Time and Work: The Magic of Teamwork 🏗️
Imagine you’re building a sandcastle on the beach. You can do it alone, or you can team up with friends. Sometimes you take turns, and sometimes you work together. This is exactly what Time and Work is all about!
🎯 The Big Idea
Work is like filling a bucket with water. The bucket is your goal. How fast you fill it depends on how efficient you are!
Work = Rate × Time
Think of Rate as your speed—how much of the bucket you fill each minute.
1. Work and Efficiency ⚡
What is Efficiency?
Efficiency is how fast someone works. A faster worker has higher efficiency.
The Magic Formula
If you can finish a job in D days, then in one day you complete:
Work per day = 1/D of the total work
🏠 Simple Example
Ravi can paint a room in 6 days. How much does he paint each day?
Ravi paints 1/6 of the room every day.
That’s his daily work rate or efficiency!
🎨 Another Example
Priya finishes reading a book in 4 days. Amit finishes the same book in 8 days. Who is more efficient?
- Priya reads 1/4 of the book per day
- Amit reads 1/8 of the book per day
1/4 > 1/8, so Priya is faster!
She’s twice as efficient as Amit.
2. Working Together 🤝
The Power of Teamwork!
When two people work together, their speeds ADD UP.
graph TD A["Person A: 1/6 per day"] --> C["Combined Work"] B["Person B: 1/3 per day"] --> C C --> D["Total: 1/6 + 1/3 = 1/2 per day"] D --> E["Finish in 2 days!"]
The Formula
If A finishes in a days and B finishes in b days:
Combined rate = 1/a + 1/b
Time together = 1 ÷ (1/a + 1/b)
🏗️ Example: Building a Treehouse
Tom can build a treehouse in 12 days. Jerry can build it in 6 days. How long if they work together?
Step 1: Find each person’s daily work
- Tom: 1/12 per day
- Jerry: 1/6 per day
Step 2: Add their rates
1/12 + 1/6 = 1/12 + 2/12 = 3/12 = 1/4
Step 3: Find total time
Time = 1 ÷ 1/4 = 4 days
Answer: 4 days together! 🎉
🍕 Real-Life Example
You can eat a pizza in 30 minutes. Your friend eats it in 20 minutes. Together?
- Your rate: 1/30 pizza per minute
- Friend’s rate: 1/20 pizza per minute
- Together: 1/30 + 1/20 = 2/60 + 3/60 = 5/60 = 1/12
- Time: 12 minutes!
3. Working Alternately 🔄
Taking Turns
Sometimes workers don’t work together—they take turns. Like tag-team wrestling!
How It Works
graph TD A["Day 1: Person A works"] --> B["Day 2: Person B works"] B --> C["Day 3: Person A works"] C --> D["Day 4: Person B works"] D --> E["...and so on"]
The Strategy
- Calculate work done in one cycle (2 days if 2 people alternate)
- Find how many complete cycles fit
- Handle the remaining work
🎮 Example: Gaming Challenge
Ali can complete a game level in 10 hours. Bob can do it in 15 hours. They play alternately, Ali starts first. When will they finish?
Step 1: Work per hour
- Ali: 1/10 per hour
- Bob: 1/15 per hour
Step 2: Work in 2 hours (one cycle)
1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6
Step 3: Complete cycles needed
For 1 unit of work: 1 ÷ 1/6 = 6 cycles
6 cycles × 2 hours = 12 hours
But wait! Let’s check:
- After 5 cycles (10 hours): 5/6 work done
- Remaining: 1/6 work
Step 4: Ali starts hour 11
- Ali does 1/10 in hour 11
- Remaining: 1/6 - 1/10 = 5/30 - 3/30 = 2/30 = 1/15
Step 5: Bob finishes in hour 12
- Bob needs to do 1/15 at rate 1/15 per hour
- Time = 1 hour
Answer: 12 hours total!
4. Pipes and Cisterns 🚰
Water Tanks = Work Problems!
A cistern is just a water tank. Pipes can either:
- Fill the tank (positive work) ➕
- Empty the tank (negative work) ➖
The Key Insight
Filling pipes ADD water = Positive rate
Emptying pipes REMOVE water = Negative rate
graph TD A["Pipe A: Fills in 4 hrs"] --> C["Tank"] B["Pipe B: Empties in 6 hrs"] --> C C --> D["Net Rate = 1/4 - 1/6"] D --> E["= 3/12 - 2/12 = 1/12"] E --> F["Fills in 12 hours"]
🛁 Example: Filling a Bathtub
Tap A fills a tub in 8 minutes. Drain B empties it in 12 minutes. If both are open, when is the tub full?
Step 1: Rates
- Tap A fills: +1/8 per minute
- Drain B empties: -1/12 per minute
Step 2: Net rate
1/8 - 1/12 = 3/24 - 2/24 = 1/24
Step 3: Time to fill
Time = 1 ÷ 1/24 = 24 minutes
Answer: 24 minutes!
🏊 Pool Example
Pipe X fills a pool in 6 hours. Pipe Y fills it in 4 hours. Pipe Z drains it in 3 hours. All pipes are open. What happens?
Step 1: Find all rates
- X fills: +1/6
- Y fills: +1/4
- Z drains: -1/3
Step 2: Net rate
1/6 + 1/4 - 1/3
= 2/12 + 3/12 - 4/12
= 1/12
Step 3: Since net rate is positive (1/12), the pool fills!
Time = 12 hours
⚠️ What if Net Rate is Negative?
If draining is faster than filling, the tank never fills! It empties instead.
Example: If Z drained in 2 hours instead:
1/6 + 1/4 - 1/2 = 2/12 + 3/12 - 6/12 = -1/12
Tank empties in 12 hours!
🧠 Quick Reference Formulas
| Scenario | Formula |
|---|---|
| Work per day | 1 ÷ (Total days) |
| Working together | 1/a + 1/b |
| Time together | ab ÷ (a+b) |
| Alternating work | Find 1 cycle, then calculate |
| Pipes (filling) | Positive rate |
| Pipes (draining) | Negative rate |
🌟 Remember This!
- Efficiency = Speed of working (1/days to complete)
- Together = Add the rates
- Alternating = Work in cycles
- Pipes = Filling is positive, Draining is negative
💡 Pro Tips
Tip 1: Always convert to “work per unit time” first!
Tip 2: When workers alternate, find work done in one complete cycle.
Tip 3: For pipes, remember: Fill = Plus, Drain = Minus
Tip 4: If the answer is negative, it means draining wins!
🎯 The Sandcastle Story Continues…
You started alone, taking 6 hours to build a sandcastle. Your friend joined, and together you finished in just 2 hours! Then the waves (like a drain pipe) started eroding it. Will you build faster than the waves destroy?
That’s Time and Work! It’s all about understanding who does what, how fast, and whether you’re adding to the goal or taking away from it.
Now you’re ready to solve any time and work problem! 🏆