Consider the numbers a, b, c and d with values 54, 71, 75 and 99,…
2022
Consider the numbers a, b, c and d with values 54, 71, 75 and 99, respectively. Find a number which may be subtracted from each of them such that a/b = c/d.
Answer: C. 3 — ConceptIf the same number x is subtracted from four numbers a, b, c and d and the resulting ratios must be equal, write (a - x)/(b - x) = (c - x)/(d - x).…
- A.
5
- B.
2
- C.
3
- D.
4
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Correct answer: C
Concept
If the same number x is subtracted from four numbers a, b, c and d and the resulting ratios must be equal, write (a - x)/(b - x) = (c - x)/(d - x).
Cross-multiplication gives an equation in x; any common quadratic term cancels.
Application
Let x be the number subtracted from 54, 71, 75 and 99. Then (54 - x)/(71 - x) = (75 - x)/(99 - x).
Cross-multiply: (54 - x)(99 - x) = (75 - x)(71 - x).
Expand both sides: 5346 - 153x + x2 = 5325 - 146x + x2.
Cancel x2 and collect terms: 5346 - 5325 = 153x - 146x, so 21 = 7x and x = 3.
Cross-check
Subtracting 3 gives 51/68 and 72/96. Both fractions simplify to 3/4, confirming the equality.
Therefore, the required number is 3.