What are Time and Work Problems?

S3-SA1-0626

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Time and Work problems are a type of math problem where you calculate how long it takes to complete a task, or how much work can be done in a certain time. These problems usually involve people or machines working together or alone to finish a job.

Simple Example

Quick Example

Imagine you need to paint a wall. If you can paint one wall in 2 hours, and your friend can paint the same wall in 3 hours, how long would it take if both of you painted together? This is a Time and Work problem!

Worked Example

Step-by-Step

PROBLEM: Rohan can finish a project in 10 days. Sohan can finish the same project in 15 days. If they work together, how many days will it take to finish the project?

STEP 1: Find Rohan's one day's work. If Rohan takes 10 days for the whole project, in one day he does 1/10 of the project.
---STEP 2: Find Sohan's one day's work. If Sohan takes 15 days for the whole project, in one day he does 1/15 of the project.
---STEP 3: Find their combined one day's work. Add their individual one day's work: 1/10 + 1/15.
---STEP 4: To add fractions, find a common denominator. For 10 and 15, the smallest common multiple (LCM) is 30. So, (3/30) + (2/30).
---STEP 5: Add the fractions: (3+2)/30 = 5/30.
---STEP 6: Simplify the fraction: 5/30 = 1/6. This means together, they complete 1/6 of the project in one day.
---STEP 7: If they complete 1/6 of the project in one day, then they will take the reciprocal of this fraction to complete the whole project. So, 6 days.
ANSWER: It will take Rohan and Sohan 6 days to finish the project together.

Why It Matters

Understanding Time and Work helps engineers plan big construction projects like bridges or buildings, making sure work is finished on time. Data scientists use similar logic to optimize tasks in computer systems. It's also useful for managers in any field to estimate project timelines and allocate resources efficiently, which can lead to successful careers in project management or even entrepreneurship.

Common Mistakes

MISTAKE: Adding the number of days directly (e.g., if A takes 10 days and B takes 15 days, thinking they take 10+15 = 25 days together). | CORRECTION: You must find the 'one day's work' (or 'one hour's work', etc.) for each person/machine first, then add those fractions. The total time will be the reciprocal of their combined work.

MISTAKE: Not finding a common denominator correctly when adding fractions for combined work. | CORRECTION: Always find the Least Common Multiple (LCM) of the denominators to make adding fractions easier and avoid errors.

MISTAKE: Forgetting to take the reciprocal at the end to find the total time. | CORRECTION: After you find the fraction of work done per unit of time (e.g., 1/6 of the work per day), remember that the total time taken is 1 divided by that fraction (e.g., 1 / (1/6) = 6 days).

Practice Questions

Try It Yourself

QUESTION: A tap can fill a tank in 8 hours. What fraction of the tank does it fill in 1 hour? | ANSWER: 1/8

QUESTION: Priya can make 50 rotis in 2 hours. If she works at a constant rate, how many rotis can she make in 3 hours? | ANSWER: 75 rotis

QUESTION: Ram can build a wall in 12 days. Shyam can build the same wall in 18 days. If they work together for 6 days, what fraction of the wall is still left to be built? | ANSWER: 1/6

MCQ

Quick Quiz

A worker can complete a task in 5 hours. What is the worker's rate of work per hour?

5 tasks per hour

1/5 task per hour

5 hours per task

1 task per 5 hours

The Correct Answer Is:

If a worker completes a task in 5 hours, their rate of work per hour is the reciprocal of the total time, which is 1/5 of the task per hour. Options A, C, and D describe the total time or an incorrect rate.

Real World Connection

In the Real World

Think about building a new metro line in a city like Delhi or Mumbai. Engineers need to calculate how many workers and machines (like tunnel boring machines) are needed to complete a section of the track by a deadline. They use Time and Work concepts to estimate project duration, allocate resources, and ensure the project stays on schedule, avoiding delays and cost overruns.

Key Vocabulary

Key Terms

WORK RATE: The amount of work done per unit of time (e.g., 1/10 of a project per day) | TOTAL WORK: The entire task that needs to be completed (usually considered as '1 unit' or 'the whole project') | EFFICIENCY: How quickly or effectively someone or something can complete a task | RECIPROCAL: The inverse of a number (e.g., the reciprocal of 1/6 is 6)

What's Next

What to Learn Next

Great job understanding Time and Work problems! Next, you can explore 'Pipes and Cisterns' problems. These are very similar but involve water filling or emptying tanks, using the same core ideas you just learned.