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The figure shows a number line. Point A is on 1, and point B is on 3, and ABCD is a rectangle with AD = 1. As the figu

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by Max@Math Revolution » Mon May 18, 2020 12:43 am

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[GMAT math practice question]

The figure shows a number line. Point A is on 1, and point B is on 3, and ABCD is a rectangle with AD = 1. As the figure shows, BD is a diagonal of ABCD and BD = BE. If x is the coordinate of point E, a is the integer part of x, and b is the decimal part of x, what is a - b/a + b?
5.18ps.png
A. 10 - 3 \(\sqrt{5}\) /2
B. 13 - 2 \(\sqrt{5}\) /3
C. 13 - 3 \(\sqrt{5}\) /3
D. 13 - 5 \(\sqrt{5}\) /2
E. 15 - 2 \(\sqrt{3}\)/ 5
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Re: The figure shows a number line. Point A is on 1, and point B is on 3, and ABCD is a rectangle with AD = 1. As the

by Max@Math Revolution » Wed May 20, 2020 3:00 am
=>

x, the coordinate of E is 3 + BD = 3 + √5.
Since we have 2 < √5 < 3, we have 5 < BD < 6 and a = 5, b = (3 + √5) – 5 = √5 - 2.
Thus, a-b/a+b
= 5 - ( √5 - 2)/5 + ( √5 - 2)
= 7 - √5/3 + √5 (adding like terms)
= (7 - √5)(3 - √5)/(3 + √5)(3 - √5)) (multiplying the numerator and denominator by the conjugate (3 - √5))
= 21 - 7 √5 - 3 √5 + 5/9 - 3 √5 + 3 √5 - 5
= 26 - 10 √5/4 (adding like terms)
= 13 - 5 √5/2 (dividing all terms by 2)

Therefore, D is the answer.
Answer: D
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