There Are Total Of 452 Cows And Goats In A Farm. 5/7 Of the Cows Are Equal To 13/27 Of The Goats In The
Understanding the composition of farm animals is essential for effective farm management, resource allocation, and planning. In this article, we explore a specific scenario involving a farm with a total of 452 cows and goats. We analyze the relationship between the numbers of cows and goats based on given fractions, solve for the individual counts, and discuss the implications of these findings. Whether you're a farm owner, student, or enthusiast interested in livestock calculations, this detailed guide will help you understand how to approach such problems systematically.
Introduction to the Problem
The problem statement provides two key pieces of information:
- The total number of animals (cows and goats) on the farm is 452.
- A relationship between the cows and goats expressed through fractions: "5/7 of the cows are equal to 13/27 of the goats."
Our goal is to determine the exact number of cows and goats on the farm based on these details.
Understanding the Fractions and Their Meaning
Before diving into calculations, let's interpret the fractions:
- 5/7 of the cows: This represents a certain subset or proportion of the total cows.
- 13/27 of the goats: Similarly, this is a subset or proportion of the total goats.
The statement "5/7 of the cows are equal to 13/27 of the goats" implies that these two quantities are numerically equal, providing a relationship between the counts of cows and goats.
Setting Up Variables and Equations
To solve the problem systematically, assign variables:
- Let C be the total number of cows.
- Let G be the total number of goats.
Given:
- Total animals:
- Fractional relationship:
This second equation captures the key relationship specified in the problem.
Formulating the Mathematical Model
Based on the above, our equations are:
\[
\begin{cases}
C + G = 452 \quad \text{(Equation 1)} \\
\frac{5}{7} C = \frac{13}{27} G \quad \text{(Equation 2)}
\end{cases}
\]
To simplify calculations, it's often easier to work with integers. Let's manipulate Equation 2:
\[
\frac{5}{7} C = \frac{13}{27} G
\]
Cross-multiplied:
\[
5 \times 27 \times C = 13 \times 7 \times G
\]
which simplifies to:
\[
135 C = 91 G
\]
Now, express G in terms of C:
\[
G = \frac{135 C}{91}
\]
Note that 91 and 135 share a common factor of 13:
- 91 = 13 × 7
- 135 = 13 × 10 + 5 (but 135 is 13 × 10 + 5, so no common factor other than 1)
Actually, 91 and 135 share no common factors besides 1, so the fraction is simplified as is.
Alternatively, express G as:
\[
G = \frac{135}{91} C
\]
And since total animals are 452:
\[
C + G = 452
\]
Substitute G:
\[
C + \frac{135}{91} C = 452
\]
Factor out C:
\[
C \left(1 + \frac{135}{91}\right) = 452
\]
Express the sum inside parentheses with a common denominator:
\[
1 + \frac{135}{91} = \frac{91}{91} + \frac{135}{91} = \frac{226}{91}
\]
So:
\[
C \times \frac{226}{91} = 452
\]
Solve for C:
\[
C = 452 \times \frac{91}{226}
\]
Simplify numerator and denominator:
- 226 = 2 × 113
- 91 = 7 × 13
Check if 452 can be simplified:
\[
452 = 4 \times 113
\]
Now, rewrite:
\[
C = 4 \times 113 \times \frac{91}{226}
\]
Since 226 = 2 × 113, substitute:
\[
C = 4 \times 113 \times \frac{91}{2 \times 113}
\]
Cancel 113:
\[
C = 4 \times \frac{91}{2}
\]
Simplify:
\[
C = 4 \times 45.5 = 182
\]
Thus, C = 182 (number of cows).
Now, find G:
\[
G = 452 - C = 452 - 182 = 270
\]
G = 270 (number of goats).
Verification of the Solution
Check the fractional relationship:
- Compute 5/7 of cows:
\[
\frac{5}{7} \times 182 = \frac{5 \times 182}{7} = \frac{910}{7} = 130
\]
- Compute 13/27 of goats:
\[
\frac{13}{27} \times 270 = \frac{13 \times 270}{27} = \frac{3510}{27} = 130
\]
Both are equal to 130, confirming the relationship holds.
Total animals check:
\[
C + G = 182 + 270 = 452
\]
Everything is consistent.
Implications and Applications
Understanding these calculations provides valuable insights:
- Farm Management: Accurate animal counts help in resource planning, feeding schedules, and healthcare management.
- Livestock Distribution: Knowing the proportions and relationships assists in breeding and sales decisions.
- Mathematical Modeling: Demonstrates how algebraic methods can solve real-world problems involving ratios and totals.
Summary of the Key Results
- Number of Cows (C): 182
- Number of Goats (G): 270
- Total Animals: 452
- Fractional Relationship: 5/7 of cows equals 13/27 of goats, both totaling 130 animals in the specified subsets.
Additional Tips for Solving Similar Problems
- Carefully interpret the fractions and relationships.
- Assign variables to unknown quantities.
- Convert fractional relationships into equations.
- Simplify equations by cross-multiplication.
- Use algebraic manipulation to isolate variables.
- Always verify solutions by plugging back into original equations.
- Check for consistency with total counts.
Conclusion
This detailed analysis illustrates how to approach and solve a livestock distribution problem involving fractions and totals. By translating the problem into algebraic equations, simplifying, and solving step-by-step, we found that the farm has 182 cows and 270 goats. These calculations are not only mathematically interesting but also practically valuable for farm management and planning. Mastering such problem-solving techniques enhances your ability to handle similar real-world challenges involving ratios, proportions, and totals efficiently.
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Keywords: farm livestock, cows and goats, algebraic problem-solving, ratios, fractions, livestock management, algebra tutorial, farm animal calculations
