Problems from Proper and improper fractions

Development:

First of all we will identify the expressions that do not correspond to fractions. We have seen that the fractions must have for the numerator and denominator integers. Of the given list of expressions, there appear two whose numerators or denominators are not entire:

$$\dfrac{0,4}{3,4},\dfrac{\sqrt{4}}{5}.$$

The first one it is not a fraction because it has decimal numbers, and in the second case, it might seem that it is not a fraction for containing a root, but we know that:

$$\sqrt{4}=2$$

And therefore:

$$\dfrac{\sqrt{4}}{5}=\dfrac{2}{5}$$

So it is a fraction.

Next we are going to look at which of them are equal to the unit. We have seen that a fraction is equal to the unit if its numerator and denominator are equal. Two fractions on the list fulfill this condition: $$\dfrac{7}{7} \ \text{ and } \ \dfrac{1}{1}.$$

To find the proper fractions, that is to say, less than the unit, we look at those whose denominator is bigger than the numerator, which are the following ones: $$\dfrac{1}{3},\dfrac{\sqrt{4}}{5}=\dfrac{2}{5},\dfrac{10}{11},\dfrac{7}{9} \ \text{ and } \ \dfrac{4}{7}.$$

If we have correctly solved the exercise, we only have improper fractions left, that is to say, ones bigger than the unit. We can identify them by seeing which ones have the numerator bigger than the denominator. In effect, the only two fractions that we have not classified fulfill this requisite: $$\dfrac{8}{5} \ \text{ and } \ \dfrac{5}{2}$$.

Solution:

• Are not a fraction: $$\dfrac{0,4}{3,4}.$$
• Equal to the unit: $$\dfrac{7}{7}=1 \ \text{ and } \ \dfrac{1}{1}=1.$$
• Proper fractions: $$\dfrac{1}{3} < 1,\dfrac{2}{5} < 1,\dfrac{10}{11} < 1,\dfrac{7}{9} < 1 \ \text{ and } \ \dfrac{4}{7} < 1.$$
• Improper fractions: $$\dfrac{8}{5}>1 \ \text{ and } \ \dfrac{5}{2}>1$$.

Hide solution and development