Back to the class

__Quiz 2__

1. Compute $\dfrac{21}{4} \cdot \dfrac{15}{3}$.

**Solution:** We multiply across and get
$$\dfrac{21}{4} \cdot \dfrac{15}{3} = \dfrac{21 \cdot 15}{4 \cdot 3} = \dfrac{315}{12}=\dfrac{105}{4}.$$
2. Compute $\dfrac{1}{3} + \dfrac{3}{4}$.

**Solution:** Find a common denominator. In this case the common denominator is 12. Now rewrite each fraction in terms of that common denominator (recall that $\dfrac{a}{a}=1$ and that $1 \cdot x = x$ for any $x$).
$$\dfrac{1}{3} = \dfrac{1}{3} \cdot 1 = \dfrac{1}{3} \cdot \dfrac{4}{4} = \dfrac{4}{3 \cdot 4} = \dfrac{4}{12}$$
and
$$\dfrac{3}{4} = \dfrac{3}{4} \cdot 1 = \dfrac{3}{4} \cdot \dfrac{3}{3} = \dfrac{9}{12}.$$
Now we may add and get
$$\dfrac{1}{3} + \dfrac{3}{4} = \dfrac{4}{12} + \dfrac{9}{12} = \dfrac{4+9}{12} = \dfrac{13}{12}.$$
3. Is $\pi$ a rational number?

**Solution:** No. We said this in class. It should have appeared in your notes.

4. Is $\dfrac{1}{2}$ an integer?

**Solution:** No. The integers are the numbers $\{\ldots,-2,-1,0,1,2,\ldots\}$. The number $\dfrac{1}{2}$ does not appear in that list.

5. Is $\dfrac{3}{7}$ a real number?

**Solution:** Yes. The real numbers are the whole real line. We did an example in class where we placed fractions on the number line -- that showed that rational numbers are real numbers.