Reason about reciprocals and unit fractions: compare 1/x values, bound a decimal between unit fractions, and determine how an interval transforms under x -> 1/x.
Free, timed SHSAT practice sets on reciprocal & unit-fraction reasoning— every question includes a full worked explanation, and there's no account required.
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The reciprocal of \(x\) is \(\frac{1}{x}\). A unit fraction has 1 on top, like \(\frac{1}{5}\).
Bigger denominator, smaller value. Cutting a pizza into more slices makes each slice smaller, so \(\frac{1}{4} > \frac{1}{7} > \frac{1}{10}\).
Closest unit fraction to a decimal: change each choice to a decimal and find the smallest gap. For 0.3: \(\frac{1}{4} = 0.25\) (gap 0.05) and \(\frac{1}{3} \approx 0.333\) (gap about 0.033), so \(\frac{1}{3}\) is closest.
Ranges flip. For positive numbers, taking the reciprocal reverses the order. If \(3 < x < 8\), then \(\frac{1}{8} < \frac{1}{x} < \frac{1}{3}\). Check with a number: \(x = 4\) gives \(\frac{1}{4} = 0.25\), which is between \(0.125\) and \(0.333\).
Common traps: picking the fraction with the biggest denominator as the biggest value, and keeping the original order of a range after you flip it.
For unit fractions, a smaller denominator means a larger value. From greatest to least: \(\frac{1}{5}\) > \(\frac{1}{6}\) > \(\frac{1}{9}\) > \(\frac{1}{12}\), so \(\frac{1}{5}\) is greatest. (Choosing the largest denominator gives \(\frac{1}{12}\), the least value. The other choices are not at either end of the list.)
Convert each unit fraction to a decimal and compare to the target: \(\frac{1}{2}\) \(\approx 0.500\), \(\frac{1}{3}\) \(\approx 0.333\), \(\frac{1}{7}\) \(\approx 0.143\), \(\frac{1}{9}\) \(\approx 0.111\). \(\frac{1}{3}\) is closest to 0.4.
Taking the reciprocal REVERSES the order: since 4 < x < 9, we get \(\frac{1}{9} < \frac{1}{x} < \frac{1}{4}\). Keeping the original order gives \(\frac{1}{4}\) to \(\frac{1}{9}\). Forgetting the reciprocals gives 4 to 9. Taking the reciprocal of only one endpoint gives \(\frac{1}{4}\) to 9.