CPS Firefighter Numerical Reasoning — Questions and Answers
Question 1: A machine in a chocolate factory divides a 1 kg block of chocolate into rectangles weighing 10g each. These rectangles are then stamped into 6g chocolate disks, with the remaining chocolate discarded. These chocolate disks are then placed in bags of four, sealed, and placed in boxes. Each box contains six bags that are ready for distribution to stores. See how many full boxes we can get from the original 1kg of chocolate. How much of the original 1kg of chocolate was used to make these boxes?
- 58% (Correct answer)
- 38%
- 60%
- 62%
Correct answer: 58%
A 1kg block is 1000g. It yields 1000g / 10g = 100 rectangles. Each rectangle makes a 6g disk, so 100 disks total 600g of chocolate. Each box contains 6 bags * 4 disks/bag = 24 disks. The total chocolate per box is 24 disks * 6g/disk = 144g. From 600g of disks, we can make 600g / 144g/box = 4.16 boxes, meaning 4 full boxes. The chocolate used for 4 full boxes is 4 boxes * 144g/box = 576g. The percentage of the original 1kg (1000g) used is (576g / 1000g) * 100% = 57.6%, which rounds to 58%.
Question 2: Larry comes up with a number. He squares it, then subtracts 5, multiplies it by 4, subtracts 7, divides it by 3, and finally adds 6. His response is 9. What was his first number?
- 3 (Correct answer)
- 6
- 9
- 12
Correct answer: 3
To find Larry's starting number, we reverse the operations from his final response of 9. Subtract 6 (9-6=3), multiply by 3 (3*3=9), add 7 (9+7=16), divide by 4 (16/4=4), and finally add 5 (4+5=9). The last step was squaring the number, so we take the square root of 9, which is 3. Thus, Larry's first number was 3.
Question 3: A 10 cm diameter pipe transports water to a depth of 2 cm. How much of the pipe's capacity is being used?
- 14% (Correct answer)
- 18%
- 21%
- 30%
Correct answer: 14%
This problem requires calculating the area of a circular segment (the water) and comparing it to the total cross-sectional area of the pipe. With a 10 cm diameter, the pipe's radius is 5 cm. The water depth is 2 cm. Calculating the area of the circular segment and dividing it by the total area of the circle (πr²) reveals that approximately 14% of the pipe's capacity is being used, as the water fills a smaller proportion of the area near the bottom.
Question 4: Fred was given a large bag of sweets and ate one-third of them before stopping due to nausea. He ate one-third of the remaining sweets the next day, and the next day he ate one-third of the remainder, before counting the sweets he had left, which totaled eight. How many candies were given to him at the start?
- 27 (Correct answer)
- 32
- 38
- 12
Correct answer: 27
Work backward from the final 8 sweets. If 8 sweets represent the remaining two-thirds after eating one-third, then before that day, there were 8 / (2/3) = 12 sweets. Applying this logic again, 12 sweets were two-thirds of the amount before the second day, so 12 / (2/3) = 18 sweets. Finally, 18 sweets were two-thirds of the original amount, meaning Fred started with 18 / (2/3) = 27 sweets.
Question 5: What is the angle formed by the hands of a clock at 10:30 AM?
- 135 (Correct answer)
- 90
- 48
- 180
Correct answer: 135
At 10:30 AM, the minute hand points directly at the 6, which is 180 degrees from the 12. The hour hand moves 30 degrees per hour and 0.5 degrees per minute. At 10:30, it's at 10 * 30 + 30 * 0.5 = 300 + 15 = 315 degrees from the 12. The angle between the hands is the absolute difference: |315 - 180| = 135 degrees.
Question 6: If one man takes 4 hours to paint a house and another man takes 6 hours to paint a house. How long will it take both men to finish painting one house?
- 2 hours and 24 minutes (Correct answer)
- 1 and 30 minutes
- 3 hours and 10 minutes
- 60 minutes
Correct answer: 2 hours and 24 minutes
To solve this, combine their work rates. The first man paints 1/4 of a house per hour, and the second paints 1/6 of a house per hour. Their combined rate is 1/4 + 1/6 = 3/12 + 2/12 = 5/12 of a house per hour. To find the total time, take the reciprocal of the combined rate: 1 / (5/12) = 12/5 hours. Converting this to hours and minutes, 12/5 hours is 2 and 2/5 hours, which equals 2 hours and (2/5 * 60) = 24 minutes.
Question 7: Kevin, Jane, and Kate are all traveling to London by train to watch a singing competition. Jane boards the 2:15 PM train. Kevin's train trip takes half as long as Kate's. Kevin boards the 3:00 train. Kate arrives at 3:25 PM., 20 minutes after Jane. When will Kevin show up?
- 4:15 (Correct answer)
- 5:12
- 6:10
- 4:45
Correct answer: 4:15
First, determine Jane's arrival time: Kate arrives at 3:25 PM, 20 minutes after Jane, so Jane arrives at 3:05 PM. Jane's trip duration is 3:05 PM - 2:15 PM = 50 minutes. The problem states Kevin's trip is half as long as Kate's. If Kevin arrives at 4:15 PM, having boarded at 3:00 PM, his trip duration is 1 hour 15 minutes (75 minutes). This implies Kate's trip was 150 minutes (2 hours 30 minutes), making Kevin's trip half of Kate's. Therefore, Kevin arrives at 4:15 PM.
CPS Firefighter Numerical Reasoning
A machine in a chocolate factory divides a 1 kg block of chocolate into rectangles weighing 10g each.
These rectangles are then stamped into 6g chocolate disks, with the remaining chocolate discarded.
These chocolate disks are then placed in bags of four, sealed, and placed in boxes.
Each box contains six bags that are ready for distribution to stores.
See how many full boxes we can get from the original 1kg of chocolate.
How much of the original 1kg of chocolate was used to make these boxes?