Question

Suppose $n^\ast$ means $\dfrac1n,$ the reciprocal of $n.$ For example, $5^\ast = \dfrac15.$ How many of the following four statements are true?

(i) $3^\ast + 6^\ast = 9^\ast$ ; (ii) $6^\ast - 4^\ast = 2^\ast$ ; (iii) $2^\ast \cdot 6^\ast = 12^\ast$ ; (iv) $10^\ast \div 2^\ast = 5^\ast$

$0$

$1$

$2$

$3$

$4$

Answer: C

Concepts:custom operation fraction

Difficulty rating: 1030

Solution:

(i) $\dfrac13 + \dfrac16 = \dfrac12,$ but $9^\ast = \dfrac19,$ so false. (ii) $\dfrac16 - \dfrac14 = -\dfrac{1}{12},$ but $2^\ast = \dfrac12,$ so false.

(iii) $\dfrac12 \cdot \dfrac16 = \dfrac{1}{12} = 12^\ast,$ true. (iv) $\dfrac{1}{10} \div \dfrac12 = \dfrac15 = 5^\ast,$ true.

So exactly $2$ of the statements are true.

Thus, the correct answer is C .

1987 AMC 8 Problem 21

Answer: C

Solution:

Problem 21 in Other Years