Problem: Define an operation * for positive real numbers as a∗b=aba+ba * b=\dfrac{a b}{a+b}a∗b=a+bab. Then 4∗(4∗4)4^{*}(4 * 4)4∗(4∗4) equals:
Answer Choices:
A. 34\dfrac{3}{4}43
B. 111
C. 43\dfrac{4}{3}34
D. 222
E. 163\dfrac{16}{3}316 Solution:
4∗4=4⋅44+4=2∴4∗(4∗4)=4⋅24+2=434 * 4=\dfrac{4 \cdot 4}{4+4}=2 \qquad \therefore 4 *(4 * 4)=\dfrac{4 \cdot 2}{4+2}=\dfrac{4}{3}4∗4=4+44⋅4=2∴4∗(4∗4)=4+24⋅2=34.