Question

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

$\mathrm{(i)}\quad 3^\ast + 6^\ast = 9^\ast$

$\mathrm{(ii)}\quad 6^\ast - 4^\ast = 2^\ast$

$\mathrm{(iii)}\quad 2^\ast \cdot 6^\ast = 12^\ast$

$\mathrm{(iv)}\quad 10^\ast \div 2^\ast = 5^\ast$

$0$

$1$

$2$

$3$

$4$

Answer: C

Concepts:custom operation fraction

Difficulty rating: 1030

Small Hint:

Replace each $n^\ast$ with $\dfrac1n$ and test the two sides of each statement

Big Hint:

For $\mathrm{(iv)},$ $10^\ast \div 2^\ast = \dfrac{1}{10} \div \dfrac12 = \dfrac{1}{10} \cdot 2$

Solution:

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

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

So exactly $2$ of the statements are true.

Thus, the correct answer is C .

1987 AMC 8 Problem 21

Answer: C

Difficulty rating: 1030

Small Hint:

Big Hint:

Solution:

Problem 21 in Other Years