Solving the Chickens and Goats Riddle
This problem involves setting up and solving a system of linear equations based on the given information about heads and legs.
Setting Up the Equations
Let $C$ represent the number of chickens and $G$ represent the number of goats.
- Heads Equation: Each animal has one head. $ C + G = 42 $
- Legs Equation: Chickens have 2 legs and goats have 4 legs. $ 2C + 4G = 138 $
Solving for the Number of Chickens ($C$)
We can use the substitution method. First, express $G$ in terms of $C$ using the heads equation:
From $C + G = 42$, we get $G = 42 - C$.
Now, substitute this expression for $G$ into the legs equation:
$ 2C + 4(42 - C) = 138 $Simplify and solve for $C$:
- Distribute the 4: $2C + 168 - 4C = 138$
- Combine the $C$ terms: $168 - 2C = 138$
- Subtract 168 from both sides: $-2C = 138 - 168$
- Calculate the difference: $-2C = -30$
- Divide by -2: $C = \frac{-30}{-2}$
- Result: $C = 15$
Therefore, there are 15 chickens.
Verification (Optional)
If $C = 15$, then $G = 42 - 15 = 27$.
Check the total legs: $(2 \times 15) + (4 \times 27) = 30 + 108 = 138$. The calculation is correct.
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What will come in the place of ‘?’ in the following equation if ‘+’ and ‘−’ are interchanged and ‘×’ and ‘÷’ are interchanged?
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