Question 24 of 58intermediate🔧 ApplyNumerical2 marks

Find three distinct rational numbers that lie strictly between −12-\frac{1}{2} and 14\frac{1}{4}.

Correct Answer

To find three distinct rational numbers between −12-\frac{1}{2} and 14\frac{1}{4}, we can use the average method. As stated, the average of two rational numbers always gives a rational number between them.

First, we find the average of −12-\frac{1}{2} and 14\frac{1}{4}: −12+142=−142=−18\frac{-\frac{1}{2} + \frac{1}{4}}{2} = \frac{-\frac{1}{4}}{2} = -\frac{1}{8}. Next, we find the average of −12-\frac{1}{2} and −18-\frac{1}{8}: −12−182=−582=−516\frac{-\frac{1}{2} - \frac{1}{8}}{2} = \frac{-\frac{5}{8}}{2} = -\frac{5}{16}. Finally, we find the average of −18-\frac{1}{8} and 14\frac{1}{4}: −18+142=182=116\frac{-\frac{1}{8} + \frac{1}{4}}{2} = \frac{\frac{1}{8}}{2} = \frac{1}{16}.

Thus, three distinct rational numbers between −12-\frac{1}{2} and 14\frac{1}{4} are −516-\frac{5}{16}, −18-\frac{1}{8}, and 116\frac{1}{16}.

Exercise: Exercise Set 3.4 | Q: 2 | (Page 12)
For More Understanding

Explanation

The textbook context explains that between any two rational numbers, you can always find another rational number by taking their average. By repeatedly applying this average method, we can find as many rational numbers as needed between the given bounds. First, the average of the given numbers gives the middle number. Then, taking the average of the lower bound and the middle number gives a second number, and the average of the middle number and the upper bound gives a third number.

Solution Steps

  1. Step 1: Find the average of −12-\frac{1}{2} and 14\frac{1}{4} to get the first rational number: −12+142=−142=−18\frac{-\frac{1}{2} + \frac{1}{4}}{2} = \frac{-\frac{1}{4}}{2} = -\frac{1}{8}

  2. Step 2: Find the average of −12-\frac{1}{2} and −18-\frac{1}{8} to get the second rational number: −12+(−18)2=−582=−516\frac{-\frac{1}{2} + (-\frac{1}{8})}{2} = \frac{-\frac{5}{8}}{2} = -\frac{5}{16}

  3. Step 3: Find the average of −18-\frac{1}{8} and 14\frac{1}{4} to get the third rational number: −18+142=182=116\frac{-\frac{1}{8} + \frac{1}{4}}{2} = \frac{\frac{1}{8}}{2} = \frac{1}{16}

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