Question

In terms of calories, 20 jamuns and 3 bananas combined are equivalent to one mango and one papaya combined, and 2 mangoes and 5 bananas combined are equivalent to 4 papayas and 80 jamuns combined. Then 8 papayas and 2 mangoes are equivalent to how many bananas?

Hint:

In systems of equations from word problems, look for convenient relationships between coefficients. Here, noticing that \(80J = 4 \times 20J\) is the key to a quick substitution. This avoids having to solve for J itself and simplifies the algebra significantly.

Answer 1

Step 1: Understanding the Concept: 
This is a word problem that can be solved by setting up and solving a system of linear equations. We need to represent the caloric value of each fruit with a variable and translate the given statements into equations. 
Step 2: Key Formula or Approach: 
Let the caloric values be: - \(J\) for one jamun - \(B\) for one banana - \(M\) for one mango - \(P\) for one papaya Translate the given information into equations: 1. \(20J + 3B = M + P\) 2. \(2M + 5B = 4P + 80J\) We need to find the value of \(x\) in the equation: \(8P + 2M = xB\). 
Step 3: Detailed Explanation: 
Our goal is to find a relationship between M, P, and B. To do this, we must eliminate the variable \(J\) from the two equations. From Equation 1, we can express \(20J\) in terms of the other fruits: \[ 20J = M + P - 3B \] Now look at Equation 2. It contains the term \(80J\), which is equal to \(4 \times (20J)\). We can substitute our expression for \(20J\) into Equation 2. \[ 2M + 5B = 4P + 4 \times (20J) \] \[ 2M + 5B = 4P + 4 \times (M + P - 3B) \] Now, let's simplify and solve for the desired expression (\(8P + 2M\)). \[ 2M + 5B = 4P + 4M + 4P - 12B \] Combine like terms on the right side: \[ 2M + 5B = (4P + 4P) + 4M - 12B \] \[ 2M + 5B = 8P + 4M - 12B \] Now, we want to isolate the term \(8P + 2M\). Let's move all terms with \(B\) to the left side and all terms with \(M\) and \(P\) to the right side. It's easier to move the \(M\) and \(B\) terms to group them as needed. Let's move the \(B\) terms to the left and the \(M\) terms to the right: \[ 5B + 12B = 8P + 4M - 2M \] \[ 17B = 8P + 2M \] 
Step 4: Final Answer: 
The resulting equation, \(17B = 8P + 2M\), gives us the exact equivalence we were looking for. It shows that 8 papayas and 2 mangoes combined are equivalent in calories to 17 bananas. Therefore, the answer is 17.