Question

In the farm-house of Bhuwan Kisan, there were total 14 deer and ducks. One day a thief entered secretely in the farmhouse and after killing all the deer and ducks he took away all the stuff except their legs. When Bhuwan noticed it he has found there 38 legs of deer and ducks. The number of deer in the farm house was: (a) 5 (b) 8 . (c) 9 (d) 10

Short answer

Answer: 5 deer

Step-by-step solution

Understand the given information

We are given that there are 14 deer and ducks in total and the total legs found are 38. We also know that deer have 4 legs and ducks have 2 legs each.

Define variables

Let's use two variables to represent the number of deer and ducks. Let D represent the number of deer and K represent the number of ducks.

Set up the equations

We have two equations based on the given information: 1. Total number of deer and ducks is 14. So, D + K = 14. 2. Total number of legs is 38. Since deer have 4 legs and ducks have 2 legs, we can represent this equation as 4D + 2K = 38.

Solve for the number of deer

From the first equation, we can write K in terms of D: K = 14 - D. Now, substitute this expression in the second equation: 4D + 2(14 - D) = 38. Now, we solve for D: 4D + 28 - 2D = 38 2D = 10 D = 5 So, there were 5 deer in the farmhouse. Thus, option (a) 5 is the correct answer.

Key concepts

Word Problems

Word problems are scenarios presented in a narrative format, requiring translation into mathematical language for solutions. In this exercise, understanding the word problem involves breaking down the story into mathematical elements. The narrative tells you Bhuwan Kisan has a mix of deer and ducks. After an unfortunate event, he counted 38 legs left at the farm. Your task is to determine how many of each animal were originally there.

To solve this, you identify crucial information:

• Total animals are 14.
• Total legs are 38.
• Deer have 4 legs, ducks have 2 legs.

The problem challenges you to convert this data into equations, a critical step in bridging the narrative to numbers. Carefully reading and comprehending the problem is key to correctly forming equations and solving them.

Simultaneous Equations

Simultaneous equations come into play when solving multiple equations together to find common solutions for the variables involved. In this problem, you need to determine the number of deer (D) and ducks (K). You create two equations based on the information:

• The sum of deer and ducks: \( D + K = 14 \)
• The sum of all legs: \( 4D + 2K = 38 \)

These equations reflect the relationships within the word problem. Solving simultaneous equations involves expressing one variable in terms of the other. Here, by expressing K as \( K = 14 - D \), you replace K in the second equation. Substituting simplifies the problem to one variable, allowing you to find a specific solution for D. These techniques are the essence of simultaneous equations, crucial for contextual multi-variable problems.

Problem Solving Techniques

Problem solving techniques are strategies employed to effectively decipher complex problems into manageable parts. One popular method is the use of substitution, as illustrated in this exercise. You start by defining variables (D and K) to represent the unknown quantities.

Next, break the problem into smaller parts to build equations. Analyze each piece of information and decide how it fits into the larger puzzle. With equations at hand, substitution helps reduce two equations to one, making it easier to solve.

A typical approach is:

• Formulate equations from the word problem.
• Express one variable in terms of the other.
• Substitute to simplify and solve the equations.

This technique provides a structured method for tackling algebraic expressions even in a narrative context, aiding in finding solutions confidently.