3A Fractions and Mixed Number Conversions

Fractions can be seen in a variety of places. For example if a pizza is cut into 10 equal-sized pieces and you eat 3 of the pieces, you have eaten 3/10 of the pizza. If you have $40 and you spend $27 you have spent 27/40 of your money. A good understanding of fractions stems from the idea of equal-sized parts of one whole. In the fraction 3/10, the number of parts (3) is the numerator and the total (10) is the denominator. One of the key ideas is that the denominator counts the equal-sized parts in 1 whole. In the pizza example, each same-sized pizza is cut into 10 equal-sized pieces so the denominator is 10. The denominator is key as it gives the size of the pieces of pizza. So each piece is 1/10 of a pizza. Notice if we cut the same-sized pizza into 5 equal-sized pieces, the pieces would each be larger. In fact each piece would be 1/5. From this example we can see that 1/10 is smaller than 1/5. Let’s go back to the pizza cut into 10 equal-sized pieces. Suppose we buy a second pizza of the same size and cut it into 10 equal-sized pieces also. If we ate a total of 13 pieces, we would have eaten 13/10. There are 20 pieces across both pizzas. To express the amount eaten in units of one pizza, use the 10 equal-sized pieces in one pizza as the denominator. So eating 13 pieces when each same-sized pizza is cut into 10 equal-sized slices is 13/10 or 1 whole pizza and 3/10. You can see why we say that $1⁤\frac{3}{10}=\frac{13}{10}$. When the numerator (13) is greater than or equal to the denominator (10), the fraction is called an improper fraction. When the numerator (3) is less than the denominator (10) we call that a proper fraction. A mixed number is a whole number plus a proper fraction $(1⁤\frac{3}{10})$. So we see the improper fraction 13/10 can also be written as a mixed number $1⁤\frac{3}{10}$. How do we convert improper fractions to mixed numbers and vice versa? Let’s review.

To convert a mixed number into an improper fraction we multiply the whole number times the denominator and add the numerator. This becomes the numerator of the improper fraction. The denominator of the improper fraction is the same as the denominator of the mixed number. Look at the mixed number $3⁤\frac{1}{6}$. To convert this into an improper fraction we multiply the whole (3) times the denominator (6) and then add the numerator (1). Hence the improper fraction will be $\frac{3\times 6+1}{6}=\frac{18+1}{6}=\frac{19}{6}$.

To convert an improper fraction into a mixed number we need to do the opposite. All fractions are divisions, so to convert the improper fraction into a mixed number simply divide the numerator by the denominator. Careful, remember the numerator always goes on the inside of the division and the denominator goes on the outside. We also do not want to convert to decimal so just leave the division as quotient and remainder. The quotient is the whole part and the remainder is the numerator of the mixed number.

Look at the improper fraction $\frac{19}{6}$. To convert into a mixed number we divide 19 by 6.

$$\begin{array}{cc}& 3\\ 6& 19\\ & -18\\ & 1\end{array}$$

Since 3 is the quotient that is the whole part. Since 1 is the remainder that is the numerator. The denominator stays the same. Hence the mixed number is $3⁤\frac{1}{6}$.

Practice the following examples with your instructor.

Practice Problems Section 3A

Convert the following mixed numbers into improper fractions.

1. $$3⁤\frac{1}{6}$$

   Check answer

   $$\frac{19}{6}$$

2. $$3⁤\frac{4}{5}$$

3. $$1⁤\frac{5}{8}$$

   Check answer

   $$\frac{13}{8}$$

4. $$4⁤\frac{3}{7}$$

5. $$8⁤\frac{1}{4}$$

   Check answer

   $$\frac{33}{4}$$

6. $$6⁤\frac{3}{5}$$

7. $$10⁤\frac{4}{9}$$

   Check answer

   $$\frac{94}{9}$$

8. $$2⁤\frac{1}{9}$$

9. $$12⁤\frac{9}{10}$$

   Check answer

   $$\frac{129}{10}$$

10. $$14⁤\frac{1}{6}$$

11. $$8⁤\frac{1}{9}$$

    Check answer

    $$\frac{73}{9}$$

12. $$14⁤\frac{3}{11}$$

13. $$2⁤\frac{11}{12}$$

    Check answer

    $$\frac{35}{12}$$

14. $$9⁤\frac{7}{10}$$

15. $$15⁤\frac{2}{7}$$

    Check answer

    $$\frac{107}{7}$$

16. $$20⁤\frac{3}{13}$$

17. $$14⁤\frac{2}{9}$$

    Check answer

    $$\frac{128}{9}$$

18. $$5⁤\frac{6}{17}$$

19. $$12⁤\frac{7}{8}$$

    Check answer

    $$\frac{103}{8}$$

20. $$19⁤\frac{3}{4}$$

21. $$7⁤\frac{19}{21}$$

    Check answer

    $$\frac{166}{21}$$

Convert the following improper fractions into mixed numbers.

22. $$\frac{15}{7}$$

23. $$\frac{13}{2}$$

    Check answer

    $$6⁤\frac{1}{2}$$

24. $$\frac{19}{5}$$

25. $$\frac{16}{3}$$

    Check answer

    $$5⁤\frac{1}{3}$$

26. $$\frac{47}{8}$$

27. $$\frac{21}{4}$$

    Check answer

    $$5⁤\frac{1}{4}$$

28. $$\frac{17}{2}$$

29. $$\frac{85}{7}$$

    Check answer

    $$12⁤\frac{1}{7}$$

30. $$\frac{87}{11}$$

31. $$\frac{93}{12}$$

    Check answer

    $$7⁤\frac{3}{4}$$

32. $$\frac{61}{7}$$

33. $$\frac{29}{4}$$

    Check answer

    $$7⁤\frac{1}{4}$$

34. $$\frac{83}{10}$$

35. $$\frac{47}{2}$$

    Check answer

    $$23⁤\frac{1}{2}$$

36. $$\frac{41}{3}$$

37. $$\frac{133}{11}$$

    Check answer

    $$12⁤\frac{1}{11}$$

38. $$\frac{25}{6}$$

39. $$\frac{98}{9}$$

    Check answer

    $$10⁤\frac{8}{9}$$

40. $$\frac{38}{23}$$

41. $$\frac{91}{18}$$

    Check answer

    $$5⁤\frac{1}{18}$$

42. $$\frac{127}{60}$$

Draw fraction diagrams and write a few sentences explaining why the following improper fractions and mixed numbers are equal.

43. $$2⁤\frac{3}{5}=\frac{13}{5}$$

    Check answer

    Each whole is broken up into 5 parts. If we have a total of 13 parts, it will make 2 whole with 3 parts left over.

44. $$3⁤\frac{5}{6}=\frac{23}{6}$$

45. $$4⁤\frac{3}{7}=\frac{31}{7}$$

    Check answer

    Each whole is broken up into 7 parts. If we have a total of 31 parts, it will make 4 whole with 3 parts left over.

46. $$6⁤\frac{1}{8}=\frac{49}{8}$$

47. $$9⁤\frac{7}{10}=\frac{97}{10}$$

    Check answer

    Each whole is broken up into 10 parts. If we have a total of 97 parts, it will make 9 whole with 7 parts left over.

48. $$5⁤\frac{2}{11}=\frac{57}{11}$$

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.