Arithmetic patterns Transitional Reasoning Practice

Questions

1. 1.

   A chemist measures the total mass of a glass beaker as equal increments of a liquid solution are added, where the total recorded mass increases in an arithmetic pattern. The total mass is 142 grams after 4 increments are added and 267 grams after 9 increments are added. What is the mass, in grams, of the empty glass beaker before any solution is added?

   • A) 25
   • B) 17
   • C) 67
   • D) 42
2. 2.

   A hardware distributor packages structural bolts into different box sizes labeled as Tier 1, Tier 2, Tier 3, and so on, where the number of bolts in each box increases in an arithmetic pattern. A quality control log shows that a Tier 4 box contains 54 bolts, while a Tier 9 box contains 89 bolts. For a framing project, a builder purchases exactly one Tier 3 box and two Tier 7 boxes. How many bolts did the builder purchase in total?

   • A) 122
   • B) 169
   • C) 197
   • D) 145
3. 3.

   A telecommunications tower uses horizontal support struts arranged from the top downward, numbered consecutively as Strut 1, Strut 2, Strut 3, and so on. The lengths of these struts increase in an arithmetic pattern, with Strut 3 measuring 4.8 meters and Strut 11 measuring 11.2 meters. Structural guidelines require a diagonal cross-brace between two consecutive struts whenever the sum of their lengths exceeds 15 meters. Which pair of consecutive struts is the first to require a diagonal cross-brace?

   • A) Struts 5 and 6
   • B) Struts 6 and 7
   • C) Struts 7 and 8
   • D) Struts 8 and 9
4. 4.

   A civil engineering team tests the water flow rate through an irrigation canal at consecutive monitoring gates numbered 1, 2, 3, and so on. Due to controlled diversion channels, the flow rate decreases from gate to gate in an arithmetic pattern. The measured flow rate is 68 liters per second at Gate 4 and drops to 44 liters per second at Gate 10. If safety protocol mandates an automated pump shutoff at the gate where the flow rate reaches exactly three-fourths of the flow rate recorded at Gate 1, at which gate is the shutoff triggered?

   • A) Gate 5
   • B) Gate 6
   • C) Gate 7
   • D) Gate 8
5. 5.

   A hydroelectric plant regulates water discharge through a control spillway across consecutive release phases numbered 1, 2, 3, and so on. The volume of water released during each phase decreases in an arithmetic pattern. The release volume in Phase 4 is 82 thousand cubic meters, and the volume released in Phase 3 is 24 thousand cubic meters greater than the volume released in Phase 7. What is the volume of water, in thousand cubic meters, released during Phase 11?

   • A) 46
   • B) 40
   • C) 34
   • D) 26
6. 6.

   An environmental science class monitors the surface area of a restored coral fragment across consecutive measurement periods numbered $n = 1, 2, 3, \dots$. The recorded surface area expands in an arithmetic pattern from one period to the next. During measurement period 4, the surface area is 41 square centimeters, and during measurement period 9, the surface area reaches 76 square centimeters. Which algebraic expression models the surface area, in square centimeters, of the coral fragment during measurement period $n$?

   • A) $7n + 13$
   • B) $7n + 20$
   • C) $5n + 21$
   • D) $7n + 6$
7. 7.

   A renewable energy facility conducts a calibration test of a wind turbine system across consecutive operational shifts. The electrical output produced during each shift increases in an arithmetic pattern. The mean output recorded across Shifts 4, 6, and 8 is 54 megawatt-hours (MWh), and the output recorded during Shift 13 is 89 MWh. What was the electrical output, in MWh, produced during Shift 1?

   • A) 24 MWh
   • B) 29 MWh
   • C) 34 MWh
   • D) 39 MWh
8. 8.

   A deep-space tracking antenna adjusts its transmission signal strength across consecutive communication windows according to an arithmetic pattern. During Window 3, the signal strength is 54 milliwatts, and during Window 7, the signal strength is 78 milliwatts. Communications protocol dictates that a window is designated as synchronized when its signal strength (in milliwatts) is numerically equal to exactly 9 times its window number. Which communication window is designated as synchronized?

   • A) Window 12
   • B) Window 10
   • C) Window 14
   • D) Window 16
9. 9.

   A landscape architect designs a series of concentric regular octagonal walking tracks around a botanical garden pavilion, numbered consecutively from Track 1 outward. The side lengths of the tracks increase in an arithmetic pattern, where Track 3 has a side length of 13 meters and Track 7 has a side length of 23 meters. What is the total perimeter, in meters, of Track 10?

   • A) 204 meters
   • B) 224 meters
   • C) 244 meters
   • D) 264 meters
10. 10.

    A materials research facility uses a cryogenic chamber to chill an alloy specimen across consecutive testing stages that follow an arithmetic pattern. The temperature of the specimen is recorded as $14^\circ\text{C}$ at Stage 3 and $-6^\circ\text{C}$ at Stage 8. If the cooling process continues at this constant rate per stage, at which stage does the specimen's temperature first drop below $-35^\circ\text{C}$?

    • A) Stage 14
    • B) Stage 15
    • C) Stage 16
    • D) Stage 17
11. 11.

    A banquet hall configures identical rectangular tables end-to-end to form one long continuous dining row. A single isolated table seats 3 guests on each of its two longer sides and 1 guest at each of its two shorter ends, accommodating 8 guests in total. When tables are pushed together end-to-end, guests can no longer sit at the joined ends, meaning a row of 2 tables seats 14 guests and a row of 3 tables seats 20 guests, continuing in this arithmetic pattern. How many tables must be joined end-to-end in this row to seat exactly 98 guests?

    • A) 13
    • B) 15
    • C) 16
    • D) 17
12. 12.

    An oceanographic research vessel lowers an acoustic sensor deeper into the water across consecutive trial stages that follow an arithmetic pattern. The sensor reaches a depth of 58 meters at Stage 4 and 123 meters at Stage 9. For the final phase of the mission, the team must deploy the sensor to a depth that is 15 meters shallower than twice the depth reached at Stage 7. At what depth, in meters, is the sensor deployed for this final phase?

    • A) 179 meters
    • B) 194 meters
    • C) 209 meters
    • D) 97 meters
13. 13.

    A community art project is assembling a stepped mosaic wall where the number of ceramic tiles in each horizontal row follows an arithmetic pattern. The bottom row contains 85 tiles, and each successive row above it contains 4 fewer tiles than the row directly below it. To preserve structural stability, the design rules state that every row must contain at least 15 tiles, and no further rows are installed once a row would drop below this minimum. How many ceramic tiles are placed in the topmost row of the completed mosaic?

    • A) 17
    • B) 13
    • C) 18
    • D) 21
14. 14.

    A civil engineer inspects the vertical suspender cables on a suspension bridge, where the cable lengths increase from the center outward according to an arithmetic pattern. The 3rd cable measures 21 meters, and the 8th cable measures 36 meters. Safety regulations require a heavy-duty vibration damper on any cable longer than 50 meters. If this bridge section contains 18 suspender cables numbered consecutively from 1 to 18, how many cables require a heavy-duty vibration damper?

    • A) 9
    • B) 8
    • C) 7
    • D) 6
15. 15.

    An agricultural research station builds terraced test plots on a hillside where each ascending level contains an arithmetic pattern of crop beds. The facility records that Level 6 contains 47 crop beds, and the combined total of crop beds on Level 3 and Level 8 is exactly 88. If the number of crop beds continues to increase by the same constant amount on each successive level, how many crop beds are located on Level 12?

    • A) 71
    • B) 77
    • C) 83
    • D) 89
16. 16.

    An environmental monitoring station tracks the depth of snowpack on a mountain ridge across consecutive weeks as snow accumulates in an arithmetic pattern. By analyzing the seasonal records, meteorologists noted that the depth recorded in week 8 was exactly double the depth recorded in week 3. If the continuous weekly accumulation resulted in a measured depth of 88 cm in week 20, what was the initial snowpack depth recorded in week 1?

    • A) 8 cm
    • B) 10 cm
    • C) 12 cm
    • D) 16 cm
17. 17.

    An automated greenhouse regulates water delivery across three consecutive growing sectors according to an arithmetic pattern. The daily water volume (in liters) supplied to Sector 1, Sector 2, and Sector 3 is modeled by the algebraic expressions $2x + 7$, $5x - 8$, and $7x - 3$, respectively. If this distribution maintains a constant difference between consecutive sectors, how many liters of water are allocated daily to Sector 3?

    • A) 137 liters
    • B) 92 liters
    • C) 47 liters
    • D) 140 liters
18. 18.

    Two regional logistics centers, Center A and Center B, expand their active daily delivery routes following distinct arithmetic patterns. Center A operates 14 routes during its first month and expands by adding 6 new routes each subsequent month. Meanwhile, Center B begins with 49 routes in its first month and expands at a slower pace by adding 1 new route each month. In which month will both logistics centers operate an identical number of delivery routes?

    • A) Month 6
    • B) Month 7
    • C) Month 8
    • D) Month 9
19. 19.

    A municipal park department plants saplings along a newly opened greenway according to an arithmetic pattern each week. The team planted 38 saplings during the 4th week and 73 saplings during the 9th week. If this linear rate of increase continues uninterrupted, during which week will the team plant exactly 115 saplings?

    • A) 13th week
    • B) 15th week
    • C) 16th week
    • D) 18th week

Answers and step-by-step solutions appear after you finish the test.