Subtraction Is Not Multiplication Normally

Algebra Level 2

If a , b , d a, b, d are positive integers such that

a b − a d = a b × a d , \frac{a}{b } - \frac{a}{d} = \frac{a}{b} \times \frac{a}{d} ,

then which of the following statements must be true?

a + b = d a + b = d b + d = a b + d = a a + d = b a + d= b a + b + d = 0 a + b + d = 0

Chung Kevin by Chung Kevin , 45, Australia

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1 solution

Pham Khanh
Mar 2, 2017

a b − a d = a b × a d \frac{a}{b}-\frac{a}{d}=\frac{a}{b} \times \frac{a}{d} ⟺ a d − a b b d = a 2 b d \iff \frac{ad-ab}{bd}=\frac{a^{2}}{bd} ⟺ a d − a b = a 2 \iff ad-ab=a^{2} ⟺ d = b + a \iff \boxed{d=b+a}

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