Question : If the ratio of two numbers is 1 : 5 and their product is 320, the difference between the squares of these two numbers is:
Option 1: 1024
Option 2: 1256
Option 3: 1536
Option 4: 1435
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Correct Answer: 1536
Solution : Let the numbers be $x$ and $5x$. According to the question, $x × 5x= 320$ $\Rightarrow 5x^2 = 320$ $\Rightarrow x^2 = 64$ $\Rightarrow x = 8$ So, the numbers are $8$ and $(8×5)=40$. Now, the difference between the squares of $8$ and $40$ $= 40^2 - 8^2$ $= 1600 - 64$ $= 1536$ Hence, the correct answer is 1536.
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Question : The square of the difference between two given natural numbers is 324, while the product of these two given numbers is 144. Find the positive difference between the squares of these two given numbers.
Option 1: 630
Option 2: 540
Option 3: 450
Option 4: 360
Question : The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the positive difference between the cubes of these two given numbers.
Option 1: 4104
Option 2: 5616
Option 3: 2160
Option 4: 5626
Question : The difference between the cubes of two given natural numbers is 6272, while the positive difference between the two given numbers is 8. What is the product of the two given numbers?
Option 1: 160
Option 2: 240
Option 3: 200
Option 4: 320
Question : The sum of the squares of three positive numbers is 323. If the sum of the squares of two numbers is twice the third, their product is:
Option 1: 255
Option 2: 260
Option 3: 275
Option 4: 270
Question : The sum of two numbers is equal to 25 and their difference is 20. The ratio of the two numbers is:
Option 1: 9 : 1
Option 2: 7 : 9
Option 3: 3 : 5
Option 4: 2 : 7
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