What Is a Reciprocal?
For nonzero numbers _x_, _y_, and _z_, the reciprocal of _x_ is twice the value of _y_ and the reciprocal of _y_ is three times the reciprocal of _z_.
| Quantity A | Quantity B |
|----|----|
| _x_ • _z_ | 1 |
Correct.
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If the reciprocal of _x_ is twice _y_, $$\frac{1}{x} = 2 y$$, then _x_ is the reciprocal of 2_y_, $$x = \frac{1}{2y}$$. If the reciprocal of _y_ is three times the reciprocal of _z_, then _z_ is 3_y_.
$$\displaystyle \frac{1}{y} = 3\cdot \frac{1}{z}$$
$$\displaystyle \frac{1}{3y} = \frac{1}{z}$$
_z_ = 3_y_. With both _x_ and _z_ in terms of _y_, you can substitute in for Quantity A.
$$\displaystyle x\cdot z = \frac{1}{2y}\cdot 3y = \frac{3y}{2y} = \frac{3}{2}$$
$$\displaystyle \frac{3}{2} >1$$
Quantity A is greater.
Incorrect.
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If you plug in 2 for _x_, then the reciprocal of 2 is 1 / 2 and _y_ is 1 / 4. The reciprocal of _y_ would be 4, so the reciprocal of _z_ is 4 / 3 and _z_ would be 3 / 4.
$$\displaystyle 2\cdot \frac{3}{4} = \frac{3}{2} < 1$$
Quantity B is not necessarily greater than Quantity A.
Incorrect.
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If the two quantities were equal, then _x_ and _z_ would be reciprocals of each other. _x_ • _z_ = 1.
$$\displaystyle x = \frac{1}{z}$$
$$\displaystyle z = \frac{1}{x}$$
This is not consistent with the given information.
Incorrect.
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You do not need the exact values for _x_, _y_, or _z_ to make this comparison. Use the relationship of _x_ with _y_ and _z_ with _y_ to find a relationship between _x_ and _z_. For example, use the first given relationship to write _x_ in terms of _y_.
$$\displaystyle \frac{1}{x} = 2y$$
Multiplying both sides by _x_ and dividing by 2_y_ will give you the value of _x_ in terms of _y_.
$$\displaystyle x = \frac{1}{2y}$$
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
