Solving Orchard Area Problems Using Basic Algebra
Understanding how to solve word problems related to the area of an orchard can be quite useful, especially for those involved in agriculture or studying mathematics.
This article will walk you through the process of solving a problem where a portion of a fruit farmer's orchard is planted with different types of trees. We will use basic algebra to find the area of the entire orchard.
Problem Statement
A fruit farmer has an orchard, and 1/6 of it is planted with apples. 1/5 of the rest is planted with other trees. 1/3 of the remaining area is planted with pear trees, and 4 hectares of the orchard are planted with pears. What is the area of the orchard?
Solution
Let's denote the area of the entire orchard by and.
Step 1: Calculating the Area Planted with Apples
1/6 of the orchard is planted with apples.
So, the area planted with apples 1/6 A.
Step 2: Calculating the Remaining Area After Apples
The remaining area after planting the apples can be calculated as follows.
Remaining area after apples A - 1/6 A 5/6 A.
Step 3: Calculating the Area Planted with Trees
1/5 of the remaining area (5/6 A) is planted with other trees.
So, the area planted with trees 1/5 (5/6 A) 1/6 A.
Step 4: Calculating the Remaining Area After Trees
Now, calculate the remaining area after planting the trees.
Remaining area after planting trees 5/6 A - 1/6 A 4/6 A 2/3 A.
Step 5: Calculating the Area Planted with Pear Trees
1/3 of the remaining area (2/3 A) is planted with pear trees.
So, the area planted with pear trees 1/3 (2/3 A) 2/9 A.
Step 6: Calculating the Remaining Area
4 hectares are planted with pears.
So, the remaining area is 4 hectares.
This means that 2/9 A 4 hectares.
Solving for A
To solve for A, we rearrange the equation:
2/9 A 4
A 4 × (9/2) 18/2 9 hectares.
Therefore, the area of the orchard is 9 hectares.
Further Explanations
Converting the planted areas to a common denominator can make the problem easier to understand. In this case, the common denominator of the first two sentences is 30. Therefore, the farmer planted 5/30s (apples) and 6/30s (trees) of his orchard. This adds up to 11/30s, leaving 19/30s of the orchard yet to be planted.
1/3 of the 19/30s (which is 19/90s) of the orchard was planted with pear trees. The remaining area, which is 2/3 of 19/30s (or 57/90s), equals the remaining 4 hectares planted with pears.
Therefore, 4 hectares 57/90 of the orchard.
Solving for the entire area:
4 × (90/57) 1 orchard 6 and 6/19s hectares or approximately 6.32 hectares.
Conclusion
Understanding how to solve problems related to the area of an orchard can be valuable in managing land and resources. Proper use of algebra and basic arithmetic can help in making informed decisions and can be key in optimization and efficiency.