🎂 Time and Motion: Ages & Partnerships
Imagine you’re a detective with a time machine. You travel through someone’s life, collecting clues about their age. And sometimes, you team up with friends to share treasures!
🌟 The Big Picture
Age Problems = Figuring out how old people are (or were, or will be) using clues.
Partnership Problems = Sharing money fairly when friends work together.
Think of it like this:
- Ages → A time detective game 🕵️
- Partnerships → Splitting pizza based on how many slices each friend earned 🍕
🕵️ PART 1: AGE PROBLEMS
What Are Age Problems?
Age problems give you clues about people’s ages and ask you to find the missing piece.
The Secret Weapon: Let the unknown age be x (a magic box that holds the answer).
🎯 Type 1: Present Age Clues
The Pattern:
“Someone is X years older/younger than another person”
Example: The Brother Mystery
Tom is 5 years older than his sister Lily. Together, their ages add up to 25. How old is each?
Step-by-Step Detective Work:
Step 1: Let Lily's age = x
Step 2: Tom's age = x + 5 (he's 5 years older)
Step 3: Together = x + (x + 5) = 25
Step 4: Solve → 2x + 5 = 25
2x = 20
x = 10
Answer: Lily is 10, Tom is 15 ✓
💡 Check: 10 + 15 = 25 ✓ and 15 - 10 = 5 ✓
🎯 Type 2: Past & Future Ages
The Time Travel Rule:
- Years ago → Subtract from current age
- Years later → Add to current age
Example: The Father-Son Time Travel
A father is 40 years old. His son is 10. In how many years will the father be twice as old as his son?
Detective Work:
Let "x" years pass...
Father's future age = 40 + x
Son's future age = 10 + x
When father = 2 × son:
40 + x = 2(10 + x)
40 + x = 20 + 2x
40 - 20 = 2x - x
20 = x
Answer: In 20 years (Father = 60, Son = 30) ✓
🎯 Type 3: Ratio of Ages
The Pattern:
“Ages are in ratio a:b” means ages are a parts and b parts
Example: The Cousin Ratio
Two cousins’ ages are in ratio 3:5. After 4 years, the ratio becomes 2:3. Find their ages.
graph TD A["Now: 3x and 5x"] --> B["After 4 years"] B --> C["3x + 4 and 5x + 4"] C --> D["New ratio = 2:3"] D --> E["Solve: #40;3x+4#41;/#40;5x+4#41; = 2/3"]
Solving:
Cross multiply:
3(3x + 4) = 2(5x + 4)
9x + 12 = 10x + 8
12 - 8 = 10x - 9x
4 = x
Answer: Ages are 12 and 20 years ✓
🔑 Age Problems: Golden Rules
| Clue Type | What It Means |
|---|---|
| “X years ago” | Subtract X from age |
| “X years hence” | Add X to age |
| “A is twice B” | A = 2 × B |
| “Sum of ages” | Add all ages together |
| “Difference of ages” | Bigger age - Smaller age |
⚠️ Important: The age difference between two people never changes! If you’re 5 years older than your sibling now, you’ll be 5 years older forever.
🤝 PART 2: PARTNERSHIP PROBLEMS
What Are Partnership Problems?
When friends start a business together, they need to share profits fairly.
The Fair Share Rule:
You get profit based on: How much you invest × How long you invest
Think of it like planting seeds:
- More seeds planted = more fruits you deserve
- Seeds planted longer = more fruits you deserve
🎯 Type 1: Simple Partnership
When everyone invests for the SAME time:
Profit Share = Investment Ratio
Example: The Lemonade Stand
Ali invests ₹3000 and Ben invests ₹2000 in a lemonade stand. They work together for the whole summer. If profit is ₹1000, how much does each get?
Investment Ratio = Ali : Ben = 3000 : 2000 = 3 : 2
Total parts = 3 + 2 = 5
Ali's share = (3/5) × 1000 = ₹600
Ben's share = (2/5) × 1000 = ₹400
Answer: Ali gets ₹600, Ben gets ₹400 ✓
🎯 Type 2: Compound Partnership
When people invest for DIFFERENT times:
Profit Ratio = (Investment × Time) for each person
Example: The Bakery Partners
Sam invests ₹5000 for 6 months. Maya invests ₹6000 for 5 months. How do they split ₹2200 profit?
graph TD A["Sam: ₹5000 × 6 = 30000"] --> C["Ratio = 30000 : 30000"] B["Maya: ₹6000 × 5 = 30000"] --> C C --> D["Simplified = 1 : 1"] D --> E["Equal share: ₹1100 each!"]
Calculation:
Sam's contribution = 5000 × 6 = 30,000
Maya's contribution = 6000 × 5 = 30,000
Ratio = 30,000 : 30,000 = 1 : 1
Each gets = 2200 ÷ 2 = ₹1100
🎯 Type 3: Partner Joins Later
When someone joins the business after it started:
Example: The Tech Startup
Ravi starts a business with ₹40,000. After 3 months, Priya joins with ₹60,000. At year end, profit is ₹14,000. Find each share.
Ravi invests for: 12 months
Priya invests for: 12 - 3 = 9 months
Ravi's contribution = 40,000 × 12 = 4,80,000
Priya's contribution = 60,000 × 9 = 5,40,000
Ratio = 480000 : 540000 = 8 : 9
Total parts = 8 + 9 = 17
Ravi's share = (8/17) × 14000 = ₹6,588
Priya's share = (9/17) × 14000 = ₹7,412
🎯 Type 4: Working Partner Bonus
When one partner also works (not just invests):
The working partner gets a salary/bonus FIRST, then remaining profit is shared.
Example: The Restaurant
Amit invests ₹30,000 and Bina invests ₹45,000. Bina runs the restaurant and gets ₹2000/month as salary. Annual profit is ₹42,000. Find each share.
Step 1: Bina's salary = 2000 × 12 = ₹24,000
Step 2: Remaining profit = 42,000 - 24,000 = ₹18,000
Step 3: Investment ratio = 30,000 : 45,000 = 2 : 3
Step 4: Amit's share = (2/5) × 18,000 = ₹7,200
Bina's share from profit = (3/5) × 18,000 = ₹10,800
Step 5: Bina's total = 24,000 + 10,800 = ₹34,800
🔑 Partnership: Golden Rules
| Situation | Formula |
|---|---|
| Same time investment | Ratio of investments |
| Different time | Investment₁ × Time₁ : Investment₂ × Time₂ |
| Partner joins late | Calculate time from joining date |
| Working partner | Deduct salary first, then share remaining |
🌈 Quick Memory Tricks
For Ages:
“PAST is LESS, FUTURE is MORE”
- Years ago = Subtract
- Years later = Add
For Partnerships:
“MONEY × MONTHS = YOUR SHARE”
- More money invested = bigger share
- Longer time invested = bigger share
🎮 Try These Yourself!
Age Challenge: A mother is 25 years older than her daughter. In 5 years, the mother will be twice as old as her daughter. What are their current ages?
👀 Click for Answer
Let daughter’s age = x Mother’s age = x + 25
After 5 years: (x + 25 + 5) = 2(x + 5) x + 30 = 2x + 10 20 = x
Daughter = 20 years, Mother = 45 years ✓
Partnership Challenge: Three friends invest ₹10,000, ₹15,000, and ₹20,000. They earn ₹9,000 profit. How much does each get?
👀 Click for Answer
Ratio = 10 : 15 : 20 = 2 : 3 : 4 Total parts = 9
- First friend: (2/9) × 9000 = ₹2,000
- Second friend: (3/9) × 9000 = ₹3,000
- Third friend: (4/9) × 9000 = ₹4,000
🏆 You’ve Got This!
Remember:
- Ages = Time detective work with algebra
- Partnerships = Fair sharing based on contribution
Both use the same superpower: Setting up equations and solving for unknowns!
🌟 “Math isn’t about numbers. It’s about finding what’s hidden and making things fair!”