is the middle coefficient, which is

-10

.

These values are required to split the middle term for factoring by grouping.

3

Find two integers that multiply to -24 and add to -10.

The numbers

-12

and

2

meet these conditions, since

(-12) \times 2 = -24

and

(-12) + 2 = -10

.

This allows us to rewrite the middle term

-10x

without changing the value of the polynomial.

4

Rewrite the middle term and factor by grouping.

8x^2 - 12x + 2x - 3

. Grouping the terms yields

(8x^2 - 12x) + (2x - 3)

. Factoring out the greatest common factor from each group gives

4x(2x - 3) + 1(2x - 3)

.

Grouping transforms the four-term expression into a format where a common binomial factor can be extracted.

5

Factor out the common binomial factor and determine the final values.

Factoring out

(2x - 3)

leaves

(4x + 1)(2x - 3)

. Comparing this to the given form

(4x + a)(2x - b)

reveals that

a = 1

and

b = 3

.

Matching the derived factorization with the required format identifies the specific constants requested by the problem.

Key Concept

Factoring non-monic quadratic trinomials using the AC method and grouping.

Question 365Question →

Find the completely factored form of the given polynomial:

-12x^2 + 30x + 72

Which of the following options shows the correct factorization?

-6(2x - 3)(x + 4)

-6(2x + 3)(x - 4)

-2(2x + 3)(3x - 12)

6(2x + 3)(x - 4)

Show answer & explanation

Answer: $-6(2x + 3)(x - 4)$

Answer

The completely factored form is

-6(2x + 3)(x - 4)

.

The polynomial requires factoring out the Greatest Common Factor (GCF) of -6, which leaves the trinomial

2x^2 - 5x - 12

. Factoring this trinomial yields the binomials

(2x + 3)

and

(x - 4)

. Multiplying these by the GCF gives the fully factored expression.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all terms in the polynomial

-12x^2 + 30x + 72

.

The GCF is -6. Factoring this out gives:

-6(2x^2 - 5x - 12)

.

Factoring out a negative GCF makes the leading coefficient of the trinomial positive, which simplifies the subsequent factoring steps.

2

Factor the remaining quadratic trinomial

2x^2 - 5x - 12

by finding two integers that multiply to

(2 \times -12) = -24

and add to the middle coefficient

-5

.

The two integers are -8 and 3.

These integers will be used to split the middle term to factor by grouping.

3

Rewrite the trinomial by splitting the middle term using the integers found, then factor by grouping.

2x^2 - 8x + 3x - 12 = 2x(x - 4) + 3(x - 4) = (2x + 3)(x - 4)

.

Grouping allows us to factor out common binomials to find the factored form of the trinomial.

4

Combine the GCF with the factored binomials to write the final completely factored expression.

-6(2x + 3)(x - 4)

A completely factored polynomial must include all extracted common factors alongside the irreducible binomial components.

Key Concept

Factoring polynomials using a Greatest Common Factor (GCF) and factoring non-monic trinomials.

Question 366Question →

A construction crew is pouring a foundation that requires $3.6$ tons of a custom gravel and sand mixture. If the mixture contains gravel and sand in a ratio of $5:4$ , how many pounds of sand are required for the foundation? ( $1$ ton $= 2,000$ pounds)

1,600

5,760

3,200

4,000

Show answer & explanation

Answer: $3,200$

Answer

The required amount of sand is

3,200

pounds.

The correct answer is found by first converting the total weight to pounds (

3.6 \times 2,000 = 7,200

pounds). The total number of parts in the ratio is

5 + 4 = 9

. Dividing the total weight by the total parts gives the weight of one part (

7,200 \div 9 = 800

pounds). Since the sand is the second component of the

5:4

ratio, it represents

4

parts. Multiplying the weight of one part by

4

gives

3,200

pounds.

Step-by-Step Solution

1

Convert the total weight of the mixture from tons to pounds.

3.6 \text{ tons} \times 2,000 \text{ pounds/ton} = 7,200 \text{ pounds}

The question asks for the final answer in pounds, so it is necessary to standardize the units first.

2

Determine the total number of parts in the ratio.

5 + 4 = 9 \text{ total parts}

To find the value of a single part, you must know how many equal parts make up the whole mixture.

3

Calculate the weight of one ratio part.

7,200 \text{ pounds} \div 9 \text{ parts} = 800 \text{ pounds/part}

Dividing the total weight by the total number of parts gives the weight of one proportional unit.

4

Calculate the total weight of the sand.

4 \text{ parts} \times 800 \text{ pounds/part} = 3,200 \text{ pounds}

The ratio of gravel to sand is

5:4

, meaning the sand accounts for exactly

4

of the

9

total parts.

Key Concept

Dividing in a Ratio

Question 367Question →

A freight broker earns a weekly base salary of $\$600$ . They also earn a $3\%$ commission on their first $\$10,000$ in freight sales for the week, and a $5\%$ commission on any sales exceeding $\$10,000$ . If the broker secures $\$24,000$ in freight sales this week, what are their total gross earnings for the week in dollars?

Show answer & explanation

Answer: 1600

Answer

1600

The total earnings consist of three parts: the fixed base salary (

600), the 3% commission on the first

10,000 of sales (

300), and the 5% commission on the remaining

14,000 of sales (

700). Adding these amounts together (

600 +

300 +

700) results in total gross earnings of $1,600.

Step-by-Step Solution

1

Calculate the commission earned on the first $10,000 of sales.

$300

The broker earns 3% on the first

10,000.

10,000 \times 0.03 = $300.

2

Calculate the remaining sales volume above the $10,000 threshold.

$14,000

Total sales were

24,000. The amount over

10,000 is

24,000 -

10,000 = $14,000.

3

Calculate the commission earned on the remaining sales.

$700

The broker earns 5% on sales exceeding

10,000.

14,000 \times 0.05 = $700.

4

Calculate total gross earnings.

$1,600

Add the base salary, first-tier commission, and second-tier commission:

600 +

300 +

700 =

1,600.

Key Concept

Calculating total gross earnings involving a base salary and a tiered commission structure.

Estimated Time:1m 30s

Question 368Question →

Which of the following expressions is equivalent to $6x^2 - 15x - 36$ and is factored completely?

3(2x + 3)(x - 4)

(2x + 3)(3x - 12)

3(2x - 3)(x + 4)

3(2x + 1)(x - 12)

Show answer & explanation

Answer: $3(2x + 3)(x - 4)$

Answer

The completely factored expression is

3(2x + 3)(x - 4)

.

The correct answer identifies the GCF of 3 and correctly factors the remaining quadratic

2x^2 - 5x - 12

into

(2x + 3)(x - 4)

. Expanding this result confirms it equals

6x^2 - 15x - 36

and no further factors can be extracted.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all three terms.

3(2x^2 - 5x - 12)

The coefficients 6, -15, and -36 are all divisible by 3.

2

Find two numbers that multiply to

ac

(where

a=2

and

c=-12

, so

2 \times -12 = -24

) and add to the middle coefficient (

b=-5

).

The numbers are

-8

and

3

.

(-8) \times 3 = -24

and

(-8) + 3 = -5

.

3

Rewrite the trinomial by splitting the middle term using these two numbers.

3(2x^2 - 8x + 3x - 12)

This sets up the polynomial for factoring by grouping.

4

Factor the trinomial by grouping.

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

Factoring out

(x - 4)

from both groups provides the final completely factored form.

Key Concept

Factoring out a GCF followed by factoring a non-monic trinomial.

Question 369Question →

Which of the following represents the completely factored form of the algebraic expression $30v^3 - 35v^2 - 15v$ ?

5v(3v + 1)(2v - 3)

5v(3v - 1)(2v + 3)

v(15v + 5)(2v - 3)

5v(v - 9)(v + 2)

Show answer & explanation

Answer: $5v(3v + 1)(2v - 3)$

Answer

The completely factored form is

5v(3v + 1)(2v - 3)

.

The correct answer accurately extracts the Greatest Common Factor of 5v from the original expression, leaving the non-monic trinomial 6v^2 - 7v - 3. Factoring this remaining trinomial by grouping correctly yields the binomials (3v + 1) and (2v - 3). Expanding these prime factors back confirms they produce the exact original expression.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) from all three terms in the polynomial

30v^3 - 35v^2 - 15v

.

5v(6v^2 - 7v - 3)

Factoring out the GCF simplifies the remaining polynomial and is a necessary first step for complete factorization.

2

Analyze the remaining quadratic trinomial

6v^2 - 7v - 3

to factor it by grouping. Find two numbers that multiply to

a \times c

(

6 \times -3 = -18

) and add to

b

(

-7

).

The two numbers are

-9

and

2

.

These numbers allow us to split the middle term (

-7v

) into two separate terms to enable factoring by grouping.

3

Rewrite the trinomial by splitting the middle term using the numbers found, then factor by grouping.

6v^2 - 9v + 2v - 3 = 3v(2v - 3) + 1(2v - 3) = (3v + 1)(2v - 3)

Grouping the terms reveals the common binomial factor of

(2v - 3)

.

4

Combine the GCF extracted in step 1 with the binomial factors found in step 3.

5v(3v + 1)(2v - 3)

This represents the original expression as a product of its simplest irreducible factors.

Key Concept

Factoring a polynomial by first extracting the greatest common factor (GCF), followed by factoring a non-monic quadratic trinomial using grouping.

Question 370Question →

A military installation allocates its annual training budget between field exercises and classroom instruction in a ratio of $5:2$ . If the total annual training budget is $\$84,000$ , how much money is allocated to classroom instruction per month?

\$2,800

\$2,000

\$24,000

\$5,000

Show answer & explanation

Answer: $\$2,000$

Answer

The correct answer is

\$2,000

.

The total budget is divided into

7

equal parts (

5 + 2

). Since classroom instruction receives

2

parts, its annual allocation is

\frac{2}{7}

of

\$84,000

, which is

\$24,000

. Dividing this annual amount by

12

months gives a monthly allocation of

\$2,000

.

Step-by-Step Solution

1

Determine the total number of parts in the ratio.

5 + 2 = 7

total parts.

To find the value of a single share of the budget, the total amount must be divided by the total number of parts.

2

Calculate the value of one part of the budget.

\$84,000 \div 7 = \$12,000

per part.

This establishes the base unit for the ratio.

3

Determine the annual budget for classroom instruction.

2 \times \$12,000 = \$24,000

.

Classroom instruction corresponds to

2

parts of the

5:2

ratio.

4

Convert the annual budget to a monthly budget.

\$24,000 \div 12 = \$2,000

.

The question specifically asks for the monthly allocation.

Key Concept

Calculating proportional shares of a total amount and converting between time periods.

Estimated Time:1m 0s

Question 371Question →

A civil engineering technician is buying supplies for a base surveying project. He purchases $4$ rolls of marking tape for $\$6.35$ each and $3$ cans of spray paint for $\$4.75$ each. If he hands the cashier a $\$50.00$ bill, how much change should he receive?

\$10.35

\$11.35

\$23.17

\$39.65

Show answer & explanation

Answer: $\$10.35$

Answer

\$10.35

The correct sequence of operations is to find the total cost of all items and then subtract that sum from the payment amount. First, multiplying

4

rolls of tape by

\$6.35

gives

\$25.40

. Next, multiplying

3

cans of paint by

\$4.75

gives

\$14.25

. Adding these amounts provides the total purchase price of

\$39.65

. Finally, subtracting

\$39.65

from the

\$50.00

bill yields exactly

\$10.35

in change.

Step-by-Step Solution

1

Calculate the total cost of the marking tape.

4 \times \$6.35 = \$25.40

To find the cost contribution of the first item.

2

Calculate the total cost of the spray paint.

3 \times \$4.75 = \$14.25

To find the cost contribution of the second item.

3

Add the costs of the two items to find the total purchase amount.

\$25.40 + \$14.25 = \$39.65

To determine the full amount owed to the cashier.

4

Subtract the total purchase amount from the amount given to the cashier.

\$50.00 - \$39.65 = \$10.35

To determine the exact amount of change to be returned.

Key Concept

Solving multi-step currency word problems requires careful decimal multiplication to find itemized totals, followed by decimal addition for the grand total, and decimal subtraction to determine change due.

Estimated Time:1m 15s

Question 372Question →

During a training exercise, a reconnaissance team travels a distance of $18$ miles in exactly $45$ minutes using all-terrain vehicles. What is their average speed in miles per hour?

Show answer & explanation

Answer: 24

Answer

24

To find the average speed in miles per hour, the travel time must first be converted into hours. Since there are

60

minutes in an hour,

45

minutes is equal to

0.75

hours (or

\frac{3}{4}

of an hour). Dividing the distance of

18

miles by the time of

0.75

hours yields a rate of

24

miles per hour.

Step-by-Step Solution

1

Convert the travel time from minutes to hours.

45 \text{ minutes} = \frac{45}{60} \text{ hours} = 0.75 \text{ hours}

(or

\frac{3}{4}

hour).

The speed is requested in miles per hour, so the time must be converted to hours to match the units of the rate.

2

Calculate the average speed using the rate formula

r = \frac{d}{t}

.

r = \frac{18 \text{ miles}}{0.75 \text{ hours}} = 24 \text{ mph}

.

Rate equals total distance divided by total time.

Key Concept

Calculating rate with necessary time unit conversions

Estimated Time:1m 0s

Question 373Question →

The interior angles of a quadrilateral are represented by the algebraic expressions $2x^\circ$ , $(3x - 15)^\circ$ , $(x + 25)^\circ$ , and $(2x + 30)^\circ$ . What is the measure of the largest angle in this quadrilateral?

135^\circ

40^\circ

110^\circ

65^\circ

Show answer & explanation

Answer: $110^\circ$

Answer

The largest angle measures

110^\circ

.

The interior angles of a quadrilateral must sum to

360^\circ

. By adding the four expressions together, we get

8x + 40 = 360

. Solving for

x

yields

x = 40

. To find the largest angle, we substitute

40

back into the original expressions:

2(40) = 80^\circ

,

3(40) - 15 = 105^\circ

,

40 + 25 = 65^\circ

, and

2(40) + 30 = 110^\circ

. Comparing these four values,

110^\circ

is the largest.

Step-by-Step Solution

1

Set up an equation based on the sum of interior angles.

2x + (3x - 15) + (x + 25) + (2x + 30) = 360

The sum of the interior angles of any quadrilateral is always

360^\circ

.

2

Combine the like terms on the left side of the equation.

8x + 40 = 360

Grouping the

x

terms (

2x + 3x + x + 2x = 8x

) and constant terms (

-15 + 25 + 30 = 40

) simplifies the algebraic expression.

3

Solve for the variable

x

.

8x = 320

, which means

x = 40

.

Subtracting 40 from both sides and then dividing by 8 isolates

x

.

4

Substitute

x = 40

into each of the original expressions to determine the measure of each angle.

The angles are

2(40) = 80^\circ

;

3(40) - 15 = 105^\circ

;

40 + 25 = 65^\circ

; and

2(40) + 30 = 110^\circ

.

The value of

x

alone does not answer the question; the specific angle measures must be evaluated.

5

Identify the largest value among the calculated angles.

The largest value is

110^\circ

.

The question specifically asks for the maximum angle measure in the quadrilateral.

Key Concept

The sum of the interior angles of a quadrilateral is always 360 degrees. Algebraic substitution is required to convert variable expressions back into geometric measurements.

Estimated Time:1m 30s

Question 374Question →

Match each polygon or quadrilateral with its defining geometric property.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Parallelogram matches with 'two pairs of parallel opposite sides', Trapezoid matches with 'exactly one pair of parallel sides', Rhombus matches with 'four sides of equal length', and Hexagon matches with 'exactly six sides'.

Each shape is matched to its fundamental geometric definition based on the number of sides, side lengths, and parallel properties. Knowing these core definitions is essential for classifying polygons and quadrilaterals correctly.

Step-by-Step Solution

1

Analyze the properties of a parallelogram.

It has two pairs of parallel opposite sides.

This is the fundamental geometric definition of a parallelogram.

2

Analyze the properties of a trapezoid.

It is defined by having exactly one pair of parallel sides.

This distinguishes a trapezoid from other quadrilaterals like parallelograms.

3

Analyze the properties of a rhombus.

It is a quadrilateral where all four sides are of equal length.

While it also has parallel sides, the equal side lengths are its distinguishing feature.

4

Identify the number of sides for a hexagon.

A hexagon is a polygon with exactly six sides.

The prefix 'hexa-' denotes six in geometry.

Key Concept

Basic Properties of Quadrilaterals and Polygons

Question 375Question →

Decompose the quadratic trinomial $6x^2 - 11x - 10$ into a product of two binomial factors structured as $(ax - b)(cx + d)$ . Assume all parameters ( $a$ , $b$ , $c$ , and $d$ ) are strictly positive integers. To ensure a unique arrangement, the leading coefficient $a$ must be strictly less than $c$ . Provide the values of the four parameters.

Fill in the blanks below

a =

,

b =

,

c =

, and

d =

.

Show answer & explanation

Answer

The parameters are

a = 2

,

b = 5

,

c = 3

, and

d = 2

.

Factoring the polynomial

6x^2 - 11x - 10

using the AC method produces

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

. Matching these two factors against the template

(ax - b)(cx + d)

under the rule that

a < c

(since

2 < 3

) perfectly determines the positive integers as

a = 2

,

b = 5

,

c = 3

, and

d = 2

.

Step-by-Step Solution

1

Identify the factoring technique for a non-monic quadratic trinomial in the form

Ax^2 + Bx + C

.

The AC method (factoring by grouping) is required because the leading coefficient

A = 6

is greater than 1.

Directly finding factors of the constant

C

that sum to

B

only works when

A = 1

.

2

Multiply the leading coefficient

A

by the constant term

C

.

6 \times (-10) = -60

.

This establishes the target product needed to properly split the middle term for grouping.

3

Find two integers that multiply to

-60

and add to the middle coefficient

B = -11

.

The correct pair of integers is

-15

and

4

.

(-15) \times 4 = -60

and

-15 + 4 = -11

.

4

Rewrite the middle term using these two integers and factor the resulting four-term expression by grouping.

6x^2 - 15x + 4x - 10 = 3x(2x - 5) + 2(2x - 5) = (2x - 5)(3x + 2)

.

Grouping the first two terms and the last two terms reveals a common binomial factor of

(2x - 5)

.

5

Match the factored expression to the required template

(ax - b)(cx + d)

while adhering to the condition

a < c

.

The factors are

(2x - 5)

and

(3x + 2)

. Since

2 < 3

,

(ax - b)

corresponds exactly to

(2x - 5)

, yielding

a=2

and

b=5

. Consequently,

(cx + d)

corresponds to

(3x + 2)

, yielding

c=3

and

d=2

.

This guarantees a unique, correctly-ordered set of strictly positive parameters.

Key Concept

Factoring non-monic quadratic trinomials via grouping

Question 376Question →

Complete the factorization of the perfect square trinomial below by filling in the missing number.

Fill in the blanks below

$x^2 + 14x + 49 = (x + )^2$

Show answer & explanation

Answer

7

A perfect square trinomial takes the general form

x^2 + 2bx + b^2 = (x + b)^2

. In the given expression

x^2 + 14x + 49

, the constant term is

49

, which is

7^2

. The middle term is

14x

, which is equal to

2 \times 7 \times x

. Therefore, the expression factors directly as

(x + 7)^2

, meaning the missing number in the blank is 7.

Step-by-Step Solution

1

Recognize the polynomial form

The expression

x^2 + 14x + 49

matches the structure of a perfect square trinomial:

a^2 + 2ab + b^2

.

The first term is a perfect square (

x^2

), and the last term is a perfect square (

49 = 7^2

).

2

Verify the middle term

Checking the middle term:

2 \times x \times 7 = 14x

. This matches the middle term of the given expression.

Confirming the middle term proves the expression can be factored exactly as

(a + b)^2

.

3

Determine the missing value

The expression factors as

(x + 7)^2

.

Using the formula

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

where

a = x

and

b = 7

.

Key Concept

Factoring perfect square trinomials

Question 377Question →

A quartermaster is allocating $2,100$ yards of heavy-duty canvas to manufacture temporary shelters, cot covers, and equipment tarps. The canvas is divided among these three projects in a ratio of $6:5:3$ , respectively. What is the difference, in yards, between the amount of canvas allocated for the temporary shelters and the amount allocated for the equipment tarps?

Show answer & explanation

Answer: 450

Answer

450

The correct answer is found by determining that the total ratio consists of 14 parts, meaning each part represents 150 yards (2100 / 14). The difference between the shelters (6 parts) and tarps (3 parts) is 3 parts, which equals 450 yards (3 * 150).

Step-by-Step Solution

1

Calculate the total number of parts in the ratio.

6 + 5 + 3 = 14

total parts.

To divide the total amount of canvas proportionally, we first need to determine the total number of equal parts.

2

Determine the value of a single part.

2,100 \div 14 = 150

yards per part.

Dividing the total yards by the total parts gives the amount of canvas for each ratio unit.

3

Calculate the difference in ratio parts between the shelters and the tarps.

6

parts (shelters) -

3

parts (tarps) =

3

parts difference.

We need the difference between the two allocations. Finding the difference in parts directly is the most efficient method.

4

Calculate the difference in yards.

3 \times 150 = 450

yards.

Multiplying the difference in ratio parts by the value of a single part yields the final answer.

Key Concept

Dividing a total quantity into a given ratio by finding the value of a single part.

Question 378Question →

A military transport helicopter flies a distance of $120$ miles between two bases. If the flight takes exactly $40$ minutes, what is the helicopter's average speed in miles per hour?

300

80

180

3

Show answer & explanation

Answer: $180$

Answer

The helicopter's average speed is

180

miles per hour.

To find the correct speed in miles per hour, the travel time must first be converted to hours. Since

40

minutes is

\frac{40}{60}

or

\frac{2}{3}

of an hour, dividing the distance of

120

miles by

\frac{2}{3}

of an hour yields

180

miles per hour.

Step-by-Step Solution

1

Convert the travel time from minutes to hours.

40 \text{ minutes} = \frac{40}{60} \text{ hours} = \frac{2}{3} \text{ hours}

.

The requested rate is in miles per hour, so the time unit must be converted to hours before calculating.

2

Set up the rate formula using the distance and the converted time.

r = \frac{120 \text{ miles}}{\frac{2}{3} \text{ hours}}

.

The formula for finding average speed is rate equals distance divided by time (

r = \frac{d}{t}

).

3

Solve for the rate by dividing the whole number by the fraction.

120 \div \frac{2}{3} = 120 \times \frac{3}{2} = \frac{360}{2} = 180 \text{ mph}

.

Dividing by a fraction is achieved by multiplying by its reciprocal.

Key Concept

Average speed is calculated by dividing total distance by total time (

r = \frac{d}{t}

). When the required rate uses different units than the provided information, a unit conversion must be performed first.

Question 379Question →

A decorative floor tile is manufactured in the shape of a regular pentagon. What is the degree measure of each interior angle of the tile?

72^\circ

108^\circ

180^\circ

540^\circ

Show answer & explanation

Answer: $108^\circ$

Answer

108^\circ

The interior angles of an

n

-sided polygon sum to

(n - 2) \times 180^\circ

. For a 5-sided pentagon, the sum is

(5 - 2) \times 180^\circ = 540^\circ

. Since the tile is a regular pentagon, all five angles are identical. Dividing the total

540^\circ

by 5 yields exactly

108^\circ

per corner.

Step-by-Step Solution

1

Identify the number of sides in a regular pentagon.

n = 5

A pentagon is defined as a geometric polygon with exactly 5 sides.

2

Calculate the sum of all interior angles using the polygon formula

(n - 2) \times 180^\circ

.

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

This formula determines the total degree measure for the interior angles of any convex polygon.

3

Divide the total sum by the number of sides to find the measure of a single interior angle.

540^\circ \div 5 = 108^\circ

Because it is a regular pentagon, all interior angles are equal in measure.

Key Concept

Interior Angles of Regular Polygons

Question 380Question →

A military fighter jet flies a mission in two segments. It travels the first $1000$ miles in $1$ hour. Then, it travels an additional $200$ miles in $15$ minutes to reach its destination. What is the average speed of the jet for the entire mission, in miles per hour?

16

75

900

960

Show answer & explanation

Answer: 960

Answer

960

The correct average speed is found by dividing the total distance traveled (

1200

miles) by the total flight time in hours (

1.25

hours). Dividing

1200

by

1.25

yields

960

miles per hour.

Step-by-Step Solution

1

Calculate the total distance traveled.

1000 \text{ miles} + 200 \text{ miles} = 1200 \text{ miles}

To find average speed for an entire trip, you first need the total distance covered.

2

Convert the total time into a single, consistent unit (hours).

1 \text{ hour} + 15 \text{ minutes} = 1 \text{ hour} + 0.25 \text{ hours} = 1.25 \text{ hours}

Since the question asks for miles per hour, the time must be expressed entirely in hours before dividing.

3

Calculate the average speed using the total distance and total time.

1200 \div 1.25 = 960 \text{ mph}

The formula for average speed is Total Distance divided by Total Time.

Key Concept

Average speed is calculated by dividing the total distance traveled by the total time taken. It is not the simple arithmetic mean of different speeds.

Estimated Time:1m 0s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept