Determine possible fruit prices and quantities that satisfy multiple mathematical constraints | Step-by-Step Solution
Problem
Find a combination of apple, banana, and coconut prices where: 1) Bob buys exactly 100 fruits in total, 2) Buys at least one of each fruit type, 3) Spends exactly 100 rupees, 4) Fruit prices are whole numbers or decimal values per fruit.
🎯 What You'll Learn
- Apply mathematical reasoning to complex multi-variable problems
- Develop strategic problem-solving skills
- Practice systematic approach to finding solution sets
Prerequisites: Algebraic equations, Linear constraint solving, Basic arithmetic
💡 Quick Summary
I can see you're working with a fascinating constraint satisfaction problem that combines algebra with real-world economics! This involves setting up a system of equations where you have multiple conditions that must all be satisfied simultaneously - Bob needs exactly 100 fruits, must spend exactly 100 rupees, and needs at least one of each type. Here's an interesting way to think about it: what if you were the store owner who gets to set the fruit prices first - what would be some reasonable, simple prices that might make the math work out nicely? Once you choose strategic prices (like keeping them as simple decimals or whole numbers), you can substitute those into your constraint equations and solve for the quantities. Think about which fruits might logically be cheaper or more expensive in real life, and how that would affect how many Bob would buy of each type. Give it a try by picking some realistic prices first, then see what quantities fall out of the math!
Step-by-Step Explanation
What We're Solving:
We need to find fruit prices and quantities where Bob buys exactly 100 fruits (at least 1 of each type), spends exactly 100 rupees, and all prices are reasonable numbers. This is a system of equations problem with multiple variables!The Approach:
This is what mathematicians call a "constraint satisfaction problem." We have conditions that must ALL be true at the same time. The key insight is that we can choose our own prices (within reason), then see what quantities work. It's like being the store owner who gets to set prices, then figuring out what Bob would buy!Step-by-Step Solution:
Step 1: Set up our variables and constraints
- Let a, b, c = quantities of apples, bananas, coconuts
- Let Pa, Pb, Pc = prices per apple, banana, coconut
- Constraints: a + b + c = 100, a·Pa + b·Pb + c·Pc = 100, a≥1, b≥1, c≥1
- Apples: 0.50 rupees each (cheap and popular)
- Bananas: 1.00 rupee each (medium price)
- Coconuts: 5.00 rupees each (expensive, so he'll buy fewer)
- a + b + c = 100 (total fruits)
- 0.5a + 1b + 5c = 100 (total cost)
Step 5: Find valid integer solutions Since a = 8c and we need a≥1, b≥1, c≥1:
- If c = 5, then a = 40, and b = 100 - 40 - 5 = 55
- Check: 40 + 55 + 5 = 100 ✓
- Check: 0.5(40) + 1(55) + 5(5) = 20 + 55 + 25 = 100 ✓
The Answer:
One valid solution is:- Prices: Apples 0.50 rupees, Bananas 1.00 rupee, Coconuts 5.00 rupees
- Quantities: 40 apples, 55 bananas, 5 coconuts
- Verification: 40 + 55 + 5 = 100 fruits, 20 + 55 + 25 = 100 rupees
Memory Tip:
Think of this as the "Price-Setting Game" - you're the store owner who gets to pick reasonable prices first, then solve for quantities. Start with simple prices (like 0.5, 1, 5) to make the algebra manageable. The expensive item (coconut) will have the smallest quantity, which makes intuitive sense!Great work tackling this multi-constraint problem! These types of problems show up in economics, business planning, and resource allocation - you're learning really practical math skills! 🍎🍌🥥
⚠️ Common Mistakes to Avoid
- Not checking all constraint conditions simultaneously
- Overlooking integer/decimal price requirements
- Failing to verify total fruit count and total cost
This explanation was generated by AI. While we work hard to be accurate, mistakes can happen! Always double-check important answers with your teacher or textbook.
Meet TinyProf
Your child's personal AI tutor that explains why, not just what. Snap a photo of any homework problem and get clear, step-by-step explanations that build real understanding.
- ✓Instant explanations — Just snap a photo of the problem
- ✓Guided learning — Socratic method helps kids discover answers
- ✓All subjects — Math, Science, English, History and more
- ✓Voice chat — Kids can talk through problems out loud
Trusted by parents who want their kids to actually learn, not just get answers.
TinyProf
📷 Problem detected:
Solve: 2x + 5 = 13
Step 1:
Subtract 5 from both sides...
Join our homework help community
Join thousands of students and parents helping each other with homework. Ask questions, share tips, and celebrate wins together.
Need help with YOUR homework?
TinyProf explains problems step-by-step so you actually understand. Join our waitlist for early access!