Solving a Word Problem with Algebra: Buying Fruits at the Market

A common task for school math classes, especially in algebra, is solving word problems. Here, we'll delve into a problem about Apples and Pears and employ algebraic techniques to find a solution.

Problem Statement

Ali buys a total of 70 Apples and Pears. He pays a total of 12 dollars for the Apples and 18 dollars for the Pears. A Pear costs twice as much as an Apple. Form an equation in terms of the number of Pears (x) and solve it.

Step-by-Step Solution

Step 1: Define Variables

Let:

a be the cost of one Apple. p be the cost of one Pear.

Form the following equations based on the problem statement:

Total number of fruits:

x y 70

Total cost:

ax py 12 18 30

Relationship between the cost of a Pear and an Apple:

p 2a

Step 2: Substitute and Simplify

Substitute p 2a into the cost equation:

ax 2ay 30

Factor out a from the left side:

a(x 2y) 30

Step 3: Express y in Terms of x

From the total number of fruits equation, express y in terms of x:

y 70 - x

Step 4: Substitute y into the Cost Equation

Substitute y 70 - x into the equation a(x 2y) 30:

a(x 2(70 - x)) 30

Expand and simplify:

a(70 - x) 30

Step 5: Solve for a

Isolate a:

a frac{30}{70 - x}

Step 6: Use the Price Relationship to Find x

Ali pays a total of 12 dollars for the Apples:

ay 12

Substitute y 70 - x into the equation ay 12:

a(70 - x) 12

Substitute a frac{30}{70 - x} into the equation:

frac{30}{70 - x}(70 - x) 12

Cancel out the 70 - x term:

30 12(70 - x)

Solve for x:

2100 - 30 840 - 12x 12x

2100 - 30 840 - 12x

2070 12x

x frac{2070}{12} 222.5

Step 7: Conclusion

The calculation results in a non-integer value, which suggests a need for review or adjustment. If you are looking for a specific integer solution, check the constraints given in the problem or consider if the costs align correctly with integer numbers of fruits.