Sample Work Problem:

Problem: Anna and Brian are both tasked with assembling gift boxes for a community event. Anna can assemble all the required gift boxes in 5 hours working alone, while Brian can do the same task in 3 hours working alone. If they work together, how long will it take them to assemble all the gift boxes?

Solution: To find out how long it will take for Anna and Brian to complete the task together, first determine their individual rates of work and then add these rates together to find their combined rate.

1. Anna’s Rate: If Anna can complete the task in 5 hours, her work rate is:

   Rate of Anna = 1 task/5 hours = 1/5 task per hour 

2. Brian’s Rate: If Brian can complete the task in 3 hours, his work rate is:

   Rate of Brian = 1 task/3 hours = 1/3 task per hour 

3. Combined Rate: Add their rates together to find the rate at which they can complete the task together:

   Combined Rate=1/5 + 1/3 = 3/15 + 5/15 = 8/15 task per hour

4. Time Required Together: To find out how long it will take them to complete one whole task working together, calculate the reciprocal of their combined rate:

   Time required=1/Combined Rate = 1/8/15 = 15/8 hours 
   15/8 hours=1 hour and 52.5 minutes 

Conclusion: Anna and Brian will be able to complete the task together in approximately 1 hour and 52.5 minutes.

Problem 2

Problem: Carla can paint a room by herself in 6 hours. Her friend, Jeff, can paint the same room in 4 hours. If they decide to paint the room together while working at their respective paces, how long will it take them to paint the room?

Solution: To solve this, first determine each person’s rate of work, then combine these rates to find the total rate when they work together.

1. Carla’s Rate: If Carla can complete the task in 6 hours:

   Rate of Carla = 1 room/6 hours = 1/6 room per hour 

2. Jeff’s Rate: If Jeff can complete the task in 4 hours:

   Rate of Jeff = 1 room/4 hours = 1/4 room per hour 

3. Combined Rate: Add their rates together to find the rate at which they can complete the room together:

   Combined Rate=1/6 + 1/4 = 2/12 + 3/12 = 5/12 room per hour 

4. Time Required Together: To find out how long it will take them to complete one whole room working together, calculate the reciprocal of their combined rate:

   Time required = 1/Combined Rate = 1/5/12 = 12/5 hours = 2.4 hours  or 2 hours and 24 minutes 

Conclusion: Carla and Jeff will be able to complete the painting of the room together in approximately 2 hours and 24 minutes.

Problem 3.

Problem: Lucy can fill a swimming pool using her hose in 5 hours. Her neighbor, Tim, has a more powerful hose and can fill the same pool in 3 hours. If they both use their hoses to fill the pool at the same time, how long will it take to fill the pool?

Solution: First, calculate each person’s rate of filling the pool, then determine the combined rate.

1. Lucy’s Rate:

   Rate of Lucy = 1 pool/5 hours = 1/5 pool per hour

2. Tim’s Rate:

   Rate of Tim = 1 pool/3 hours = 1/3 pool per hour 

3. Combined Rate: Sum up their rates to get the total filling rate:

   Combined Rate = 1/5 + 1/3 = 3/15 + 5/15 = 8/15 pool per hour 

4. Time Required Together: Find the reciprocal of their combined rate:

   Time required = 1/Combined Rate = 1/8/15 = 15/8 hours = 1.875 hours or 1 hour and 52.5 minutes 

Conclusion: Using both hoses, Lucy and Tim can fill the pool together in approximately 1 hour and 52.5 minutes.

Problem 4

A photocopier makes $y$ copies in $k$ hours and 45 minutes. What is its average operating speed in copies per hour?

Choices:

• A. 4𝑦/4𝑘+3  copies per hour
• B. 𝑦/𝑘+0.75 copies per hour
• C. 𝑦/4𝑘+3  copies per hour
• D. $4y$ copies per hour

Answer: B. 𝑦/𝑘+0.75​ copies per hour

Solution:
Convert 45 minutes to hours, which is 0.75 hours. Thus, the total time is $k+0.75$ hours. The average speed is calculated as 𝑦 copies/𝑘+0.75 hours.

Problem 5.

Sarah and Kaye, working together, can finish binding 1000 books in 4 ½ days. Sarah working alone, can bind the 1000 books in 10 days. How long will it take Kaye to do the same job working alone?

Choices:

A. 6 2/11days

B. 7 2/11 days

C. 8 2/11 days

D. 9 2/11 days

1. Given:

   • Sarah and Kaye together can finish 1000 books in 4.5 days.
   • Sarah alone can finish 1000 books in 10 days.

2. Step 1: Calculate the rate at which Sarah and Kaye together can bind the books per day:

   Rate Sarah + Kaye=1 job/4.5 days ​

3. Step 2: Calculate the rate at which Sarah can bind the books per day:

   Rate Sarah = 1 job/10 days ​

4. Step 3: Use the formula for the sum of individual rates to find Kaye’s rate:

   Rate Kaye = Rate Kaye + Sarah − Rate Sarah
   Rate Kaye = 1/4.5 – 1/10

   
   ​

5. Step 4: Calculate the rate difference:

   Rate Kaye = 10/45 − 4.5/45 = 5.5/45 = 11/90  ​

6. Step 5: Determine the time it takes for Kaye to complete the job alone:

   Time Kaye = 1/ job Rate Kaye = 1/11/90 = 90/11 ≈ 8 2/11 days