All practice questions

Question 721Question →

If $(x + 3)^2 = 25$ and $x > 0$ , what is the value of $x^2 - 1$ ?

3

4

15

24

63

Show answer & explanation

Answer: 3

Answer

3

Taking the square root of both sides of

(x + 3)^2 = 25

gives

x + 3 = \pm 5

. Since

x > 0

, we choose

x + 3 = 5

, which gives

x = 2

. Substituting

x = 2

into

x^2 - 1

produces

2^2 - 1 = 3

.

Step-by-Step Solution

1

Take the square root of both sides of the given quadratic equation

x + 3 = 5

or

x + 3 = -5

The equation

(x + 3)^2 = 25

implies

x + 3

can be either positive or negative 5.

2

Solve for

x

under the constraint

x > 0

x = 2

From

x + 3 = 5

, we obtain

x = 2

. The second root

x = -8

is discarded because

x > 0

.

3

Substitute

x = 2

into the target expression

x^2 - 1

2^2 - 1 = 3

Evaluating

x^2 - 1

at

x = 2

yields

4 - 1 = 3

.

Key Concept

Solving Quadratic Equations by Taking Square Roots

Estimated Time:1m 0s

Question 722Question →

For all non-zero real numbers $x$ and $y$ , the custom operation $\star$ is defined by $x \star y = \frac{x}{y} - \frac{y}{x}$ . If $p$ and $q$ are positive numbers such that $p + q = 5$ and $p \star q = \frac{5}{6}$ , what is the value of $(p + 1) \star q$ ?

\frac{3}{2}

-\frac{3}{2}

\frac{24}{5}

0

\frac{21}{10}

Show answer & explanation

Answer: $\frac{3}{2}$

Answer

\frac{3}{2}

By simplifying

p \star q = \frac{p^2 - q^2}{pq} = \frac{(p - q)(p + q)}{pq}

, we substitute

p + q = 5

to get

\frac{5(p - q)}{pq} = \frac{5}{6}

, which reduces to

6(p - q) = pq

. Substituting

q = 5 - p

yields the quadratic equation

p^2 + 7p - 30 = 0

. Since

p > 0

, we find

p = 3

and

q = 2

. Evaluating

(3 + 1) \star 2 = 4 \star 2 = \frac{4}{2} - \frac{2}{4} = \frac{3}{2}

.

Step-by-Step Solution

1

Express the custom operation using algebraic fractions and factor the numerator

p \star q = \frac{p}{q} - \frac{q}{p} = \frac{p^2 - q^2}{pq} = \frac{(p - q)(p + q)}{pq}

Finding a common denominator converts the expression into a form where known identities can be substituted.

2

Substitute

p + q = 5

and

p \star q = \frac{5}{6}

into the simplified expression

\frac{5(p - q)}{pq} = \frac{5}{6} \implies \frac{p - q}{pq} = \frac{1}{6} \implies 6(p - q) = pq

Simplifying by dividing both sides by 5 establishes a direct relation between the difference and product of p and q.

3

Substitute

q = 5 - p

to form and solve a quadratic equation for

p

6(2p - 5) = p(5 - p) \implies 12p - 30 = 5p - p^2 \implies p^2 + 7p - 30 = 0 \implies (p + 10)(p - 3) = 0

Since p and q are positive numbers, p must be positive, giving p = 3 and q = 2.

4

Evaluate

(p + 1) \star q

with

p = 3

and

q = 2

(3 + 1) \star 2 = 4 \star 2 = \frac{4}{2} - \frac{2}{4} = 2 - \frac{1}{2} = \frac{3}{2}

Applying the custom operator definition to inputs 4 and 2 yields the final value.

Key Concept

Custom Operator Evaluation and Quadratic System Solving

Question 723Question →

What is the sum of all distinct real solutions to the equation

\frac{x^2 - 8x + 15}{x - 3} = (x - 5)^2

?

6

9

11

14

1

Show answer & explanation

Answer: 11

Answer

11

Factoring the numerator

x^2 - 8x + 15

gives

(x - 3)(x - 5)

. For all

x \neq 3

, the original equation simplifies to

x - 5 = (x - 5)^2

. Rearranging into standard factored form

(x - 5)(x - 6) = 0

yields two distinct real solutions:

x = 5

and

x = 6

. Neither value violates the restriction

x \neq 3

. Adding these values together yields

5 + 6 = 11

.

Step-by-Step Solution

1

Determine the domain restriction for the rational expression

The expression

\frac{x^2 - 8x + 15}{x - 3}

requires that the denominator

x - 3 \neq 0

, so

x \neq 3

.

Division by zero is undefined in real numbers.

2

Factor the numerator and simplify the left side of the equation

Since

x^2 - 8x + 15 = (x - 3)(x - 5)

, for

x \neq 3

,

\frac{(x - 3)(x - 5)}{x - 3} = x - 5

.

Canceling the common non-zero factor

(x - 3)

simplifies the rational equation.

3

Solve the simplified quadratic equation for

x

x - 5 = (x - 5)^2 \implies (x - 5)^2 - (x - 5) = 0 \implies (x - 5)(x - 5 - 1) = 0 \implies (x - 5)(x - 6) = 0

. Thus,

x = 5

or

x = 6

.

Factoring out the common factor

(x - 5)

preserves all valid solutions.

4

Verify solutions against the domain restriction and calculate the sum

Both

x = 5

and

x = 6

satisfy

x \neq 3

. The sum of all distinct real solutions is

5 + 6 = 11

.

Combining the valid distinct roots gives the required total.

Key Concept

Polynomial factoring, domain restrictions on rational expressions, and avoiding root loss when solving quadratic equations.

Estimated Time:2m 0s

Question 724Question →

A specialized mapping drone, Model X, can complete an aerial survey of a solar farm in $20$ hours operating alone at a constant rate. A second drone, Model Y, can complete the exact same survey in $30$ hours operating alone at a constant rate. If both drones operate simultaneously at their respective constant rates, how many hours will it take them to complete the aerial survey together?

Show answer & explanation

Answer: 12

Answer

It will take 12 hours for both drones operating together to complete the survey.

Model X completes

\frac{1}{20}

of the job per hour and Model Y completes

\frac{1}{30}

of the job per hour. Combined, their rate is

\frac{1}{20} + \frac{1}{30} = \frac{5}{60} = \frac{1}{12}

of the job per hour. Taking the reciprocal yields a total time of 12 hours.

Step-by-Step Solution

1

Calculate individual work rates

Model X rate =

\frac{1}{20}

job/hr, Model Y rate =

\frac{1}{30}

job/hr

Work rate is the reciprocal of the total time required to complete one full job.

2

Calculate the combined rate of work

Combined rate =

\frac{1}{20} + \frac{1}{30} = \frac{5}{60} = \frac{1}{12}

job/hr

When working simultaneously, individual rates add together.

3

Compute total time required for combined work

Total time =

\frac{1}{1/12} = 12

hours

Time is equal to total work (1 job) divided by the combined work rate.

Key Concept

Combined Work Rate formula:

\text{Rate}_{\text{total}} = \text{Rate}_1 + \text{Rate}_2

, and

\text{Time}_{\text{total}} = \frac{1}{\text{Rate}_{\text{total}}}

.

Question 725Question →

A rectangular garden has an area of $48$ square meters. If the length of the garden is $2$ meters greater than its width, what is the perimeter of the garden, in meters?

Show answer & explanation

Answer: 28

Answer

The perimeter of the garden is 28 meters.

By setting up the area equation

w(w + 2) = 48

, we obtain the quadratic equation

w^2 + 2w - 48 = 0

. Factoring gives

(w + 8)(w - 6) = 0

. Since width must be positive,

w = 6

meters. The length is

6 + 2 = 8

meters. Thus, the perimeter is

2(6 + 8) = 28

meters.

Step-by-Step Solution

1

Define variables for width and length.

Width =

w

, Length =

w + 2

.

The length is given as 2 meters greater than the width.

2

Formulate and rearrange the quadratic equation for area.

w^2 + 2w - 48 = 0

Area is length multiplied by width, set equal to 48.

3

Factor the quadratic polynomial.

(w + 8)(w - 6) = 0

Find two numbers that multiply to -48 and add up to +2.

4

Determine the valid physical dimension.

w = 6

meters and length

l = 8

meters.

A physical dimension cannot be negative, so

w = -8

is discarded.

5

Compute the perimeter.

Perimeter =

2(6 + 8) = 28

meters.

Perimeter of a rectangle is twice the sum of its length and width.

Key Concept

Solving quadratic equations by factoring in word problem contexts.

Question 726Question →

For any real number $x$ , the function $h$ is defined by $h(x) = 3x^2 - 5x + 4$ . What is the value of $h(3)$ ?

Show answer & explanation

Answer: 16

Answer

16

To find

h(3)

, substitute

x = 3

into the function

h(x) = 3x^2 - 5x + 4

. First, compute

3^2 = 9

. Multiplying by 3 yields 27. Subtracting

5(3) = 15

gives 12, and adding 4 results in the final value of 16.

Step-by-Step Solution

1

Substitute the value

x = 3

into the definition of

h(x)

.

h(3) = 3(3)^2 - 5(3) + 4

To evaluate a function at a specific value, replace all instances of the input variable with that value.

2

Evaluate the exponent and carry out the multiplications.

h(3) = 3(9) - 15 + 4 = 27 - 15 + 4

Follow standard order of operations: compute powers before performing multiplication.

3

Perform the remaining addition and subtraction.

h(3) = 16

Complete the calculation from left to right.

Key Concept

Function Evaluation

Question 727Question →

A graphic design department has senior designers and junior designers in a ratio of $3 : 4$ . If there are $24$ junior designers in the department, what is the total number of senior and junior designers in the department?

Show answer & explanation

Answer: 42

Answer

The total number of senior and junior designers in the department is 42.

The ratio of senior to junior designers is

3 : 4

. Given that there are

24

junior designers,

4

ratio parts equal

24

, making

1

ratio part equal to

6

. The total department consists of

3 + 4 = 7

ratio parts. Therefore, the total number of designers is

7 \times 6 = 42

.

Step-by-Step Solution

1

Determine the value of one ratio unit using the given number of junior designers.

Since

4

ratio parts correspond to

24

junior designers,

1

ratio part

= 24 / 4 = 6

.

The given count of

24

aligns with the

4

parts of junior designers in the

3 : 4

ratio.

2

Calculate the total ratio parts and find the total count of designers.

Total ratio parts

= 3 + 4 = 7

. Total designers

= 7 \times 6 = 42

.

The total department size is represented by the sum of the ratio components.

Key Concept

Ratio and Proportion Word Problems

Question 728Question →

If $x^2 - 9x + 18 = 0$ , and $x_1$ and $x_2$ are the real roots of the equation such that $x_1 > x_2$ , what is the value of $2x_1 - x_2$ ?

Show answer & explanation

Answer: 9

Answer

The value of

2x_1 - x_2

is 9.

Factoring the quadratic equation

x^2 - 9x + 18 = 0

yields

(x - 6)(x - 3) = 0

. The roots are

x = 6

and

x = 3

. Given that

x_1 > x_2

, we assign

x_1 = 6

and

x_2 = 3

. Evaluating

2x_1 - x_2

gives

2(6) - 3 = 9

.

Step-by-Step Solution

1

Factor the quadratic expression

(x - 6)(x - 3) = 0

Find two numbers that multiply to

18

and add to

-9

, which are

-6

and

-3

.

2

Solve for the roots of the equation

x = 6

or

x = 3

Set each factor equal to zero:

x - 6 = 0 \implies x = 6

and

x - 3 = 0 \implies x = 3

.

3

Assign root values based on the condition

x_1 > x_2

x_1 = 6

and

x_2 = 3

Since

6 > 3

, the larger root is assigned to

x_1

and the smaller root to

x_2

.

4

Evaluate the target expression

9

Substitute

x_1 = 6

and

x_2 = 3

into

2x_1 - x_2 = 2(6) - 3 = 12 - 3 = 9

.

Key Concept

Solving Quadratic Equations by Factoring

Question 729Question →

At a textile weaving mill, high-grade cotton, synthetic polyester, and natural silk threads are blended to produce a specialized fabric. Initially, the weights of cotton, polyester, and silk in the blending vat are in the ratio of $5 : 3 : 2$ , respectively. To meet a custom order requirement, $12$ kilograms of silk are added to the vat, and $4$ kilograms of polyester are removed, while the amount of cotton remains unchanged. If the new ratio of cotton weight to silk weight is $5 : 4$ , how many kilograms of synthetic polyester are in the blending vat after these adjustments?

Show answer & explanation

Answer: 14

Answer

14 kg

Representing the initial weights as

5x

,

3x

, and

2x

, the addition of

12

kg of silk updates the silk quantity to

2x + 12

. Setting the unchanged cotton quantity (

5x

) over the new silk quantity equal to

\frac{5}{4}

yields

x = 6

. The initial weight of polyester was

3(6) = 18

kg, and removing

4

kg gives a final polyester weight of

14

kg.

Step-by-Step Solution

1

Express initial component weights using a common ratio multiplier

x

Cotton =

5x

, Polyester =

3x

, Silk =

2x

The given initial ratio of cotton to polyester to silk is

5 : 3 : 2

.

2

Set up a proportion for the updated Cotton-to-Silk ratio

\frac{5x}{2x + 12} = \frac{5}{4}

Adding

12

kg of silk increases its weight to

(2x + 12)

kg, while cotton remains at

5x

kg.

3

Solve the algebraic proportion for

x

x = 6

Cross-multiplying gives

20x = 5(2x + 12) = 10x + 60

, which simplifies to

10x = 60

, so

x = 6

.

4

Calculate the final weight of synthetic polyester

14 kg

Initial polyester was

3(6) = 18

kg. Subtracting the

4

kg removed leaves

18 - 4 = 14

kg.

Key Concept

Multi-part ratios and algebraic setting of updated proportional quantities

Question 730Question →

An investment consultancy allocates capital across three asset classes: Equities, Fixed Income, and Real Estate. Initially, the ratio of Equities to Fixed Income is $3 : 4$ , and the ratio of Fixed Income to Real Estate is $2 : 3$ . Following a market expansion, the total value of the portfolio increases by $50\%$ . To comply with updated risk guidelines, the firm reallocates the expanded capital such that the amount in Fixed Income decreases by $25\%$ , and the remaining portfolio value is divided between Equities and Real Estate in the ratio $3 : 2$ , respectively. If the new allocation in Real Estate is $\$18,000$ greater than its initial allocation, what was the initial total value of the investment portfolio?

\$130,000

\$300,000

\$390,000

\$520,000

\$585,000

Show answer & explanation

Answer: $\$390,000$

Answer

The initial total value of the investment portfolio was

\$

390,000.

Combining the initial ratios gives an Equities to Fixed Income to Real Estate ratio of

3 : 4 : 6

, making the initial total

13x

. After a

50\%

increase in total value (

19.5x

) and a

25\%

decrease in Fixed Income (

3x

), the remaining

16.5x

yields a new Real Estate value of

\frac{2}{5} \times 16.5x = 6.6x

. The difference

6.6x - 6x = 0.6x = 18,000

gives

x = 30,000

, resulting in an initial total portfolio of

13 \times 30,000 = \$390,000

.

Step-by-Step Solution

1

Combine the initial two-part ratios into a single three-part ratio for Equities (

E_1

), Fixed Income (

F_1

), and Real Estate (

R_1

).

E_1 : F_1 = 3 : 4

and

F_1 : R_1 = 2 : 3 = 4 : 6

, giving

E_1 : F_1 : R_1 = 3 : 4 : 6

.

A common term for Fixed Income (

4

) is needed to express all three asset classes in terms of a single variable

x

.

2

Express initial amounts and initial total portfolio value in terms of

x

.

E_1 = 3x

,

F_1 = 4x

,

R_1 = 6x

, and Total Initial Portfolio

T_1 = 3x + 4x + 6x = 13x

.

Summing the ratio units gives the total initial quantity representation.

3

Calculate the updated total portfolio value (

T_2

) and updated Fixed Income amount (

F_2

).

T_2 = 1.50 \times 13x = 19.5x

, and

F_2 = (1 - 0.25) \times 4x = 3x

.

The overall portfolio increases by

50\%

while Fixed Income decreases by

25\%

.

4

Determine the remaining portfolio value for Equities and Real Estate, and calculate the new Real Estate value (

R_2

).

Remaining Value

= 19.5x - 3x = 16.5x

.

R_2 = \frac{2}{3 + 2} \times 16.5x = \frac{2}{5} \times 16.5x = 6.6x

.

The remaining capital is divided between Equities and Real Estate in a

3 : 2

ratio.

5

Set up the change equation for Real Estate to solve for

x

and determine

T_1

.

R_2 - R_1 = 6.6x - 6x = 0.6x = 18,000 \implies x = 30,000

. Therefore,

T_1 = 13 \times 30,000 = 390,000

.

The problem states the final Real Estate amount is

\$

18,000 greater than its initial value.

Key Concept

Combining compound ratios and applying sequential percentage modifications to part-to-part and part-to-whole relationships.

Question 731Question →

At an architectural firm, the ratio of senior architects to junior architects to structural engineers was initially $3 : 7 : 4$ , respectively. After the firm hired $6$ additional senior architects and $4$ structural engineers left the firm, the ratio of senior architects to structural engineers became $3 : 2$ , while the number of junior architects remained unchanged. What was the total number of senior architects, junior architects, and structural engineers combined initially at the firm?

Show answer & explanation

Answer: 56

Answer

The total number of senior architects, junior architects, and structural engineers combined initially at the firm was 56.

Let the initial numbers of senior architects, junior architects, and structural engineers be

3x

,

7x

, and

4x

, respectively. After adding

6

senior architects and subtracting

4

structural engineers, the updated counts are

3x + 6

and

4x - 4

. Setting their ratio to

3 : 2

yields

\frac{3x + 6}{4x - 4} = \frac{3}{2}

. Cross-multiplying gives

6x + 12 = 12x - 12

, which simplifies to

6x = 24

, so

x = 4

. The total number of initial employees is

3x + 7x + 4x = 14x = 14(4) = 56

.

Step-by-Step Solution

1

Define initial quantities using ratio multiplier x

Senior Architects

= 3x

, Junior Architects

= 7x

, Structural Engineers

= 4x

Ratios specify relative proportions, so multiplying each term by

x

gives actual counts.

2

Formulate new quantities after staff changes

Senior Architects

= 3x + 6

, Structural Engineers

= 4x - 4

6 senior architects joined and 4 structural engineers left.

3

Set up ratio equation and solve for x

\frac{3x + 6}{4x - 4} = \frac{3}{2} \implies 2(3x + 6) = 3(4x - 4) \implies 6x + 12 = 12x - 12 \implies 6x = 24 \implies x = 4

Cross-multiplication converts the proportion into a linear equation.

4

Compute total initial staff count

Initial Total

= 3x + 7x + 4x = 14x = 14 \times 4 = 56

Summing all 3 initial ratio parts and substituting

x = 4

yields the total initial count.

Key Concept

Multi-part ratio formulation and linear equation setup based on additions and subtractions

Estimated Time:2m 0s

Question 732Question →

For all real numbers $x$ and $y$ , the custom operation $\Delta$ is defined by $x \Delta y = (x + 1)(y - 2)$ . If the function $g$ is defined by $g(n) = n^2 - 3$ for all real numbers $n$ , what is the value of $g(3) \Delta g(2)$ ?

Show answer & explanation

Answer: -7

Answer

The value of

g(3) \Delta g(2)

is

-7

.

Evaluating

g(3)

gives

3^2 - 3 = 6

, and evaluating

g(2)

gives

2^2 - 3 = 1

. Substituting these values as

x = 6

and

y = 1

into the custom operator definition

x \Delta y = (x + 1)(y - 2)

yields

(6 + 1)(1 - 2) = 7 \times (-1) = -7

.

Step-by-Step Solution

1

Evaluate the inner function

g(n)

at

n = 3

g(3) = 3^2 - 3 = 9 - 3 = 6

Before evaluating the custom operator, the inputs

g(3)

and

g(2)

must be explicitly calculated.

2

Evaluate the inner function

g(n)

at

n = 2

g(2) = 2^2 - 3 = 4 - 3 = 1

Determine the second operand for the custom operation.

3

Apply the custom operation

\Delta

to the results from Step 1 and Step 2

6 \Delta 1 = (6 + 1)(1 - 2) = 7 \times (-1) = -7

Substitute

x = 6

and

y = 1

into the formula

x \Delta y = (x + 1)(y - 2)

.

Key Concept

Evaluating functions and applying non-commutative custom operators

Question 733Question →

For how many integer values of $k$ does the quadratic equation $(k - 2)x^2 - 2(k + 1)x + (k + 3) = 0$ have two distinct real roots $\alpha$ and \beta $such that$ \frac{1}{\alpha} + \frac{1}{\beta} < 1$?

2

3

5

6

7

Show answer & explanation

Answer: 3

Answer

There are exactly 3 integer values of

k

that satisfy all given conditions.

To find the number of valid integer values for

k

, we apply the condition for distinct real roots (

D > 0

), which yields

k > -7

with

k \neq 2

. Expressing the sum of reciprocals as

\frac{\alpha + \beta}{\alpha\beta} = \frac{2(k+1)}{k+3} < 1

, solving

\frac{k-1}{k+3} < 0

yields

-3 < k < 1

. The integer values within this range are

-2, -1,

and

0

, giving a total of 3 valid integer values.

Step-by-Step Solution

1

Ensure the given equation is a valid quadratic equation and determine the condition for two distinct real roots.

For the equation to be quadratic, the coefficient of

x^2

must be non-zero, so

k - 2 \neq 0 \implies k \neq 2

. For distinct real roots, the discriminant

D > 0

.

D = [-2(k+1)]^2 - 4(k-2)(k+3) = 4(k^2 + 2k + 1) - 4(k^2 + k - 6) = 4(k + 7)

. Thus,

4(k+7) > 0 \implies k > -7

.

A quadratic equation has two distinct real roots if and only if its discriminant is strictly positive and the leading coefficient is non-zero.

2

Apply Vieta's formulas to express the sum and product of the roots in terms of

k

.

\alpha + \beta = \frac{2(k+1)}{k-2}

and

\alpha\beta = \frac{k+3}{k-2}

. Note that for

\frac{1}{\alpha} + \frac{1}{\beta}

to exist, neither root can be 0, which requires

\alpha\beta
eq 0 \implies k
eq -3$.

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

3

Simplify the condition

\frac{1}{\alpha} + \frac{1}{\beta} < 1

using Vieta's relations.

\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{\frac{2(k+1)}{k-2}}{\frac{k+3}{k-2}} = \frac{2(k+1)}{k+3}

. We need

\frac{2k+2}{k+3} < 1$.

Combining the fractions over a common denominator allows direct substitution of the root sum and product.

4

Solve the algebraic inequality

\frac{2k+2}{k+3} < 1

.

\frac{2k+2}{k+3} - 1 < 0 \implies \frac{(2k+2) - (k+3)}{k+3} < 0 \implies \frac{k-1}{k+3} < 0

. The fraction is negative when the numerator and denominator have opposite signs, which gives the interval

-3 < k < 1$.

Subtracting 1 to get 0 on one side prevents sign errors that occur when cross-multiplying variable expressions.

5

Combine all constraints on

k

and count the valid integer values.

Constraints on

k

: (1)

k > -7

, (2)

k \neq 2

, (3)

k \neq -3

, and (4)

-3 < k < 1

. Intersecting these gives

-3 < k < 1

. The integer values in this open interval are

k = -2, -1, 0

. Total count = 3.

All structural and domain restrictions must be satisfied simultaneously.

Key Concept

Quadratic Equations and Polynomial Factoring

Estimated Time:3m 0s

Question 734Question →

In a game, a fair wheel lands on a single integer from $1$ to $100$ , inclusive, with each integer being equally likely to be selected. What is the probability that the integer selected is either a prime number or leaves a remainder of $1$ when divided by $5$ , but not both?

\frac{7}{20}

\frac{17}{50}

\frac{2}{5}

\frac{9}{20}

\frac{1}{20}

Show answer & explanation

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

Answer

The correct answer is 7/20.

The total number of possible outcomes is 100. There are 25 prime numbers in the range 1 to 100, and 20 numbers that leave a remainder of 1 when divided by 5. Exactly 5 numbers (11, 31, 41, 61, 71) meet both conditions. To find the numbers that meet either condition but NOT both, subtract the 5 overlapping numbers from each category: 20 numbers are prime only, and 15 numbers leave a remainder of 1 only. The number of favorable outcomes is 20 + 15 = 35. Therefore, the single-event probability is 35/100 = 7/20.

Step-by-Step Solution

1

Determine the total sample space size.

The total number of possible outcomes is

N = 100

.

Each integer from 1 to 100 inclusive is equally likely.

2

Identify and count all prime numbers in the set

\{1, 2, \dots, 100\}

.

There are 25 prime numbers:

\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\}

.

Note that 1 is not a prime number.

3

Identify and count all integers in

\{1, 2, \dots, 100\}

that leave a remainder of 1 when divided by 5.

There are 20 such integers:

\{1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96\}

.

These integers take the form

5k + 1

for integers

k \ge 0

where

1 \le 5k + 1 \le 100

.

4

Find the intersection of both sets (numbers that are prime AND leave a remainder of 1 when divided by 5).

The 5 numbers satisfying both conditions are

\{11, 31, 41, 61, 71\}

.

Any integer leaving a remainder of 1 when divided by 5 must end in 1 or 6. No prime ends in 6. The primes ending in 1 in this range are 11, 31, 41, 61, and 71.

5

Calculate the number of favorable outcomes satisfying 'either condition, but not both' (symmetric difference).

Favorable outcomes =

(25 - 5) + (20 - 5) = 20 + 15 = 35

.

We must count numbers that are only prime (20) plus numbers that are only remainder 1 (15), excluding the 5 overlapping numbers.

6

Calculate the single-event probability.

P = \frac{35}{100} = \frac{7}{20}

.

Divide favorable outcomes by total sample space size.

Key Concept

Basic single-event probability with set operations and number properties (prime numbers, remainders).

Estimated Time:2m 0s

Question 735Question →

A pharmaceutical production facility utilizes three automated synthesis modules—Module A, Module B, and Module C—to produce a specific batch of medication. Working alone at its constant rate, Module A can complete a full batch in $15$ hours. Modules B and C, working together at their respective constant rates, can complete a full batch in $20$ hours. A production run begins with Module A and Module B working together for $4$ hours. At that point, Module A is turned off for recalibration, and Module C immediately joins Module B to complete the remaining portion of the batch. If Modules B and C work together for $12$ hours to finish the batch, how many hours would it take Module B to complete a full batch working alone?

20

hours

25

hours

30

hours

40

hours

60

hours

Show answer & explanation

Answer: $30$ hours

Answer

Module B would take

30

hours to complete a full batch working alone.

The correct answer is

30

hours. In Stage 2, Modules B and C work together for

12

hours at their combined rate of

\frac{1}{20}

batch per hour, completing

\frac{12}{20} = \frac{3}{5}

of the entire batch. This leaves

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

of the batch that was completed during Stage 1 by Modules A and B working together for

4

hours. Setting up the Stage 1 equation:

4(r_A + r_B) = \frac{2}{5}

. Substituting Module A's rate

r_A = \frac{1}{15}

, we get

4\left(\frac{1}{15} + r_B\right) = \frac{2}{5} \implies \frac{4}{15} + 4r_B = \frac{6}{15} \implies 4r_B = \frac{2}{15} \implies r_B = \frac{1}{30}

. Therefore, Module B takes

30

hours operating alone.

Step-by-Step Solution

1

Express the individual and combined rates of the modules

Rate of Module A (

r_A

) =

\frac{1}{15}

batch/hr; Combined rate of Modules B and C (

r_B + r_C

) =

\frac{1}{20}

batch/hr.

Work rate is defined as the fraction of the job completed per unit of time.

2

Calculate the work completed during Stage 2

Work in Stage 2 =

12 \times (r_B + r_C) = 12 \times \frac{1}{20} = \frac{3}{5}

of the batch.

Modules B and C worked together for

12

hours at their known combined rate of

\frac{1}{20}

batch/hr.

3

Calculate the work done in Stage 1 and solve for Module B's rate (

r_B

)

Total Work = Stage 1 Work + Stage 2 Work =

1

. Thus,

4(r_A + r_B) + \frac{3}{5} = 1 \implies 4\left(\frac{1}{15} + r_B\right) = \frac{2}{5} \implies \frac{4}{15} + 4r_B = \frac{6}{15} \implies 4r_B = \frac{2}{15} \implies r_B = \frac{1}{30}

batch/hr.

The sum of the work performed across both stages must equal

1

full batch.

4

Convert Module B's rate into total time required working alone

Time for Module B alone =

\frac{1}{r_B} = 30

hours.

Total time working alone is the reciprocal of the individual work rate.

Key Concept

Work Rate and Combined Work

Question 736Question →

At an international conference of $200$ delegates, every delegate speaks at least one of three languages: French, German, or Spanish. Exactly $115$ delegates speak French, $85$ speak German, and $67$ speak Spanish. Furthermore, exactly $15$ delegates speak all three languages. If the number of delegates who speak only one language is four times the number of delegates who speak exactly two languages, how many delegates speak exactly two of these three languages?

Show answer & explanation

Answer: 37

Answer

The number of delegates who speak exactly two of the three languages is 37.

By dividing the 200 delegates into three disjoint groups—those speaking exactly 1 language (

E_1

), exactly 2 languages (

E_2

), and all 3 languages (

E_3 = 15

)—we establish that

E_1 + E_2 + 15 = 200

, giving

E_1 + E_2 = 185

. Given

E_1 = 4E_2

, substituting yields

5E_2 = 185

, which gives

E_2 = 37

.

Step-by-Step Solution

1

Define exact-count variables for the overlapping sets

Let

E_1

be the number of delegates speaking exactly one language,

E_2

be the number speaking exactly two languages, and

E_3 = 15

be the number speaking all three languages.

Categorizing elements into mutually exclusive subsets (exactly 1, exactly 2, exactly 3) simplifies 3-set inclusion-exclusion problems.

2

Set up the total population equation

E_1 + E_2 + E_3 = 200 \implies E_1 + E_2 + 15 = 200 \implies E_1 + E_2 = 185

Since every delegate speaks at least one language, the sum of all mutually exclusive regions equals the total number of delegates.

3

Apply the given relationship between

E_1

and

E_2

Substitute

E_1 = 4E_2

into

E_1 + E_2 = 185

to get

4E_2 + E_2 = 185 \implies 5E_2 = 185

The problem establishes that the number of single-language speakers is four times the number of dual-language speakers.

4

Solve for the target quantity

E_2

E_2 = 37

Dividing 185 by 5 yields the exact count of delegates speaking exactly two languages.

Key Concept

Three-Set Inclusion-Exclusion Principle using Exact-Overlap Subsets

Question 737Question →

A software engineering team uses three automated refactoring scripts—Script A, Script B, and Script C—to update a large codebase. Working alone at their respective constant rates, Script A can finish the entire codebase in $10$ hours, Script B in $15$ hours, and Script C in $30$ hours. All three scripts begin running simultaneously. After $2$ hours, Script A encounters an error and stops running. Script B and Script C continue running together without interruption until the codebase is completely refactored. How many additional hours will it take for Script B and Script C to complete the remaining work?

Show answer & explanation

Answer: 6

Answer

6 hours

The rate of Script A is

\frac{1}{10}

of the codebase per hour, Script B is

\frac{1}{15}

per hour, and Script C is

\frac{1}{30}

per hour. Working together for

2

hours, all three scripts complete

2 \times \left(\frac{1}{10} + \frac{1}{15} + \frac{1}{30}\right) = 2 \times \frac{1}{5} = \frac{2}{5}

of the codebase, leaving

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

of the task remaining. With Script A offline, the combined rate of Script B and Script C is

\frac{1}{15} + \frac{1}{30} = \frac{1}{10}

per hour. The additional time required to complete the remaining work is

\frac{3/5}{1/10} = 6

hours.

Step-by-Step Solution

1

Determine the individual work rates of Script A, Script B, and Script C.

Rate of Script A =

\frac{1}{10}

codebase/hour, Rate of Script B =

\frac{1}{15}

codebase/hour, Rate of Script C =

\frac{1}{30}

codebase/hour.

Work rate is the reciprocal of the total time required to complete one entire job working alone.

2

Calculate the combined work rate of all three scripts during the initial 2-hour period.

Combined Rate(A+B+C) =

\frac{1}{10} + \frac{1}{15} + \frac{1}{30} = \frac{3 + 2 + 1}{30} = \frac{6}{30} = \frac{1}{5}

codebase/hour.

When multiple agents work together simultaneously, their individual work rates are added.

3

Determine the fraction of the codebase refactored during the first 2 hours.

Work Completed =

2 \text{ hours} \times \frac{1}{5} \text{ codebase/hour} = \frac{2}{5}

of the total codebase.

Total work done equals rate multiplied by time.

4

Calculate the remaining fraction of the codebase left to be processed.

Remaining Work =

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

of the total codebase.

Subtracting the completed fraction from 1 gives the uncompleted portion of the job.

5

Calculate the combined rate of Script B and Script C after Script A stops.

Rate(B+C) =

\frac{1}{15} + \frac{1}{30} = \frac{2 + 1}{30} = \frac{3}{30} = \frac{1}{10}

codebase/hour.

Only Script B and Script C contribute to the remaining work.

6

Calculate the additional time required for Script B and Script C to finish the remaining work.

Additional Time =

\frac{3/5}{1/10} = \frac{3}{5} \times 10 = 6

hours.

Time equals remaining work divided by the active combined rate.

Key Concept

Work Rate and Combined Work

Estimated Time:1m 30s

Question 738Question →

If $x = 1$ is a root of the cubic polynomial $P(x) = x^3 + ax^2 - 11x + 12$ , where $a$ is a constant, what is the value of the largest real root of $P(x)$ ?

Show answer & explanation

Answer: 4

Answer

The largest real root of the polynomial

P(x)

is

4

.

Evaluating

P(1) = 0

yields

1 + a - 11 + 12 = 0

, giving

a = -2

. Factoring the polynomial

x^3 - 2x^2 - 11x + 12

by

(x - 1)

produces the quadratic expression

x^2 - x - 12

. Factoring this quadratic gives

(x - 4)(x + 3) = 0

. The roots of the polynomial are

-3

,

1

, and

4

. Comparing these values, the maximum root is

4

.

Step-by-Step Solution

1

Find the constant

a

using the given root

x = 1

.

a = -2

Since

x = 1

is a root,

P(1) = 1^3 + a(1)^2 - 11(1) + 12 = 0

, which simplifies to

a + 2 = 0

.

2

Factor

(x - 1)

out of

P(x) = x^3 - 2x^2 - 11x + 12

.

P(x) = (x - 1)(x^2 - x - 12)

Dividing

x^3 - 2x^2 - 11x + 12

by

(x - 1)

yields the quotient

x^2 - x - 12

.

3

Factor the quadratic factor

x^2 - x - 12

.

(x - 4)(x + 3)

The quadratic factors into two linear terms whose roots are

4

and

-3

.

4

Determine the largest of all real roots of

P(x)

.

4

The complete set of real roots is

\{-3, 1, 4\}

, of which

4

is the greatest.

Key Concept

Polynomial Factoring and Remainder/Factor Theorem

Question 739Question →

An agricultural processing facility uses three independent conveyor systems—Conveyor $X$ , Conveyor $Y$ , and Conveyor $Z$ —to fill a grain storage silo. Working together at their respective constant rates, Conveyors $X$ and $Y$ can fill the empty silo in $12$ hours, while Conveyors $Y$ and $Z$ working together can fill the empty silo in $20$ hours. Initially, Conveyors $X$ and $Z$ work together for $5$ hours, completing exactly $\frac{1}{3}$ of the silo. How many hours would it take Conveyor $Y$ operating alone to fill the remaining $\frac{2}{3}$ of the silo?

Show answer & explanation

Answer: 20

Answer

It would take Conveyor

Y

operating alone

20

hours to fill the remaining

\frac{2}{3}

of the silo.

By representing the rate of each pair of conveyors as a fraction of the total job per hour, we find

R_X + R_Y = 1/12

,

R_Y + R_Z = 1/20

, and

R_X + R_Z = 1/15

. Summing these three equations yields

2(R_X + R_Y + R_Z) = 1/5

, so

R_X + R_Y + R_Z = 1/10

. Subtracting

R_X + R_Z = 1/15

gives

R_Y = 1/30

silo per hour. To fill the remaining

2/3

of the silo, Conveyor

Y

requires

(2/3) / (1/30) = 20

hours.

Step-by-Step Solution

1

Express the combined rate of each pair of conveyors as a fraction of the silo filled per hour.

R_X + R_Y = \frac{1}{12}

,

R_Y + R_Z = \frac{1}{20}

, and

R_X + R_Z = \frac{1/3}{5} = \frac{1}{15}

.

Work rate equals work done divided by time taken.

2

Sum the three pairwise rates to determine the combined rate of all three conveyors working together.

2(R_X + R_Y + R_Z) = \frac{1}{12} + \frac{1}{20} + \frac{1}{15} = \frac{12}{60} = \frac{1}{5} \implies R_X + R_Y + R_Z = \frac{1}{10}

.

Adding pairwise rates counts each conveyor's individual rate twice.

3

Subtract the combined rate of Conveyors

X

and

Z

from the total rate of all three conveyors to isolate the rate of Conveyor

Y

.

R_Y = \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30}

of the silo per hour.

Subtracting

(R_X + R_Z)

from

(R_X + R_Y + R_Z)

yields

R_Y

directly.

4

Divide the remaining fraction of work by Conveyor

Y

's individual rate to find the required time.

\text{Time} = \frac{2/3}{1/30} = 20

hours.

Time required equals remaining work divided by the individual work rate.

Key Concept

Solving systems of simultaneous work rate equations by summing pairwise rates.

Question 740Question →

A security analyst needs to generate a 6-symbol access code using all of the following characters: $A, A, A, B, B,$ and $C$ . How many distinct 6-symbol access codes can be formed such that the two $B$ 's are not adjacent to each other?

20

24

40

60

120

Show answer & explanation

Answer: 40

Answer

40 distinct 6-symbol access codes can be formed.

First arrange the four symbols that have no adjacency restrictions (

A, A, A,

and

C

). The number of unique ways to arrange these 4 symbols with 3 identical

A

's is

\frac{4!}{3!} = 4

. Placing these 4 symbols in a row creates 5 available spaces (one before the first character, three between characters, and one after the last character). To ensure the two

B

's are not adjacent, choose 2 distinct spaces out of the 5 for the

B

's. The number of ways to select 2 spaces out of 5 is

\binom{5}{2} = 10

. Multiplying the arrangement of base characters by the space selections yields

4 \times 10 = 40

total valid codes.

Step-by-Step Solution

1

Arrange the non-restricted characters

There are 4 ways to arrange the remaining characters

A, A, A,

and

C

.

The number of distinct arrangements of 4 characters where 3 are identical is

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

.

2

Determine the available slots for the restricted character

Arranging 4 characters creates 5 distinct slots (including the two ends): _ X _ X _ X _ X _.

To ensure no two

B

's are adjacent, each

B

must occupy a distinct slot around or between the arranged characters.

3

Calculate the combinations of slot positions

There are

\binom{5}{2} = 10

ways to choose 2 slots for the two identical

B

's.

Since the two

B

's are identical, order of selection within the slots does not matter.

4

Apply the Fundamental Counting Principle

4 \times 10 = 40

distinct codes.

Multiply the number of ways to arrange the base characters by the number of ways to place the

B

's into the available slots.

Key Concept

Counting with restrictions using the slotting method for non-adjacent elements

Answer

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