Question

1. nathan multiplied $\frac{1}{2}$ by another number. the product was less than $\frac{1}{2}$. which type of number could he have multiplied by? a a whole number b a mixed number c a fraction less than 1 d a fraction greater than 1

Explanation:

Step1: Analyze Option A

Whole numbers are \(0, 1, 2, 3, \dots\). If we multiply \(\frac{1}{2}\) by a whole number like \(1\), we get \(\frac{1}{2}\times1 = \frac{1}{2}\), and if we multiply by \(2\), we get \(\frac{1}{2}\times2 = 1\) which is greater than \(\frac{1}{2}\). So whole numbers (except maybe 0, but 0 times \(\frac{1}{2}\) is 0 which is less, but generally whole numbers start from 0 and include positive integers. However, when we consider non - zero whole numbers, the product is not always less than \(\frac{1}{2}\), so A is not correct.

Step2: Analyze Option B

A mixed number is of the form \(a\frac{b}{c}\) where \(a\geq1\) and \(\frac{b}{c}\) is a proper fraction. So a mixed number is greater than \(1\). If we multiply \(\frac{1}{2}\) by a mixed number, say \(1\frac{1}{2}=\frac{3}{2}\), then \(\frac{1}{2}\times\frac{3}{2}=\frac{3}{4}\) which is greater than \(\frac{1}{2}\). So multiplying by a mixed number will give a product greater than \(\frac{1}{2}\), so B is incorrect.

Step3: Analyze Option C

Let the fraction less than \(1\) be \(\frac{x}{y}\) where \(0<\frac{x}{y}<1\). Then \(\frac{1}{2}\times\frac{x}{y}=\frac{x}{2y}\). Since \(\frac{x}{y}<1\), then \(\frac{x}{2y}<\frac{1}{2}\) (because we are multiplying \(\frac{1}{2}\) by a number less than \(1\)). For example, if we take \(\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\) which is less than \(\frac{1}{2}\).

Step4: Analyze Option D

A fraction greater than \(1\) is of the form \(\frac{x}{y}\) where \(x > y>0\). If we multiply \(\frac{1}{2}\) by a fraction greater than \(1\), say \(\frac{3}{2}\), then \(\frac{1}{2}\times\frac{3}{2}=\frac{3}{4}\) which is greater than \(\frac{1}{2}\). So D is incorrect.

Answer:

C. a fraction less than 1