Question

1. leslie will use more than \\(\frac{1}{2}\\) cup but less than 1 whole cup of flour for a recipe. what fraction of a cup might leslie use? explain.

Explanation:

Step1: Understand the range

We need a fraction \( f \) such that \( \frac{1}{2}

Step2: Choose a denominator and find numerator

Let's take denominator \( 3\). Then half of \( 3\) is \( \frac{3}{2}=1.5\). So we need a numerator \( n\) such that \( 1.5 < n<3\). The integer \( n = 2\) works. So the fraction \( \frac{2}{3}\) is a candidate. Let's check: \( \frac{1}{2}=\frac{3}{6}\) and \( \frac{2}{3}=\frac{4}{6}\). Since \( \frac{3}{6}<\frac{4}{6}<\frac{6}{6} = 1\), \( \frac{2}{3}\) satisfies \( \frac{1}{2}<\frac{2}{3}<1\). Another example: \( \frac{3}{4}\). \( \frac{1}{2}=\frac{2}{4}\) and \( \frac{3}{4}\) is between \( \frac{2}{4}\) and \( \frac{4}{4}=1\).

Step3: General explanation

Any fraction where the numerator is greater than half of the denominator and less than the denominator (for positive proper fractions) or a mixed number less than \( 1\) (but in the form of a fraction, proper fraction with \( \text{numerator}>\frac{1}{2}\times\text{denominator}\) and \( \text{numerator}<\text{denominator}\)) will work. For example, \( \frac{2}{3}\): compare \( \frac{1}{2}=\frac{3}{6}\) and \( \frac{2}{3}=\frac{4}{6}\), \( \frac{3}{6}<\frac{4}{6}\) and \( \frac{4}{6}<\frac{6}{6} = 1\). So \( \frac{2}{3}\) is more than \( \frac{1}{2}\) (because \( \frac{2}{3}-\frac{1}{2}=\frac{4 - 3}{6}=\frac{1}{6}>0\)) and less than \( 1\) (because \( 1-\frac{2}{3}=\frac{1}{3}>0\)).

Answer:

A possible fraction is \( \frac{2}{3}\) (or other fractions like \( \frac{3}{4},\frac{5}{8}\) etc.). For example, \( \frac{2}{3}\) is more than \( \frac{1}{2}\) (since \( \frac{2}{3}=\frac{4}{6}\) and \( \frac{1}{2}=\frac{3}{6}\), \( \frac{4}{6}>\frac{3}{6}\)) and less than \( 1\) (since \( \frac{2}{3}<\frac{3}{3} = 1\)).