Solving a Real-Life Arithmetic Problem: The Cost of Apples and Bananas
The problem presented here involves a common mathematical task, where we need to solve a system of linear equations to determine the cost of each individual item. This type of problem is not only a great way to practice algebraic skills but also applicable in real-life scenarios such as shopping.
Introduction to the Problem
Joe bought 5 apples and 4 bananas for $6, while Dawn bought 3 apples and 6 bananas for $6.30. The goal is to determine the cost of each apple and banana.
Setting Up the Equations
Let's denote:
The price of an apple as a The price of a banana as bWe can set up two equations based on the given information:
Equation 1:
5 apples and 4 bananas cost $6:
5a 4b 6
Equation 2:
3 apples and 6 bananas cost $6.30:
3a 6b 6.30
Solving the System of Linear Equations
Let's solve the system of equations step-by-step:
Step 1: Isolate one variable
From Equation 2, we can isolate a:
3a 6b 6.30
Subtract 6b from both sides:
3a 6.30 - 6b
Divide by 3:
a (6.30 - 6b)/3
Simplify the right side:
a 2.1 - 2b
Step 2: Substitute into the other equation
Substitute (a 2.1 - 2b) into Equation 1:
5(2.1 - 2b) 4b 6
Expand and simplify:
10.5 - 10b 4b 6
Combine like terms:
10.5 - 6b 6
Subtract 10.5 from both sides:
-6b 6 - 10.5
Simplify the right side:
-6b -4.5
Divide both sides by -6:
b 0.75
Step 3: Solve for the other variable
Now that we know (b 0.75), substitute this value back into (a 2.1 - 2b):
a 2.1 - 2(0.75)
Calculate:
a 2.1 - 1.5
a 0.60
Conclusion
Thus, the cost of an apple is 60 cents and the cost of a banana is 75 cents. We can verify this solution by plugging these values into the original equations.
Verification:
5 apples and 4 bananas at 60 cents and 75 cents respectively:
5 × 0.60 4 × 0.75 3.00 3.00 6.00
3 apples and 6 bananas at 60 cents and 75 cents respectively:
3 × 0.60 6 × 0.75 1.80 4.50 6.30
This confirms that our solution is correct.
Final Answer
The cost of an apple is 60 cents and the cost of a banana is 75 cents.
Reusable Equations
For reference, here are the manipulated equations:
Equation 1:
5 × apple 4 × banana 600
Equation 2:
1 × apple 2 × banana 210
Multiplying the second equation by 5 and the first equation by 1, we get:
1 × A 2 × B → apple 0.60
Multiplying the first equation by 5 and the second by 1, we get:
5 × B 1 × A → banana 0.75
Related Keywords
arithmetic problem linear equations shopping math cost calculationTags: #arithmetic #linearEquations #shoppingMath #costCalculation
If you enjoyed this problem and solution, feel free to explore more arithmetic and algebra problems for practice and improvement.
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