1987 AMC 8 Problem 21

Attempt Problem 21 of the 1987 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1987 AMC 8 solutions, or check the answer key.

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21.

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 .