sat suite question viewer

Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

$\frac{20}{p} = \frac{20}{q} - \frac{20}{r} - \frac{20}{s}$

The given equation relates the positive variables $p$ , $q$ , $r$ , and $s$ . Which of the following is equivalent to $q$ ?

Back question 133 of 148 Next

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148

Explanation

Choice C is correct. Multiplying each side of the given equation by $\frac{1}{20}$ yields $\frac{1}{20} (\frac{20}{p}) = \frac{1}{20} (\frac{20}{q} - \frac{20}{r} - \frac{20}{s})$ . Distributing $\frac{1}{20}$ on each side of this equation yields $\frac{20}{20 p} = \frac{20}{20 q} - \frac{20}{20 r} - \frac{20}{20 s}$ , or $\frac{1}{p} = \frac{1}{q} - \frac{1}{r} - \frac{1}{s}$ . Adding $\frac{1}{r} + \frac{1}{s}$ to each side of this equation yields $\frac{1}{s} + \frac{1}{r} + \frac{1}{p} = \frac{1}{q}$ . Multiplying $StartFraction 1 Over s EndFraction$ by $\frac{p r}{p r}$ , $StartFraction 1 Over r EndFraction$ by $\frac{p s}{p s}$ , and $StartFraction 1 Over p EndFraction$ by $\frac{r s}{r s}$ yields $\frac{p r}{p r s} + \frac{p s}{p r s} + \frac{r s}{p r s} = \frac{1}{q}$ , which is equivalent to $\frac{p r + p s + r s}{p r s} = \frac{1}{q}$ . Since $\frac{p r + p s + r s}{p r s} = \frac{1}{q}$ , and it's given that $p$ , $q$ , $r$ , and $s$ are positive, it follows that the reciprocals of each side of this equation are also equal. Thus, $\frac{p r s}{p r + p s + r s} = \frac{q}{1}$ , or $\frac{p r s}{p r + p s + r s} = q$ . Therefore, $\frac{p r s}{p r + p s + r s}$ is equivalent to $q$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.