What is Work and Time Problems using Algebra?

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Grade Level:

Class 7

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

Definition

What is it?

Work and Time problems using Algebra help us figure out how long it takes for individuals or groups to complete a task, or how much work they can do in a certain time. We use variables (like 'x' or 'y') to represent unknown quantities and form equations to solve them.

Simple Example

Quick Example

Imagine you and your friend are helping your parents clean the house for Diwali. If you can clean one room in 2 hours and your friend can clean the same room in 3 hours, how long will it take both of you to clean it together? Algebra helps us find this 'together' time.

Worked Example

Step-by-Step

PROBLEM: Rohan can paint a wall in 6 hours. Sohan can paint the same wall in 3 hours. How long will it take them to paint the wall if they work together?
---STEP 1: Find the work done by each person in 1 hour.
Rohan's 1-hour work = 1/6 of the wall
Sohan's 1-hour work = 1/3 of the wall
---STEP 2: Let 't' be the time they take to paint the wall together.
In 't' hours, Rohan does t/6 of the work.
In 't' hours, Sohan does t/3 of the work.
---STEP 3: When they work together, their combined work completes 1 whole wall.
So, (t/6) + (t/3) = 1
---STEP 4: Find a common denominator for the fractions (which is 6).
Multiply the second fraction by 2/2: (t/6) + (2t/6) = 1
---STEP 5: Add the fractions.
(t + 2t)/6 = 1
3t/6 = 1
---STEP 6: Simplify the fraction.
t/2 = 1
---STEP 7: Solve for 't'.
t = 1 * 2
t = 2
---ANSWER: It will take Rohan and Sohan 2 hours to paint the wall together.

Why It Matters

Understanding Work and Time problems is super useful in many fields! Engineers use it to plan large construction projects, like building a flyover, figuring out how many workers are needed and how long it will take. In project management, it helps managers schedule tasks and allocate resources efficiently, ensuring projects finish on time and within budget.

Common Mistakes

MISTAKE: Adding the times directly (e.g., if one takes 2 hours and another takes 3 hours, students might think it takes 2+3=5 hours together). | CORRECTION: You cannot add times directly. Instead, you add the 'rates of work' (the fraction of work done per unit of time) because when people work together, the task gets done faster, not slower.

MISTAKE: Forgetting that '1' represents the completion of the entire task. | CORRECTION: Always remember that the sum of the work done by individuals or groups must equal 1 (representing 100% of the task completed) when they finish the job together.

MISTAKE: Mixing up units, for example, calculating work per hour for one person and work per day for another. | CORRECTION: Ensure all work rates are in the same unit of time (e.g., all in 'work per hour' or 'work per day') before adding them.

Practice Questions

Try It Yourself

QUESTION: A tap can fill a tank in 10 hours. Another tap can fill the same tank in 15 hours. How much of the tank will be filled in 1 hour if both taps are opened? | ANSWER: 1/6 of the tank

QUESTION: Arijit can assemble a toy car in 4 minutes. If he works with his sister, who is a bit slower, they can assemble the same car in 3 minutes. How long would it take Arijit's sister to assemble the toy car by herself? | ANSWER: 12 minutes

QUESTION: Three friends, Priya, Qasim, and Ritu, are preparing for a school play. Priya can make all the props in 6 days. Qasim can make them in 8 days. If all three work together, they can finish in 2 days. How many days would Ritu take to make all the props alone? | ANSWER: 24 days

MCQ

Quick Quiz

A painter can paint a room in 5 hours. What fraction of the room can he paint in 2 hours?

1/5

2/5

1/2

5/2

The Correct Answer Is:

If the painter takes 5 hours to paint 1 room, in 1 hour he paints 1/5 of the room. So, in 2 hours, he will paint 2 * (1/5) = 2/5 of the room.

Real World Connection

In the Real World

Imagine you're helping your parents plan a renovation for your house in Delhi. A contractor uses Work and Time calculations to figure out how many painters, plumbers, and electricians are needed and how long the whole project will take to finish before the next festive season. This ensures your home is ready on time!

Key Vocabulary

Key Terms

WORK RATE: The amount of work done per unit of time (e.g., 1/5 of a wall per hour) | TOTAL WORK: Usually represented as '1' (meaning the entire task is completed) | ALGEBRAIC EQUATION: A mathematical statement with variables and an equals sign used to solve for unknowns | RECIPROCAL: Flipping a fraction (e.g., the reciprocal of 5 hours is 1/5 work per hour)

What's Next

What to Learn Next

Great job understanding Work and Time problems! Next, you can explore 'Pipes and Cisterns' problems. These are very similar but apply to filling or emptying tanks with water, using the same algebraic principles you've just learned. Keep practicing!