Question 169
Sil has a lot of cards, which are yellow on one side and blue on the other. Most cards have a number on both sides. If two cards have the same number on the yellow side, then they have the same number on the blue side. There are also cards with a $\times$ on the yellow side and a $+$ on the blue side. Finally, there are cards which have an $=$ sign on both sides. If you put down a correct multiplication with some of the yellow cards and then turn over these cards, then you get a correct addition in blue. Cards with a $2$ on the yellow side have a $2$ on the blue side, cards with a $3$ on the yellow side have a $3$ on the blue side, and cards with a $4$ on the yellow side have a $4$ on the blue side.
Cards with a $5$ on the yellow side have a $5$ on the blue side.
All cards are lying on the table with the yellow side facing up. Sil tries to discover what is on the blue side, without turning over the cards. For example, cards with a $6$ on the yellow side have a $5$ on the blue side, because the yellow expression $2 \times 3 = 6$ must have $2+3=5$ on the blue back. Cards with a $20$ on the yellow side have a $9$ on the blue back side, because the yellow expression $2 \times 2 \times 5 = 20$ becomes $2+2+5=9$ in blue. The back of a yellow card containing a fraction, for example $\frac{5}{3}$, can be determined using $\frac{5}{3} \times 3 = 5$, which becomes $2+3=5$ when flipped; hence on the blue side is a $2$.
For which of the following numbers on the yellow side will there be a negative number on the blue side?
A) $\frac{9}{8}$
B) $\frac{25}{27}$
C) $\frac{32}{27}$
D) $\frac{64}{81}$
E) $\frac{128}{125}$
Cards with a $5$ on the yellow side have a $5$ on the blue side.
All cards are lying on the table with the yellow side facing up. Sil tries to discover what is on the blue side, without turning over the cards. For example, cards with a $6$ on the yellow side have a $5$ on the blue side, because the yellow expression $2 \times 3 = 6$ must have $2+3=5$ on the blue back. Cards with a $20$ on the yellow side have a $9$ on the blue back side, because the yellow expression $2 \times 2 \times 5 = 20$ becomes $2+2+5=9$ in blue. The back of a yellow card containing a fraction, for example $\frac{5}{3}$, can be determined using $\frac{5}{3} \times 3 = 5$, which becomes $2+3=5$ when flipped; hence on the blue side is a $2$.
For which of the following numbers on the yellow side will there be a negative number on the blue side?
A) $\frac{9}{8}$
B) $\frac{25}{27}$
C) $\frac{32}{27}$
D) $\frac{64}{81}$
E) $\frac{128}{125}$