Q. The total cost of 4 oranges, 6 mangoes and 8 apples is equal to twice the total cost of 1 orange, 2 mangoes and 5 apples.
Consider the following statements:
1. The total cost of 3 oranges, 5 mangoes and 9 apples is equal to the total cost of 4 oranges, 6 mangoes and 8 apples.
2. The total cost of one orange and one mango is equal to the cost of one apple.
Which of the statements given above is/are correct?
a) 1 only
b) 2 only
c) Both 1 and 2
d) Neither 1 nor 2
Correct answer: c) Both 1 and 2
Question from UPSC Prelims 2024 CSAT
Explanation :
Let’s analyze the problem step by step.
Let:
o = cost of one orange
m = cost of one mango
a = cost of one apple
Given:
The total cost of 4 oranges, 6 mangoes, and 8 apples is equal to twice the total cost of 1 orange, 2 mangoes, and 5 apples.
This can be written as:
4o + 6m + 8a = 2(1o + 2m + 5a)
4o + 6m + 8a = 2o + 4m + 10a
Simplifying:
Subtract 2o + 4m + 10a from both sides:
2o + 2m – 2a = 0
Divide by 2:
o + m = a
This confirms Statement 2: The total cost of one orange and one mango is equal to the cost of one apple.
Now, let’s evaluate Statement 1:
“The total cost of 3 oranges, 5 mangoes, and 9 apples is equal to the total cost of 4 oranges, 6 mangoes, and 8 apples.”
Expressed mathematically:
3o + 5m + 9a = 4o + 6m + 8a
Rearrange:
3o + 5m + 9a – 4o – 6m – 8a = 0
-o – m + a = 0
o + m = a
This is the same as what we derived earlier, hence Statement 1 is also correct.
Conclusion: Both statements 1 and 2 are correct.