Solving a System of Equations: A Fruitful Challenge

In this article, we will explore a fascinating problem involving fruit prices and how to solve it systematically using basic algebra. Let's dive into the details and uncover the hidden truth behind the value of plums and the price of a few fruits.

Problem Statement

Given the following information: if an apple and 3 pears are worth 1.7, and a pear and 2 plums are worth 1, how much is 5 plums worth?

Defining Variables

First, let's denote the price of an apple as A, the price of a pear as P, and the price of a plum as M. We can set up the equations based on the given information:

A 3P 1.7 (Equation 1)

P 2M 1 (Equation 2)

Solving the Equations Step by Step

Step 1: Solve for P in terms of M

From Equation 2, we can express P in terms of M as follows:

P 1 - 2M

Step 2: Substitute P into Equation 1

Now substitute P 1 - 2M into Equation 1:

A 3(1 - 2M) 1.7

Expanding and simplifying:

A 3 - 6M 1.7

A - 6M 1.7 - 3

A - 6M -1.3 (Equation 3)

Step 3: Express A in terms of M

From Equation 3, we can express A in terms of M:

A 6M - 1.3

Step 4: Substitute A back into Equation 1

Now use Equation 1 again:

(6M - 1.3) 3(1 - 2M) 1.7

Expanding and simplifying:

6M - 1.3 3 - 6M 1.7

1.7 1.7

This shows that the equations are consistent but we need to find a specific value for M.

Step 5: Isolate M

Using the relationship established, we find:

P 1 - 2M

Let's use the value M 0.5.

Substitution Method

Assuming M 0.5:

P 2(0.5) 1 gives P 0.

Substituting P 0 into A 3P 1.7 gives A 1.7.

Checking if these values satisfy both equations:

0 2(0.5) 1 is satisfied.

1.7 3(0) 1.7 is satisfied.

Final Calculation

Given M 0.5:

5M 5(0.5) 2.5.

Thus, the cost of 5 plums is 2.5.

Alternative Approach

Let's also consider the scenario where the apple is free. In this case, the three pears are worth 1.7/3, so each pear is worth 1.7/3/3. Since a pear and 2 plums are worth 1, each plum is worth 1 - 1.7/3/3/2. Therefore, five plums are worth 5(1 - 1.7/3/3/2).

However, without knowing the exact value of an apple, we cannot determine the exact price of the plums. The price of an apple is somewhere between free and 1.7 each.

Conclusion

The problem, as written, is poorly worded and unanswerable in its current form. However, by adding the detail that the apple can be considered free, we can solve for the price of the plums. The answer is that 5 plums are worth 2.5.