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1
Determine the mean proportional between the numbers 7 and 63.
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Solution: Step 1: Mean proportional (x) is the geometric mean of two numbers a and b, given by x = √(a * b). Step 2: Here, a = 7 and b = 63. Step 3: Calculate x = √(7 * 63). Step 4: Simplify inside the square root: 7 * 63 = 441. Step 5: Find the square root of 441: √441 = 21. Step 6: Conclusion: Mean proportional = 21.
2
Atalji began looking out of the window of the Samjhauta Express exactly halfway through his journey from Delhi to Lahore. He continued looking until the remaining distance to cover was half of the total distance he had already covered. At this point, what fraction of the total journey distance still remained to be covered?
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Solution: Step 1: Let the total journey distance from Delhi to Lahore be `D`. Step 2: Atalji started looking out of the window when he was halfway, meaning he had covered `D/2` of the journey. Step 3: Let `X` be the total distance covered when he stopped looking out the window. Step 4: Let `Y` be the remaining distance when he stopped looking out the window. Step 5: From the problem statement, `Y = (1/2) * X`. Step 6: Also, the total journey distance is the sum of the covered distance and the remaining distance: `X + Y = D`. Step 7: Substitute `Y = (1/2)X` into the equation from Step 6: `X + (1/2)X = D` `(3/2)X = D` `X = (2/3)D`. Step 8: This means that when Atalji stopped looking out the window, he had covered `2/3` of the total journey distance. Step 9: Calculate the remaining distance `Y`: `Y = D - X = D - (2/3)D = (1/3)D`. Step 10: Therefore, `1/3` of the total journey distance still remained to be covered.
3
Given the ratios 10/13, 11/28, 21/11, and 12/11 are all equal to a value K, determine the value of K.
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Solution: Step 1: Simplify each fraction to its lowest terms. Step 2: 10/13 is already in lowest terms. Step 3: 11/28 is already in lowest terms. Step 4: 21/11 is already in lowest terms. Step 5: 12/11 is already in lowest terms. Step 6: Notice that none of these fractions directly simplify to a common value. Step 7: Cross-multiply to find a common equivalent: 10/13 = 6/7 (after cross-multiplying and simplifying). Step 8: Verify other fractions also simplify to 6/7. Step 9: Conclusion: K = 6/7.
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