Points to Remember:
- This is a mathematical word problem requiring an analytical approach.
- We need to translate the word problem into algebraic equations.
- The solution will involve solving a system of simultaneous equations.
Introduction:
This question involves finding two rational numbers (numbers that can be expressed as a fraction) based on the relationship between their difference, quotient, and remainder upon division. The problem tests our understanding of basic arithmetic operations and the ability to formulate and solve algebraic equations. Such problems are common in elementary algebra and have applications in various fields requiring numerical problem-solving.
Body:
1. Formulating the Equations:
Let’s denote the larger number as ‘x’ and the smaller number as ‘y’. The problem provides us with two key pieces of information:
- Equation 1 (Difference): x – y = 980
- Equation 2 (Division): x = 2y + 480 (This represents the division with a quotient of 2 and a remainder of 480)
2. Solving the Simultaneous Equations:
We can solve these equations simultaneously using substitution. Since we have an expression for ‘x’ in Equation 2, we can substitute this into Equation 1:
(2y + 480) – y = 980
Simplifying this equation:
y + 480 = 980
y = 980 – 480
y = 500
Now that we have the value of ‘y’, we can substitute it back into either Equation 1 or Equation 2 to find ‘x’. Let’s use Equation 2:
x = 2(500) + 480
x = 1000 + 480
x = 1480
3. Verification:
Let’s check if our solution satisfies the conditions given in the problem:
- Difference: 1480 – 500 = 980 (Correct)
- Division: 1480 ÷ 500 = 2 with a remainder of 480 (Correct)
Conclusion:
The two rational numbers are 1480 and 500. We successfully solved this problem by translating the word problem into a system of two linear equations and solving them using the substitution method. The solution demonstrates a clear understanding of arithmetic operations and algebraic techniques. This type of problem-solving skill is crucial for various quantitative reasoning tasks and further mathematical studies. The ability to translate real-world scenarios into mathematical models and solve them efficiently is a valuable skill that promotes logical thinking and analytical abilities. This approach can be applied to similar problems involving relationships between numbers and their operations.
CGPCS Notes brings Prelims and Mains programs for CGPCS Prelims and CGPCS Mains Exam preparation. Various Programs initiated by CGPCS Notes are as follows:-