Equivalent expressions and the distributive property Transitional Reasoning Download PDF

Questions

1. 1.

   An electrical contractor brings a master spool containing $8(8x + 5)$ meters of industrial wire to a construction site. During the first phase of the installation, workers cut 3 wiring runs, each measuring $4(3x - 2)$ meters in length. During the second phase, workers cut 2 additional wiring runs, each measuring $5(2x + 3)$ meters in length. Which expression represents the remaining length of wire, in meters, left on the master spool?

   • A) $8x + 34$
   • B) $8x + 46$
   • C) $120x + 46$
   • D) $8x + 4$
2. 2.

   A public library organizes a youth summer reading challenge across three participant groups: Junior, Intermediate, and Senior. The Junior group reads a total of $3(4x - 1)$ books during the summer. The Intermediate group reads 2 times as many books as the Junior group, while the Senior group reads $4(2x + 5)$ fewer books than the Intermediate group. Which expression represents the total number of books read by all three groups combined?

   • A) $52x - 35$
   • B) $52x + 5$
   • C) $40x - 32$
   • D) $68x + 5$
3. 3.

   A custom framing workshop creates a rectangular display frame around a decorative print. The outer boundary of the frame has a width of 8 centimeters and a length of $5(2x + 3)$ centimeters. The print placed in the center covers an area of $6(4x - 1)$ square centimeters. Which expression represents the area, in square centimeters, of the frame surrounding the print?

   • A) $56x + 114$
   • B) $104x + 114$
   • C) $56x + 126$
   • D) $104x + 126$
4. 4.

   A digital climate-control system in a research greenhouse calculates an automated ventilation index based on an ambient temperature variable, $x$. The system starts with a baseline reading of $4(3x + 5)$, doubles this entire quantity during peak daylight hours, and adds an auxiliary booster term of $5(2 - x)$. To prevent over-ventilation, a safety routine then subtracts $6(2x - 3)$ from the accumulated total. Which expression represents the final ventilation index determined by the system?

   • A) $7x + 32$
   • B) $7x + 48$
   • C) $7x + 68$
   • D) $27x + 68$
5. 5.

   A municipal eco-park generates electricity using solar inverters installed across two campus buildings, Building A and Building B. Building A operates 8 inverters, each generating $5x - 3$ kilowatt-hours of energy per day. Building B operates 12 inverters, each generating $2x + 4$ kilowatt-hours less energy per day than an inverter in Building A. Which expression represents the total daily energy generation, in kilowatt-hours, of both buildings combined?

   • A) $76x - 108$
   • B) $76x - 12$
   • C) $64x + 24$
   • D) $124x - 12$
6. 6.

   A mobile gaming platform awards reward tokens to players who complete daily challenges across two different tiers. Each completed Silver challenge grants $4(3x + 2)$ tokens, and each completed Gold challenge grants $6(2x - 1)$ tokens. During a weekend tournament, a player finishes 3 Silver challenges and 4 Gold challenges, but spends $5(4x - 3)$ tokens to unlock custom avatar items. Which expression represents the net number of reward tokens the player retains from the tournament?

   • A) $64x - 15$
   • B) $4x + 17$
   • C) $104x - 15$
   • D) $64x + 15$
7. 7.

   A municipal water treatment facility operates two main pumps connected to an emergency storage reservoir. During each hour of routine operation, Intake Pump A fills the reservoir with $6(3x + 4)$ liters of water, while Drainage Pump B discharges $4(2x - 5)$ liters of rinse water. The facility runs both pumps simultaneously for 3 hours, after which an auxiliary intake system pumps an additional $5(4x + 8)$ liters into the reservoir. Which expression represents the net change in water volume, in liters, inside the reservoir at the end of this entire process?

   • A) $50x + 52$
   • B) $30x + 84$
   • C) $98x + 52$
   • D) $50x + 172$
8. 8.

   A municipal bike-share program manages bicycle fleets across two main transit stations, Station A and Station B. Station A initially holds $5(4x - 3)$ bicycles, and Station B holds $3(2x + 7)$ bicycles. During the morning rebalancing routine, a transport van moves $\frac{2}{5}$ of the bicycles from Station A to Station B. Which expression represents the total number of bicycles at Station B after the transfer?

   • A) $10x + 18$
   • B) $14x + 15$
   • C) $14x + 27$
   • D) $14x + 18$
9. 9.

   A logistics distribution center receives 5 identical crates of merchandise, each containing $6x - 2$ items. Upon inspection, workers identify that a total of $4(3x - 5)$ damaged items must be removed across the entire shipment. The remaining undamaged items are then distributed equally into 2 storage containers. Which expression represents the number of items placed into each storage container?

   • A) $9x - 15$
   • B) $9x + 5$
   • C) $18x + 10$
   • D) $9x - 5$
10. 10.

    An agricultural cooperative deploys 4 autonomous seeding drones to plant a clover cover crop across several fields. Each drone is initially loaded with $7(2x + 3)$ grams of seed mix in its main hopper. Before entering the fields, each drone discharges $3(3x - 1)$ grams of seed mix during a mandatory calibration test. Which expression represents the total amount of seed mix, in grams, remaining in all 4 drones after calibration?

    • A) $20x + 72$
    • B) $20x + 96$
    • C) $92x + 72$
    • D) $5x + 24$
11. 11.

    A botanical research center designs a greenhouse complex with a total floor area of $72x - 48$ square meters. The lead botanist reserves $\frac{3}{4}$ of this total floor area for climate-controlled research exhibits. This reserved space is then divided equally among 6 identical cultivation zones. Which expression represents the floor area, in square meters, of a single cultivation zone?

    • A) $9x - 6$
    • B) $9x - 8$
    • C) $12x - 8$
    • D) $18x - 12$
12. 12.

    An environmental engineering team monitors the hourly carbon dioxide emissions of a prototype bio-filtration unit during two operational stages. In Stage 1, the unit produces emissions at a rate of $18x - 24$ grams per hour, while in Stage 2, an active chemical scrubber reduces the emission rate to $\frac{2}{3}$ of the Stage 1 rate. The unit operates continuously for 2 hours in Stage 1 and then for 5 hours in Stage 2. Which expression represents the total carbon dioxide emissions, in grams, generated by the unit across both stages?

    • A) $96x - 128$
    • B) $96x - 168$
    • C) $96x + 32$
    • D) $78x - 104$
13. 13.

    A wildlife conservation team monitors water quality across two protected wetland reserves using specialized testing kits. The number of testing kits required for the first reserve is represented by $\frac{1}{5}(25x - 15)$, while the second reserve requires $\frac{2}{3}(9x + 12)$ kits. Which expression is equivalent to the total number of testing kits required for both reserves?

    • A) $11x + 5$
    • B) $11x + 11$
    • C) $11x + 1$
    • D) $11x - 3$
14. 14.

    An educational software platform monitors the daily server workload, in processing cycles, for a school district with $n$ active classrooms. The morning instruction consumes $5(4n + 3)$ cycles, the afternoon session consumes $3(6n - 1)$ cycles, and an automated system routine releases $2(7n - 9)$ unused cycles back to the network. Which expression represents the net server workload, in processing cycles, consumed by the district each day?

    • A) $6(4n + 5)$
    • B) $6(4n - 1)$
    • C) $2(26n - 3)$
    • D) $3(8n + 7)$
15. 15.

    An urban planner is designing a community sports park enclosed by a perimeter fence. The park is laid out in the shape of a rectangle with a length of $4(3x - 2)$ meters and a width of $2(5x + 7)$ meters. Which expression represents the total length, in meters, of fencing required to completely enclose the park?

    • A) $44x + 12$
    • B) $22x + 6$
    • C) $44x + 44$
    • D) $44x + 10$
16. 16.

    A robotics engineering team constructs a multi-segment sensor track by connecting three straight rail sections end to end. The length of the first rail section is $5(2x + 3)$ centimeters, the length of the second rail section is $2(4x - 1)$ centimeters, and the length of the third rail section is $3(5 - 2x)$ centimeters. Which expression is equivalent to the total length, in centimeters, of the connected sensor track?

    • A) $4(3x + 4)$
    • B) $4(3x + 7)$
    • C) $4(3x + 8)$
    • D) $4(6x + 7)$
17. 17.

    A community sports complex features two adjacent rectangular recreational zones that share a common dividing boundary of width 8 meters. The play lawn has an area of $24x + 56$ square meters, and the adjacent sport court has an area of $16x - 32$ square meters. Which expression represents the total combined length, in meters, along the outer edge of both zones?

    • A) $5x + 11$
    • B) $10x + 6$
    • C) $5x - 3$
    • D) $5x + 3$
18. 18.

    A solar energy company offers two commercial maintenance plans for office buildings, Plan A and Plan B. The annual service cost, in dollars, depends on the number of installed panel arrays, $x$. Plan A costs $8(3x + 5)$ dollars per year, whereas Plan B costs $6(2x - 7)$ dollars per year. Which expression represents how much more, in dollars, a customer pays per year for Plan A than for Plan B?

    • A) $12x - 2$
    • B) $12x + 82$
    • C) $12x + 12$
    • D) $36x - 2$
19. 19.

    A logistics company packages promotional gift sets into master crates. Each master crate contains 4 identical gift sets, and each gift set consists of 3 branded water bottles and 2 insulated tumblers. If each bottle weighs $2x - 1$ ounces and each tumbler weighs $5x + 4$ ounces, which expression represents the total weight, in ounces, of the items in one master crate?

    • A) $32x + 10$
    • B) $64x + 5$
    • C) $64x - 20$
    • D) $64x + 20$
20. 20.

    A landscaping team is creating 4 identical community garden zones. Each zone includes a rectangular planting bed with a width of 3 meters and a length of $2x + 5$ meters, along with an adjacent paved walkway with an area of $4x - 6$ square meters. Which expression represents the total combined area, in square meters, of all 4 garden zones?

    • A) $28x + 54$
    • B) $40x + 36$
    • C) $40x - 4$
    • D) $40x + 9$