Algebraic Word Problems and Modeling Alıştırma Soruları — GRE General Test — Sayfa 3

Soru 41Soru →

A logistics company uses two delivery vehicles, Vehicle P and Vehicle Q, to transport cargo between two warehouses that are $240$ miles apart. Vehicle P travels at a constant average speed of $r$ miles per hour ( $r > 0$ ). Vehicle Q travels at a constant average speed that is $20$ miles per hour faster than that of Vehicle P. In addition to driving time, Vehicle P requires a flat setup time of $1$ hour before departure, while Vehicle Q requires a flat setup time of $2$ hours before departure.

Let $T_P(r)$ and $T_Q(r)$ represent the total elapsed time in hours (including setup time) required for Vehicle P and Vehicle Q to complete the trip, respectively.

Which of the following statements are true? Select all such statements.

Geçerli olan tümünü seçin

The total elapsed time for Vehicle P, in hours, as a function of its speed

r

, is given by

T_P(r) = \frac{240 + r}{r}

.

Vehicle P and Vehicle Q take the exact same total elapsed time to complete the trip when

r = 60

miles per hour.

For all positive speeds

r > 0

, Vehicle Q always requires strictly less total elapsed time than Vehicle P to complete the trip.

If

r = 30

miles per hour, the travel time (excluding setup time) for Vehicle Q is half the travel time (excluding setup time) for Vehicle P.

Doubling the speed of Vehicle P from

r = 40

miles per hour to

r = 80

miles per hour reduces its total elapsed time

T_P(r)

by

50\%

.

Cevabı ve açıklamayı göster

Cevap: The total elapsed time for Vehicle P, in hours, as a function of its speed $r$ , is given by $T_P(r) = \frac{240 + r}{r}$ .; Vehicle P and Vehicle Q take the exact same total elapsed time to complete the trip when $r = 60$ miles per hour.

Cevap

The correct statements are the algebraic expression for Vehicle P's total time as a function of speed and the equal total elapsed time condition at a speed of 60 miles per hour.

The expression for Vehicle P's total time accurately combines

1

hour of setup time with

\frac{240}{r}

driving hours to get

\frac{240+r}{r}

. Setting

T_P(r) = T_Q(r)

yields the quadratic equation

r^2 + 20r - 4800 = 0

, which correctly solves to

r = 60

mph, at which point both vehicles require exactly

5

hours total.

Adım Adım Çözüm

1

Model total elapsed time functions

T_P(r)

and

T_Q(r)

using setup time plus travel time.

T_P(r) = 1 + \frac{240}{r} = \frac{240 + r}{r}

and

T_Q(r) = 2 + \frac{240}{r + 20}

.

Total elapsed time is the sum of fixed pre-departure setup overhead and variable driving time.

2

Determine the speed

r

where total elapsed times are equal by setting

T_P(r) = T_Q(r)

.

1 + \frac{240}{r} = 2 + \frac{240}{r + 20} \implies \frac{240}{r} - \frac{240}{r + 20} = 1 \implies 240 \cdot 20 = r(r + 20) \implies r^2 + 20r - 4800 = 0

. Factoring gives

(r - 60)(r + 80) = 0

, so

r = 60

mph.

Solving the rational equation identifies the exact breakeven speed where higher travel efficiency balances extra setup time.

3

Evaluate the remaining candidate assertions against the derived model.

For

r > 60

,

T_Q(r) > T_P(r)

, invalidating the claim that Vehicle Q is always faster. At

r = 40

,

T_P(40) = 7

hours and

T_P(80) = 4

hours, giving a decrease of

\frac{3}{7} \approx 42.86\%

, invalidating the

50\%

reduction claim.

Fixed setup costs distort simple constant-proportion and percentage changes in total time.

Anahtar Kavram

Algebraic Modeling of Combined Time with Fixed Overhead and Variable Rates

Soru 42Soru →

A water purification facility uses a primary filtration system and a secondary filtration system to process untreated water. The primary system operates at a constant rate that is $25\%$ faster than the secondary system. Working together at their normal constant rates, both systems can process a full reservoir of $36,000$ gallons in $8$ hours.

On a day when the primary system operates at only $80\%$ of its normal rate due to maintenance while the secondary system operates at its normal rate, both systems work together for $6$ hours. At that point, the primary system is shut down completely. How many additional hours will it take the secondary system, working alone at its normal rate, to process the remainder of the reservoir?

4.5

hours

5.0

hours

6.0

hours

7.2

hours

8.0

hours

Cevabı ve açıklamayı göster

Cevap: $6.0$ hours

Cevap

6.0

hours

The correct answer of

6.0

hours is derived by establishing that the combined normal processing rate is

4,500

gal/hr. Since the primary system is

25\%

faster than the secondary system (

P = 1.25S

), the secondary rate is

2,000

gal/hr and the primary rate is

2,500

gal/hr. At

80\%

efficiency, the primary system operates at

2,000

gal/hr, making the joint rate

4,000

gal/hr. In

6

hours,

24,000

gallons are processed, leaving

12,000

gallons. The secondary system working alone at

2,000

gal/hr processes the remaining volume in exactly

6.0

hours.

Adım Adım Çözüm

1

Determine the combined normal operating rate and set up individual rates.

Combined rate =

4,500

gal/hr; Secondary rate =

2,000

gal/hr; Primary rate =

2,500

gal/hr.

The combined rate is

\frac{36,000\text{ gallons}}{8\text{ hours}} = 4,500\text{ gal/hr}

. Let

S

be the secondary rate. The primary rate is

1.25S

. Thus,

S + 1.25S = 2.25S = 4,500

, yielding

S = 2,000\text{ gal/hr}

and

P = 2,500\text{ gal/hr}

.

2

Calculate the reduced primary rate and the total water processed in the first 6 hours.

24,000

gallons processed in the first 6 hours.

During maintenance, the primary system operates at

80\%

of

2,500\text{ gal/hr}

, which is

0.80 \times 2,500 = 2,000\text{ gal/hr}

. The combined rate during this period is

2,000 + 2,000 = 4,000\text{ gal/hr}

. Over 6 hours, the volume processed is

4,000 \times 6 = 24,000\text{ gallons}

.

3

Find the remaining volume of water to process.

12,000

gallons remaining.

Subtract the volume processed from the total reservoir capacity:

36,000 - 24,000 = 12,000\text{ gallons}

.

4

Calculate the time required for the secondary system alone to process the remainder.

6.0

hours.

Divide the remaining volume by the normal secondary system rate

S = 2,000\text{ gal/hr}

:

\frac{12,000\text{ gallons}}{2,000\text{ gal/hr}} = 6.0\text{ hours}

.

Anahtar Kavram

Linear rate modeling and percentage rate adjustment in combined work problems

Tahmini Süre:2m 30s

Soru 43Soru →

A theater sold a total of $500$ tickets for an evening performance, consisting of VIP tickets priced at $\$80$ each and General Admission tickets priced at $\$50$ each. On the day of the show, a promotional discount of $20\%$ was applied to all General Admission tickets, while VIP ticket prices remained unchanged. If the total revenue collected from ticket sales was $\$31,000$ , which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

More than 70% of the total revenue was generated from VIP ticket sales.

The number of VIP tickets sold was 50 greater than the number of General Admission tickets sold.

The ratio of VIP tickets sold to General Admission tickets sold was 11 to 7.

Applying the 20% discount to General Admission tickets reduced the total potential revenue by 20%.

The number of General Admission tickets sold was 250.

Cevabı ve açıklamayı göster

Cevap: More than 70% of the total revenue was generated from VIP ticket sales.; The number of VIP tickets sold was 50 greater than the number of General Admission tickets sold.

Cevap

The statements confirming that more than 70% of total revenue came from VIP ticket sales and that 50 more VIP tickets were sold than General Admission tickets are correct.

The system of equations

V + G = 500

and

80V + 40G = 31,000

yields

V = 275

VIP tickets and

G = 225

General Admission tickets. The VIP revenue is

22,000, which is approximately 70.97% of the total

31,000 revenue (greater than 70%). Additionally, the difference

275 - 225 = 50

confirms that 50 more VIP tickets were sold than General Admission tickets.

Adım Adım Çözüm

1

Define variables and determine the discounted price of General Admission tickets.

Let

V

be the number of VIP tickets and

G

be the number of General Admission tickets. The discounted price for General Admission tickets is

\$50 \times (1 - 0.20) = \$40

.

Establishing correct variable representations and effective unit prices is required to build the revenue model.

2

Set up and solve the system of linear equations.

V + G = 500

and

80V + 40G = 31,000

. Substituting

G = 500 - V

gives

80V + 40(500 - V) = 31,000 \implies 40V = 11,000 \implies V = 275

. Thus,

G = 225

.

Solving the linear system determines the exact quantity of each ticket type sold.

3

Calculate revenue shares and verify each statement.

VIP revenue =

275 \times \$80 = \$22,000

. Revenue share of VIP =

\frac{22,000}{31,000} \approx 70.97\% > 70\%

. Ticket difference =

275 - 225 = 50

. Ratio

V:G = 275:225 = 11:9

.

Evaluating each calculated metric against the given statements determines which statements must be true.

Anahtar Kavram

Linear word problems involving systems of equations and percentage modifications.

Tahmini Süre:2m 0s

Soru 44Soru →

A manufacturing plant operates two automated production lines, Assembly Line X and Assembly Line Y. Assembly Line Y operates at a standard rate that is $25\%$ faster than the standard rate of Assembly Line X. During a specific shift, Assembly Line X operated for $4$ hours at its standard rate, after which its processing rate decreased by $20\%$ for an additional $2$ hours due to maintenance. Assembly Line Y began operating $1$ hour after Line X started; it operated at its standard rate for $4$ hours, and then operated for another $2$ hours at $80\%$ of its standard rate. If the two lines produced a combined total of $2,772$ units during this shift, what was the standard operating rate of Assembly Line X, in units per hour?

Cevabı ve açıklamayı göster

Cevap: 220

Cevap

The standard operating rate of Assembly Line X is 220 units per hour.

Assembly Line X produces

4r

units in its first 4 hours and

1.6r

units in its final 2 hours, totaling

5.6r

units. Assembly Line Y produces

4 \times 1.25r = 5r

units in its first 4 hours and

2 \times (0.80 \times 1.25r) = 2r

units in its final 2 hours, totaling

7r

units. The sum of their outputs is

5.6r + 7r = 12.6r = 2,772

. Dividing

2,772

by

12.6

gives

r = 220

.

Adım Adım Çözüm

1

Set up rate expressions for both assembly lines using a single variable

Standard rate of Line X =

r

; Standard rate of Line Y =

1.25r

Line Y is 25% faster than Line X, so its rate is

r + 0.25r = 1.25r

.

2

Calculate the total work done by Assembly Line X

Line X output =

4(r) + 2(0.80r) = 5.6r

units

Line X worked 4 hours at 100% rate and 2 hours at 80% rate.

3

Calculate the total work done by Assembly Line Y

Line Y output =

4(1.25r) + 2(0.80 \times 1.25r) = 5r + 2r = 7r

units

Line Y worked 4 hours at full rate

1.25r

and 2 hours at 80% of

1.25r

, which equals rate

r

.

4

Equate combined production to 2,772 units and solve for

r

5.6r + 7r = 12.6r = 2,772 \implies r = 220

Dividing the total combined output by the total rate multiplier 12.6 yields the baseline standard rate.

Anahtar Kavram

Algebraic Modeling of Staggered Work and Variable Production Rates

Tahmini Süre:2m 30s

Soru 45Soru →

An agricultural facility uses two automated irrigation systems, System P and System Q, to water a field. System P pumps water at a constant rate of $120$ gallons per hour after requiring an initial setup overhead of $30$ minutes ( $0.5$ hours). System Q pumps water at a constant rate of $180$ gallons per hour after requiring an initial setup overhead of $45$ minutes ( $0.75$ hours). Both systems begin their setup process at the exact same time and operate continuously until a combined total of $1,605$ gallons of water has been pumped. Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

The total elapsed time from when setup began until the target volume was reached is

6

hours.

System Q pumped

285

more gallons of water than System P.

System P was actively pumping water for exactly

5

hours and

15

minutes.

System Q accounted for more than

60\%

of the total water pumped.

The ratio of the volume of water pumped by System P to that pumped by System Q is

44:63

.

Cevabı ve açıklamayı göster

Cevap: The total elapsed time from when setup began until the target volume was reached is $6$ hours.; System Q pumped $285$ more gallons of water than System P.; The ratio of the volume of water pumped by System P to that pumped by System Q is $44:63$ .

Cevap

The correct statements are: the total elapsed time from when setup began is 6 hours, System Q pumped 285 more gallons than System P, and the ratio of water pumped by System P to System Q is 44:63.

Solving the combined linear rate model yields a total elapsed time of 6 hours. With this duration, System P operates for 5.5 hours producing 660 gallons, and System Q operates for 5.25 hours producing 945 gallons. This confirms that the total time is 6 hours, System Q produces 285 more gallons than System P (945 - 660 = 285), and the ratio of System P's volume to System Q's volume is 660:945, which simplifies to 44:63.

Adım Adım Çözüm

1

Define the variable for total time and establish time expressions for active pumping.

Let

t

be the total elapsed time in hours since both systems began setup (

t \geq 0.75

). System P actively pumps for

(t - 0.5)

hours, and System Q actively pumps for

(t - 0.75)

hours.

Setup overhead delays the start of pumping, so active pumping duration equals total elapsed time minus setup time.

2

Set up and solve the linear rate equation for combined total volume.

120(t - 0.5) + 180(t - 0.75) = 1605 \implies 120t - 60 + 180t - 135 = 1605 \implies 300t - 195 = 1605 \implies 300t = 1800 \implies t = 6

hours.

The sum of the volumes produced by both systems must equal the target total of 1,605 gallons.

3

Calculate individual pumping times and volumes produced by each system.

System P: active time

= 5.5

hours, volume

= 120 \times 5.5 = 660

gallons. System Q: active time

= 5.25

hours, volume

= 180 \times 5.25 = 945

gallons.

Individual volumes are required to evaluate statements regarding volume differences, percentages, and ratios.

4

Evaluate each given statement against the calculated values.

Elapsed time is

6

hours (True). Volume difference is

945 - 660 = 285

gallons (True). Active pumping time for P is

5.5

hours, not

5.25

hours (False). Percentage for Q is

945 / 1605 \approx 58.88\% < 60\%

(False). Ratio P to Q is

660 : 945 = 44 : 63

(True).

Determines which of the statements must be selected.

Anahtar Kavram

Linear Modeling with Combined Work Rates and Staggered Start Times

Soru 46Soru →

A laboratory processes two types of chemical samples, Type A and Type B. Processing each Type A sample requires $40$ minutes and costs $\$30$ , while processing each Type B sample requires $30$ minutes and costs $\$50$ . On a given day, the laboratory spent a total of $20$ hours processing these two types of samples at a total cost of $\$1,450$ . How many more Type B samples were processed than Type A samples?

5

10

15

20

35

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5 more Type B samples were processed than Type A samples.

The correct answer is 5. By defining

x

as the number of Type A samples and

y

as the number of Type B samples, we construct the time equation

40x + 30y = 1,200

and cost equation

30x + 50y = 1,450

. Solving this system yields

x = 15

and

y = 20

. The difference

y - x = 20 - 15 = 5

.

Adım Adım Çözüm

1

Define variables and convert units to maintain consistency.

Let

x

be the number of Type A samples and

y

be the number of Type B samples. Total time available is

20 \text{ hours} \times 60 \text{ minutes/hour} = 1,200 \text{ minutes}

.

Time specifications for individual samples are given in minutes, so total time must also be in minutes.

2

Set up a system of two linear equations representing total time and total cost.

Time equation:

40x + 30y = 1,200 \implies 4x + 3y = 120

. Cost equation:

30x + 50y = 1,450 \implies 3x + 5y = 145

.

The total processing time is the sum of time spent on each sample type, and total cost is the sum of costs for each sample type.

3

Solve the system of equations using elimination.

Multiply the time equation by 3:

12x + 9y = 360

. Multiply the cost equation by 4:

12x + 20y = 580

. Subtracting the first from the second gives

11y = 220 \implies y = 20

. Substitute

y = 20

into

4x + 3(20) = 120 \implies 4x = 60 \implies x = 15

.

Eliminating

x

allows direct calculation of

y

, which then yields

x

.

4

Calculate the difference requested by the problem.

Difference =

y - x = 20 - 15 = 5

.

The question specifically asks for how many more Type B samples were processed than Type A samples.

Anahtar Kavram

Formulating and solving systems of linear equations from real-world rate and budget constraints.

Tahmini Süre:1m 45s

Soru 47Soru →

A commercial bakery uses two automated ovens, Oven X and Oven Y, to bake identical orders of bread. Working alone at its constant rate, Oven X bakes a full order of bread in $8$ hours, while Oven Y, working alone at its constant rate, bakes the same order in $12$ hours. Both ovens begin baking a full order together at 8:00 a.m. At 10:00 a.m., Oven X shuts down due to a maintenance alert, and Oven Y continues working alone at its constant rate until the order is completed. How many total hours, from 8:00 a.m. until completion, does it take to finish the order?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The total time required from 8:00 a.m. to complete the order is

9

hours.

Working together for

2

hours at a combined rate of

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

per hour completes

\frac{5}{12}

of the order. The remaining

\frac{7}{12}

of the order takes Oven Y

7

hours to complete at its rate of

\frac{1}{12}

per hour. Adding the initial

2

hours gives a total time of

9

hours.

Adım Adım Çözüm

1

Determine the individual hourly work rates.

Oven X completes

\frac{1}{8}

of the job per hour, and Oven Y completes

\frac{1}{12}

of the job per hour.

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

2

Calculate the fraction of the job completed in the first

2

hours.

Combined rate is

\frac{1}{8} + \frac{1}{12} = \frac{5}{24}

job per hour. In

2

hours, they complete

2 \times \frac{5}{24} = \frac{5}{12}

of the job.

Both ovens work simultaneously for

2

hours before Oven X stops.

3

Determine the remaining fraction of the job.

1 - \frac{5}{12} = \frac{7}{12}

of the job remains.

The full order represents

1

whole unit of work.

4

Find the additional time required for Oven Y to finish the remaining job alone.

\text{Time} = \frac{7/12}{1/12} = 7

hours.

Time equals remaining work divided by Oven Y's individual work rate.

5

Calculate the total time elapsed from start to completion.

2 + 7 = 9

hours.

The total time includes the

2

hours of combined work plus the

7

hours Oven Y worked alone.

Anahtar Kavram

Combined Work Rates and Modeling Staggered Work

Soru 48Soru →

A technology company manages data storage across three servers: Server X, Server Y, and Server Z. Initially, the amount of data stored on Server Y is $25\%$ greater than the amount stored on Server X, and Server Z stores $40$ gigabytes less data than Server Y. During a system reorganization, the data on Server X increases by $20\%$ , the data on Server Y decreases by $20\%$ , and the data on Server Z increases by $60$ gigabytes. If the total amount of data stored across all three servers after the reorganization is $10\%$ greater than the total initial amount of data, what was the initial amount of data, in gigabytes, stored on Server X?

120

150

160

192

200

Cevabı ve açıklamayı göster

Cevap: 160

Cevap

160 gigabytes

Let

x

be the initial storage on Server X in gigabytes. Then Server Y initially stores

1.25x

gigabytes and Server Z stores

1.25x - 40

gigabytes. The initial total storage is

T_1 = x + 1.25x + (1.25x - 40) = 3.5x - 40

. After reorganization, Server X stores

1.20x

, Server Y stores

0.80(1.25x) = 1.00x

, and Server Z stores

(1.25x - 40) + 60 = 1.25x + 20

. The new total storage is

T_2 = 1.20x + 1.00x + 1.25x + 20 = 3.45x + 20

. Since

T_2

is

10\%

greater than

T_1

, we have

3.45x + 20 = 1.10(3.5x - 40)

. Expanding the right side gives

3.45x + 20 = 3.85x - 44

. Rearranging terms yields

0.40x = 64

, which simplifies to

x = 160

gigabytes.

Adım Adım Çözüm

1

Define variables for the initial storage on each server in terms of the initial storage on Server X,

x

.

Server X =

x

, Server Y =

1.25x

, Server Z =

1.25x - 40

.

Server Y is 25% greater than Server X (

1 + 0.25 = 1.25

), and Server Z is 40 gigabytes less than Server Y.

2

Sum the initial storage values to find the total initial storage

T_1

.

T_1 = x + 1.25x + (1.25x - 40) = 3.5x - 40

.

Combining like terms gives the overall starting storage equation.

3

Express the storage on each server after the reorganization in terms of

x

.

Server X' =

1.20x

, Server Y' =

0.80(1.25x) = 1.00x

, Server Z' =

(1.25x - 40) + 60 = 1.25x + 20

.

Server X increases by 20%, Server Y decreases by 20% of its initial value, and Server Z gains 60 gigabytes.

4

Sum the updated storage values to find the new total storage

T_2

.

T_2 = 1.20x + 1.00x + (1.25x + 20) = 3.45x + 20

.

Adding the three updated amounts gives the new total expression.

5

Set up and solve the equation representing the

10\%

overall increase (

T_2 = 1.10 T_1

).

3.45x + 20 = 1.10(3.5x - 40) \implies 3.45x + 20 = 3.85x - 44 \implies 64 = 0.40x \implies x = 160

.

Solving the linear algebraic model yields the value for initial storage on Server X.

Anahtar Kavram

Linear algebraic modeling of dynamic multi-variable systems involving percentage changes.

Tahmini Süre:2m 30s

Soru 49Soru →

A logistics company operates a delivery van and a cargo drone along a straight route of $270$ kilometers connecting Hub X and Hub Y. The delivery van departs from Hub X toward Hub Y at a constant speed of $60$ kilometers per hour. Exactly $2$ hours later, the cargo drone departs from Hub Y toward Hub X along the same route at a constant speed of $90$ kilometers per hour. How many hours after the delivery van departs will the van and the cargo drone meet?

1.8 hours

2.6 hours

3.0 hours

4.5 hours

5.0 hours

Cevabı ve açıklamayı göster

Cevap: 3.0 hours

Cevap

3.0 hours

The option specifying '3.0 hours' is correct. During the 2 hours before the drone departs, the van covers

60 \text{ km/h} \times 2 \text{ h} = 120 \text{ km}

. The remaining distance between the vehicles is

270 \text{ km} - 120 \text{ km} = 150 \text{ km}

. Once the drone departs, the two vehicles close the gap at a combined rate of

60 + 90 = 150 \text{ km/h}

. The time needed to cover the remaining

150 \text{ km}

is

\frac{150 \text{ km}}{150 \text{ km/h}} = 1 \text{ hour}

. Adding the van's initial

2

hours gives a total travel time of

3.0

hours.

Adım Adım Çözüm

1

Define variables for travel time and compute distance covered by the van before the drone departs.

In

2

hours at

60

km/h, the van covers

60 \times 2 = 120

kilometers.

The van travels alone for the first two hours.

2

Determine the remaining distance separating the two vehicles when the drone starts moving.

Remaining distance =

270 - 120 = 150

kilometers.

The total distance between the hubs is

270

kilometers.

3

Calculate the relative speed of approach and the additional time required to meet.

Combined speed =

60 + 90 = 150

km/h. Additional time =

\frac{150}{150} = 1

hour.

Since the vehicles travel toward each other, their speeds add up.

4

Find the total time elapsed since the van departed.

Total time =

2 + 1 = 3.0

hours.

The question asks for total hours since the van's departure.

Anahtar Kavram

Distance-Rate-Time linear modeling with staggered start times and opposing directions

Tahmini Süre:1m 45s

Soru 50Soru →

An investment firm allocated a total principal of $\$120,000$ between two accounts, Account X and Account Y. Account X earned a simple annual interest rate of $6\%$ , and Account Y earned a simple annual interest rate of $10\%$ . At the end of one year, the combined total interest earned from both accounts was $\$9,200$ . An annual administrative fee was then deducted from the interest: a $25\%$ fee on the interest earned from Account X, and a $10\%$ fee on the interest earned from Account Y. What was the net amount of interest, in dollars, remaining after these administrative fees were deducted?

\$7,530

\$7,650

\$7,750

\$8,775

\$10,750

Cevabı ve açıklamayı göster

Cevap: $\$7,650$

Cevap

The net amount of interest remaining after administrative fees were deducted is

\$7,650

.

The correct answer is

\$7,650

. Solving the system

x + y = 120,000

and

0.06x + 0.10y = 9,200

gives

\$70,000

invested in Account X and

\$50,000

in Account Y. Account X produces

\$4,200

in interest, from which a

25\%

fee (

\$1,050

) is deducted. Account Y produces

\$5,000

in interest, from which a

10\%

fee (

\$500

) is deducted. Subtracting total fees of

\$1,550

from total interest of

\$9,200

yields

\$7,650

.

Adım Adım Çözüm

1

Set up a system of linear equations for the principal amounts invested in Account X and Account Y.

Let

x

be the amount invested in Account X and

y

be the amount invested in Account Y. The total principal is

x + y = 120,000

. The total interest earned is

0.06x + 0.10y = 9,200

.

Modeling the word problem using two variables captures both total principal and total combined interest.

2

Solve the system of equations for

x

and

y

.

Multiplying

x + y = 120,000

by

0.06

gives

0.06x + 0.06y = 7,200

. Subtracting this equation from

0.06x + 0.10y = 9,200

yields

0.04y = 2,000

, so

y = 50,000

. Substituting back gives

x = 70,000

.

Determining individual account principals allows calculation of interest generated by each account.

3

Calculate the interest earned from each account and the corresponding administrative fees.

Interest from X =

0.06 \times 70,000 = \$4,200

. Interest from Y =

0.10 \times 50,000 = \$5,000

. Fee for X =

0.25 \times 4,200 = \$1,050

. Fee for Y =

0.10 \times 5,000 = \$500

. Total fees =

1,050 + 500 = \$1,550

.

Each fee rate must be multiplied by the specific interest amount earned by that account.

4

Subtract the total administrative fees from the total interest earned.

Net Interest =

\$9,200 - \$1,550 = \$7,650

.

Deducting fees yields the net interest remaining.

Anahtar Kavram

System of Linear Equations for Mixture and Interest Word Problems

Tahmini Süre:2m 0s

Soru 51Soru →

A commercial print shop uses two high-speed printing presses, Press Alpha and Press Beta. Press Alpha operates at a constant rate of $120$ pages per minute, while Press Beta operates at a constant rate of $180$ pages per minute. Press Alpha begins printing a job of $15,000$ pages at 9:00 AM. At 9:15 AM, Press Beta is turned on to assist Press Alpha, and both presses continue printing simultaneously at their respective constant rates until the job is completed. How many total minutes after 9:00 AM will the entire $15,000$ -page job be finished?

Cevabı ve açıklamayı göster

Cevap: 59

Cevap

The entire 15,000-page job will be finished 59 minutes after 9:00 AM.

Press Alpha operates alone for the first 15 minutes, completing

15 \times 120 = 1,800

pages. That leaves

15,000 - 1,800 = 13,200

pages. Once Press Beta joins at 9:15 AM, the combined rate becomes

120 + 180 = 300

pages per minute. The remaining pages require

13,200 / 300 = 44

minutes. Summing the 15-minute initial period and the 44-minute joint period gives a total of 59 minutes after 9:00 AM.

Adım Adım Çözüm

1

Find the work completed by Press Alpha during the 15-minute staggered start period.

1,800 pages completed.

Press Alpha ran alone for 15 minutes at 120 pages per minute.

2

Determine the remaining work to be done after 9:15 AM.

13,200 pages remaining.

Subtract the completed pages from the total batch size of 15,000 pages.

3

Calculate the combined work rate of Press Alpha and Press Beta.

300 pages per minute.

When working together, rates add linearly: 120 + 180 = 300.

4

Calculate time needed to complete the remaining pages.

44 minutes.

Divide remaining work (13,200 pages) by combined rate (300 pages/min).

5

Calculate total elapsed time from 9:00 AM.

59 minutes.

Combine the 15 initial minutes with the 44 subsequent minutes.

Anahtar Kavram

Linear work-rate equations with staggered initial start times

Tahmini Süre:1m 30s

Soru 52Soru →

Two industrial processing units, Unit X and Unit Y, process liquid solution at constant rates of $40$ liters per hour and $60$ liters per hour, respectively. Unit X begins processing a $1,100$ -liter batch of solution alone. After $5$ hours, Unit Y is activated and joins Unit X, working simultaneously at their respective constant rates until the entire batch is completely processed. What is the total time, in hours, from when Unit X started processing until the entire batch was completely processed?

9

11

14

16

20

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14 hours

Unit X works alone for 5 hours, processing

40 \times 5 = 200

liters. This leaves

1,100 - 200 = 900

liters of solution remaining. When Unit Y joins, the units process solution at a joint rate of

40 + 60 = 100

liters per hour. The remaining 900 liters require

\frac{900}{100} = 9

hours of combined work. Summing the 5 hours of solo work and 9 hours of combined work gives a total time of 14 hours.

Adım Adım Çözüm

1

Calculate the volume processed by Unit X during its solo operation

40 \text{ liters/hour} \times 5 \text{ hours} = 200 \text{ liters}

Unit X operates alone for the first 5 hours at a constant rate of 40 liters per hour.

2

Determine the remaining batch volume to be processed

1,100 \text{ liters} - 200 \text{ liters} = 900 \text{ liters}

Subtract the volume already completed from the total batch size.

3

Calculate the combined processing rate of Unit X and Unit Y

40 \text{ liters/hour} + 60 \text{ liters/hour} = 100 \text{ liters/hour}

Both units work together simultaneously after the first 5 hours.

4

Find the time needed for both units to process the remaining volume

\frac{900 \text{ liters}}{100 \text{ liters/hour}} = 9 \text{ hours}

Divide the remaining volume by the combined rate.

5

Calculate the total elapsed time from the start

5 \text{ hours (solo)} + 9 \text{ hours (combined)} = 14 \text{ hours}

Add the initial solo work time to the combined work time.

Anahtar Kavram

Algebraic modeling of combined work rates with staggered start times

Tahmini Süre:1m 30s

Soru 53Soru →

A cyclist travels from City $X$ to City $Y$ at a constant speed of $24$ miles per hour. On the return trip from City $Y$ to City $X$ along the exact same route, adverse weather conditions reduce the cyclist's average speed by $25\%$ . If the total time for the entire round trip is $7$ hours, what is the distance, in miles, between City $X$ and City $Y$ ?

73.5

96

72

126

144

Cevabı ve açıklamayı göster

Cevap: $72$

Cevap

The distance between City

X

and City

Y

is

72

miles.

To find the one-way distance

d

, first calculate the return speed:

24 \times 0.75 = 18

mph. Express the total time spent traveling as the sum of the time for each leg:

\frac{d}{24} + \frac{d}{18} = 7

. Combining the fractions over a common denominator of

72

gives

\frac{7d}{72} = 7

, which simplifies to

d = 72

miles.

Adım Adım Çözüm

1

Determine the return speed

Return speed =

24 \times (1 - 0.25) = 18

miles per hour.

Adverse weather reduces the outbound speed of

24

mph by

25\%

.

2

Formulate the total time equation in terms of distance

d

\frac{d}{24} + \frac{d}{18} = 7

Time equals distance divided by rate. The sum of the outbound time and return time is

7

hours.

3

Solve the algebraic equation for

d

\frac{3d + 4d}{72} = 7 \implies \frac{7d}{72} = 7 \implies d = 72

Finding a common denominator of

72

allows combining the fractional time expressions.

Anahtar Kavram

Distance, Rate, and Time Word Problems

Tahmini Süre:1m 30s

Soru 54Soru →

A theater sells student tickets for $\$20$ each and adult tickets for $\$35$ each. For a specific performance, the theater sold a total of $150$ tickets and collected total revenue of $R$ dollars. If at least $40$ student tickets were sold and at most $90$ adult tickets were sold, which of the following values could be the total revenue $R$ ? Select all such values.

Geçerli olan tümünü seçin

\$2{}850

\$3{}450

\$4{}200

\$4{}400

\$4{}650

Cevabı ve açıklamayı göster

Cevap: $\$3{}450$ ; $\$4{}200$

Cevap

The possible values for the total revenue

R

are

\$3{}450

and

\$4{}200

.

The linear revenue model is

R = 5250 - 15s

. Considering both conditions (

s \ge 40

and

a = 150 - s \le 90 \implies s \ge 60

), the valid range for student tickets is

60 \le s \le 150

. This restricts the possible revenue

R

to multiples of

\$15

between

\$3{}000

and

\$4{}350

. Both

\$3{}450

and

\$4{}200

fall within this valid range and correspond to integer ticket quantities (

s = 120, a = 30

and

s = 70, a = 80

, respectively).

Adım Adım Çözüm

1

Set up equations for the total number of tickets and revenue.

Let

s

be the number of student tickets and

a

be the number of adult tickets. Then

s + a = 150

, so

a = 150 - s

. Total revenue

R = 20s + 35a = 20s + 35(150 - s) = 5250 - 15s

.

Expressing revenue in terms of a single variable

s

simplifies finding the domain and range.

2

Determine the constraints on the variable

s

.

We are given

s \ge 40

and

a \le 90

. Substituting

a = 150 - s \le 90

yields

s \ge 60

. Combining constraints gives

60 \le s \le 150

.

The number of adult tickets being at most

90

forces the number of student tickets to be at least

60

.

3

Calculate the upper and lower bounds for the revenue

R

.

Maximum revenue occurs when

s = 60

:

R_{\text{max}} = 5250 - 15(60) = \$4{}350

. Minimum revenue occurs when

s = 150

:

R_{\text{min}} = 5250 - 15(150) = \$3{}000

.

Since

R = 5250 - 15s

is a decreasing linear function of

s

, the maximum revenue occurs at the minimum valid value of

s

and vice versa.

4

Evaluate the given choices against the range and divisibility requirements.

R

must be an integer multiple of

15

subtracted from

5250

, meaning

R

must be between

\$3{}000

and

\$4{}350

inclusive, and

(5250 - R)

must be divisible by

15

.

\$3{}450

(where

s = 120

) and

\$4{}200

(where

s = 70

) are both valid.

\$2{}850

is below the minimum bound,

\$4{}400

does not yield an integer value for

s

, and

\$4{}650

violates the adult ticket upper bound.

Only options meeting both inequality constraints and integer ticket requirements are valid.

Anahtar Kavram

Linear modeling of word problems under linear system constraints and inequalities

Tahmini Süre:1m 30s

Soru 55Soru →

A clean energy technology company manufactures two models of solar panels: Model X and Model Y. Model X produces $150$ kilowatt-hours (kWh) of electricity per day and costs $\$400$ to manufacture, while Model Y produces $200$ kWh of electricity per day and costs $\$550$ to manufacture. A solar energy project purchased a total of $50$ panels for a total manufacturing cost of $\$24,500$ . What is the total daily electricity production, in kilowatt-hours, of all $50$ panels combined?

8,500

8,750

9,000

9,500

10,000

Cevabı ve açıklamayı göster

Cevap: $9,000$

Cevap

The total daily electricity production of all

50

panels combined is

9,000

kWh.

The correct option is

9,000

kWh. Defining

x

as the number of Model X panels and

y

as the number of Model Y panels gives the equations

x + y = 50

and

400x + 550y = 24,500

. Substituting

y = 50 - x

yields

400x + 550(50 - x) = 24,500

, which simplifies to

-150x = -3,000

, so

x = 20

and

y = 30

. Computing the total output yields

20(150) + 30(200) = 3,000 + 6,000 = 9,000

kWh.

Adım Adım Çözüm

1

Define variables for the quantities of each panel model.

Let

x

represent the number of Model X panels and

y

represent the number of Model Y panels.

Establishing explicit variables allows formulating a system of linear equations from the word problem.

2

Set up equations for total panel quantity and total manufacturing cost.

System equations:

x + y = 50

and

400x + 550y = 24,500

.

The total number of panels is

50

, and the combined manufacturing cost equals

\$24,500

.

3

Solve the system using substitution.

Substitute

y = 50 - x

into the cost equation:

400x + 550(50 - x) = 24,500 \implies 400x + 27,500 - 550x = 24,500 \implies -150x = -3,000 \implies x = 20

. Consequently,

y = 50 - 20 = 30

.

Determining

x = 20

and

y = 30

gives the exact number of Model X and Model Y panels purchased.

4

Calculate the total daily electricity production.

Total production

= 20(150) + 30(200) = 3,000 + 6,000 = 9,000

kWh.

Multiply the quantity of each panel model by its respective daily output rate and sum the products.

Anahtar Kavram

Formulating and solving a system of two linear equations from a real-world scenario to find unknown quantities and evaluate a secondary combination function.

Tahmini Süre:1m 30s

Soru 56Soru →

A courier service calculates the shipping cost for a package based on its weight $w$ , in pounds. For packages weighing up to $20$ pounds, the cost is a flat fee of $\$12.00$ plus $\$2.50$ per pound. For packages weighing more than $20$ pounds, the cost is a flat fee of $\$20.00$ plus $\$2.00$ per pound. If the total shipping cost for a package was strictly greater than $\$45.00$ and at most $\$70.00$ , which of the following could be the weight of the package, in pounds? Select all such weights.

Geçerli olan tümünü seçin

12

pounds

15

pounds

20

pounds

24

pounds

26

pounds

Cevabı ve açıklamayı göster

Cevap: $15$ pounds; $20$ pounds; $24$ pounds

Cevap

The valid weights of the package are 15 pounds, 20 pounds, and 24 pounds.

The weight of the package must result in a total cost

C(w)

such that

\$45.00 < C(w) \le \$70.00

. Evaluating the options:
- For 15 pounds:

12 + 2.50(15) = \$49.50

, which falls within the range.
- For 20 pounds:

12 + 2.50(20) = \$62.00

, which falls within the range.
- For 24 pounds:

20 + 2.00(24) = \$68.00

, which falls within the range.

Adım Adım Çözüm

1

Formulate the cost function

C(w)

as a piecewise model.

C(w) = 12 + 2.50w

for

0 < w \le 20

, and

C(w) = 20 + 2.00w

for

w > 20

.

The cost structure changes depending on whether the weight exceeds

20

pounds.

2

Set up and solve the inequality

45 < C(w) \le 70

for packages up to

20

pounds.

45 < 12 + 2.50w \le 70 \implies 33 < 2.50w \le 58 \implies 13.2 < w \le 23.2

. Restricting to

w \le 20

gives

13.2 < w \le 20

.

This determines the valid weight interval for packages billed under the first tier.

3

Set up and solve the inequality

45 < C(w) \le 70

for packages over

20

pounds.

45 < 20 + 2.00w \le 70 \implies 25 < 2.00w \le 50 \implies 12.5 < w \le 25

. Restricting to

w > 20

gives

20 < w \le 25

.

This determines the valid weight interval for packages billed under the second tier.

4

Combine the valid intervals and evaluate each option.

The overall valid weight interval is

13.2 < w \le 25

. Testing the values: 12 pounds is outside the range; 15 pounds, 20 pounds, and 24 pounds are within the range; 26 pounds is outside the range.

Any weight strictly greater than

13.2

pounds and up to

25

pounds produces a cost between

\$45.00

and

\$70.00

.

Anahtar Kavram

Algebraic Modeling of Piecewise Functions and Compound Inequalities

Soru 57Soru →

A logistics company operates two delivery vans, Van A and Van B. Van A consumes fuel at a rate of $1$ gallon for every $15$ miles driven, and Van B consumes fuel at a rate of $1$ gallon for every $25$ miles driven. On a specific trip, the two vans were driven a total combined distance of $450$ miles and together consumed exactly $22$ gallons of fuel. What was the total distance, in miles, driven by Van A?

100 miles

150 miles

225 miles

250 miles

300 miles

Cevabı ve açıklamayı göster

Cevap: 150 miles

Cevap

The total distance driven by Van A was 150 miles.

The correct answer is 150 miles. By setting up the total fuel consumption equation as

\frac{d_A}{15} + \frac{450 - d_A}{25} = 22

, multiplying by the least common multiple

75

gives

5d_A + 1350 - 3d_A = 1650

, which simplifies to

2d_A = 300

and

d_A = 150

miles.

Adım Adım Çözüm

1

Define variables for the distance traveled by each van.

Let

d_A

be the distance driven by Van A. Then the distance driven by Van B is

450 - d_A

.

The total combined distance for both vans is

450

miles.

2

Express the total fuel consumed in terms of

d_A

.

\frac{d_A}{15} + \frac{450 - d_A}{25} = 22

Fuel used equals distance divided by miles per gallon for each vehicle.

3

Clear denominators by multiplying the entire equation by the common denominator 75.

5d_A + 3(450 - d_A) = 1650

Simplifies fractions to solve the linear algebraic equation easily.

4

Solve for

d_A

.

5d_A + 1350 - 3d_A = 1650 \implies 2d_A = 300 \implies d_A = 150

Isolates the target variable

d_A

to find the distance driven by Van A.

Anahtar Kavram

Linear Equations and Rate-Distance Modeling

Soru 58Soru →

A laboratory technician has $12$ liters of a solution containing $15\%$ salt by weight. The technician wants to increase the salt concentration to $25\%$ by adding a second solution that contains $40\%$ salt by weight. How many liters of the $40\%$ solution must be added?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8 liters of the 40% solution must be added.

The total amount of salt contributed by the two solutions is

0.15(12) + 0.40x = 1.8 + 0.40x

. The final mixture volume is

(12 + x)

liters, and its concentration must be

25\%

. Equating the total salt to

25\%

of total volume yields

1.8 + 0.40x = 0.25(12 + x)

. Expanding gives

1.8 + 0.40x = 3.0 + 0.25x

, which simplifies to

0.15x = 1.2

, giving

x = 8

liters.

Adım Adım Çözüm

1

Define the unknown variable and calculate the initial solute quantity.

Let

x

be the volume of the

40\%

solution added (in liters). Salt in initial solution =

0.15 \times 12 = 1.8

liters.

Establishing the mass balance of the salt solute is necessary to construct the algebraic equation.

2

Set up the algebraic concentration equation.

1.8 + 0.40x = 0.25(12 + x)

The combined salt from both solutions must equal

25\%

of the total combined liquid volume

(12 + x)

liters.

3

Solve the linear equation for

x

.

1.8 + 0.40x = 3.0 + 0.25x \implies 0.15x = 1.2 \implies x = 8

Isolating

x

yields the exact number of liters required.

Anahtar Kavram

Algebraic mixture problems using mass balance equations.

Tahmini Süre:1m 30s

Soru 59Soru →

A municipal water treatment facility uses two intake pipes, Pipe $X$ and Pipe $Y$ , to fill a main reservoir. Operating alone at its constant rate, Pipe $X$ can fill the empty reservoir in $10$ hours. Operating alone at its constant rate, Pipe $Y$ can fill the empty reservoir in $15$ hours. Pipe $X$ is turned on first and operates alone for $3$ hours. Then, Pipe $Y$ is also turned on, and both pipes operate together until the reservoir is completely full. What is the total number of hours Pipe $X$ operates from the moment it is turned on until the reservoir is completely filled?

4.2

4.8

6.0

7.2

7.5

Cevabı ve açıklamayı göster

Cevap: $7.2$

Cevap

The total number of hours Pipe X operates is

7.2

hours.

The correct answer is

7.2

hours. Pipe X completes

3/10

of the reservoir in 3 hours, leaving

7/10

of the reservoir to be filled. The combined rate of Pipe X and Pipe Y is

1/10 + 1/15 = 1/6

per hour. The joint time required to fill the remaining

7/10

is

(7/10) / (1/6) = 4.2

hours. Adding the initial 3 hours Pipe X worked alone yields a total of

3 + 4.2 = 7.2

hours.

Adım Adım Çözüm

1

Determine the individual hourly work rates of Pipe X and Pipe Y.

Rate of Pipe X =

\frac{1}{10}

reservoir per hour; Rate of Pipe Y =

\frac{1}{15}

reservoir per hour.

The rate is the reciprocal of the total time required to complete the job individually.

2

Calculate the fraction of the reservoir filled by Pipe X during its initial 3-hour solo operation.

Work done in first 3 hours =

3 \times \frac{1}{10} = \frac{3}{10}

of the reservoir.

Multiplying Pipe X's rate by its solo operating time gives the completed portion of the work.

3

Find the remaining fraction of the reservoir that needs to be filled.

Remaining work =

1 - \frac{3}{10} = \frac{7}{10}

of the reservoir.

Subtracting the completed portion from the whole (1) leaves the uncompleted portion.

4

Determine the combined hourly rate when both pipes operate simultaneously.

Combined rate =

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

of the reservoir per hour.

Simultaneous operation rates are additive.

5

Calculate the time

t

during which both pipes operate together to finish the remaining work.

t = \frac{\frac{7}{10}}{\frac{1}{6}} = \frac{7}{10} \times 6 = \frac{42}{10} = 4.2

hours.

Dividing the remaining work by the combined rate yields the joint operation time.

6

Add Pipe X's solo time to the joint operation time to get Pipe X's total operating time.

Total time =

3 + 4.2 = 7.2

hours.

Pipe X was active during both the initial solo period and the joint operating period.

Anahtar Kavram

Combined Rate and Staggered Work Modeling

Tahmini Süre:2m 0s

Soru 60Soru →

An investor allocated a total principal of $\$10,000$ between two accounts, Account X and Account Y. Account X pays simple annual interest at a rate of $6\%$ , while Account Y pays simple annual interest at a rate of $8\%$ . Let $x$ represent the amount of money, in dollars, invested in Account X, and let $d$ represent the total annual interest earned, in dollars, from both accounts after one year. Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The total annual interest

d

satisfies

600 \le d \le 800

.

The amount invested in Account X can be represented as

x = \frac{800 - d}{0.02}

.

If

d = 750

, the amount invested in Account X is greater than the amount invested in Account Y.

If equal amounts were invested in both accounts, the total annual interest earned is

\$700

.

If

d = 680

, the ratio of the amount invested in Account X to the amount invested in Account Y is

2

to

3

.

Cevabı ve açıklamayı göster

Cevap: The total annual interest $d$ satisfies $600 \le d \le 800$ .; The amount invested in Account X can be represented as $x = \frac{800 - d}{0.02}$ .; If equal amounts were invested in both accounts, the total annual interest earned is $\$700$ .

Cevap

The true statements are that the total annual interest

d

satisfies

600 \le d \le 800

, the amount in Account X is given by

x = \frac{800 - d}{0.02}

, and equal investment in both accounts yields

\$700

in total interest.

The total interest equation

d = 800 - 0.02x

dictates all valid relationships. Because

0 \le x \le 10,000

, the bounds for

d

are strictly between

600

and

800

. Rearranging the equation yields

x = \frac{800 - d}{0.02}

, which correctly models the Account X investment. Substituting

x = 5,000

yields

d = 700

, confirming the equal-allocation scenario.

Adım Adım Çözüm

1

Formulate the total interest model as a linear equation in terms of

x

.

d = 0.06x + 0.08(10,000 - x) = 800 - 0.02x

The interest earned from Account X is

0.06x

and the interest from Account Y is

0.08(10,000 - x)

.

2

Determine the range of possible interest values for

0 \le x \le 10,000

.

When

x = 10,000

,

d = 600

. When

x = 0

,

d = 800

. Thus

600 \le d \le 800

.

The minimum and maximum interest values occur at the extreme allocation bounds.

3

Rearrange the interest equation to express

x

in terms of

d

.

0.02x = 800 - d \implies x = \frac{800 - d}{0.02}

Isolating

x

provides a formula for calculating the Account X principal directly from the total interest.

4

Evaluate the specific numerical cases given in the options.

Equal investment (

x = 5,000

) gives

d = 800 - 100 = 700

. For

d = 750

,

x = 2,500

(Account Y =

7,500

). For

d = 680

,

x = 6,000

(Account Y =

4,000

, ratio

3:2

).

Substituting specific values tests the validity of each conditional statement.

Anahtar Kavram

Linear Modeling and Algebraic Rate Allocation

Algebraic Word Problems and Modeling

Cevap

Adım Adım Çözüm

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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