3B Simplifying and Equivalent Fractions

Another idea we need to review is the idea of equivalent fractions. Suppose I have $40 total and I spend $20 of it. So I have spent 20/40 of my money. But isn’t $20 half of $40? Of course it is. In fact $\frac{20}{40}=\frac{1}{2}$. This is an example of equivalent fractions (fractions that look different but are actually equal). If you recall, ½ is called “simplest form” or “lowest terms” because its numerator and denominator have no common positive whole-number factor greater than 1. Let’s see if we can make sense of the relationship between equal fractions. For a fraction with a nonzero denominator, multiplying its numerator and denominator by the same nonzero number gives an equal fraction. So we can see why ½ is the same as 20/40.

$\frac{1}{2}=\frac{1\times 20}{2\times 20}=\frac{20}{40}$. You can convert any fraction to an equal fraction with this principle.

Let’s convert $\frac{5}{6}$ into an equal fraction with a denominator of 54. The denominator is 6 so what do we have to multiply 6 by to get 54? 9 of course. The key is that if I multiply the denominator by 9, then we should also multiply the numerator by 9. So we will get our equal fraction.

$$\frac{5}{6}=\frac{5\times 9}{6\times 9}=\frac{45}{54}$$

This is an important skill when we add and subtract fractions.

Try a couple of examples with your instructor.

We saw above that $\frac{45}{54}=\frac{5}{6}$, but which is in simplest form? 5/6 of course, but if we are given a fraction like 45/54, how can we find the simplest form? The key is dividing the complete numerator and denominator by the same nonzero common factor. For the positive whole-number numerators and denominators used here, a fraction is in simplest form when they have no common positive whole-number factor greater than 1.

$$\frac{45}{54}=\frac{45÷9}{54÷9}=\frac{5}{6}$$

Look at the following example. Write the fraction $\frac{24}{72}$ in simplest form. Are there any positive whole-number factors greater than 1 that divide both 24 and 72? These are common factors. How about 8? So we can divide out the 8 and the fraction will be simpler. The key is that you have to divide the complete numerator and denominator by the same nonzero common factor.

$$\frac{24}{72}=\frac{24÷8}{72÷8}=\frac{3}{9}$$

Is this in simplest form though? The fraction left is 3/9. Can you think of any numbers that go into both 3 and 9? What about 3? Then we can divide out the 3 as well.

$\frac{3}{9}=\frac{3÷3}{9÷3}=\frac{1}{3}$. There is no positive whole-number factor greater than 1 that divides both 1 and 3. So we are at the simplest form.

Try the following examples with your instructor.

Practice Problems Section 3B

1. Convert $\frac{3}{4}$ into an equal fraction with a denominator of 28.

   Check answer

   $$\frac{21}{28}$$

2. Convert $\frac{3}{5}$ into an equal fraction with a denominator of 15.

3. Convert $\frac{2}{3}$ into an equal fraction with a denominator of 36.

   Check answer

   $$\frac{24}{36}$$

4. Convert $\frac{5}{7}$ into an equal fraction with a denominator of 21.

5. Convert $\frac{3}{11}$ into an equal fraction with a denominator of 66.

   Check answer

   $$\frac{18}{66}$$

6. Convert $\frac{7}{9}$ into an equal fraction with a denominator of 72.

7. Convert $\frac{6}{13}$ into an equal fraction with a denominator of 39.

   Check answer

   $$\frac{18}{39}$$

8. Convert $\frac{4}{7}$ into an equal fraction with a denominator of 63.

9. Convert $\frac{5}{8}$ into an equal fraction with a denominator of 24.

   Check answer

   $$\frac{15}{24}$$

10. Convert $\frac{8}{15}$ into an equal fraction with a denominator of 75.

11. Convert $\frac{11}{21}$ into an equal fraction with a denominator of 84.

    Check answer

    $$\frac{44}{84}$$

12. Convert $\frac{20}{27}$ into an equal fraction with a denominator of 243.

13. Convert $\frac{1}{16}$ into an equal fraction with a denominator of 64.

    Check answer

    $$\frac{4}{64}$$

14. Convert $\frac{4}{49}$ into an equal fraction with a denominator of 98.

15. Convert $\frac{5}{32}$ into an equal fraction with a denominator of 192.

    Check answer

    $$\frac{30}{192}$$

16. Convert $\frac{2}{13}$ into an equal fraction with a denominator of 104.

17. Convert $\frac{12}{23}$ into an equal fraction with a denominator of 69.

    Check answer

    $$\frac{36}{69}$$

18. Convert $\frac{17}{19}$ into an equal fraction with a denominator of 76.

19. Convert $\frac{1}{30}$ into an equal fraction with a denominator of 120.

    Check answer

    $$\frac{4}{120}$$

20. Convert $\frac{5}{22}$ into an equal fraction with a denominator of 154.

21. Convert $\frac{1}{28}$ into an equal fraction with a denominator of 56.

    Check answer

    $$\frac{2}{56}$$

22. Convert $\frac{14}{15}$ into an equal fraction with a denominator of 45.

23. Convert $\frac{25}{31}$ into an equal fraction with a denominator of 310.

    Check answer

    $$\frac{250}{310}$$

24. Convert $\frac{16}{27}$ into an equal fraction with a denominator of 108.

Convert the following fractions into simplest form.

25. $$\frac{5}{20}$$

    Check answer

    $$\frac{1}{4}$$

26. $$\frac{14}{21}$$

27. $$\frac{12}{24}$$

    Check answer

    $$\frac{1}{2}$$

28. $$\frac{25}{30}$$

29. $$\frac{35}{40}$$

    Check answer

    $$\frac{7}{8}$$

30. $$\frac{49}{84}$$

31. $$\frac{24}{90}$$

    Check answer

    $$\frac{4}{15}$$

32. $$\frac{30}{120}$$

33. $$\frac{126}{144}$$

    Check answer

    $$\frac{7}{8}$$

34. $$\frac{96}{108}$$

35. $$\frac{180}{240}$$

    Check answer

    $$\frac{3}{4}$$

36. $$\frac{320}{4000}$$

37. $$\frac{34}{85}$$

    Check answer

    $$\frac{2}{5}$$

38. $$\frac{16}{56}$$

39. $$\frac{121}{132}$$

    Check answer

    $$\frac{11}{12}$$

40. $$\frac{18}{36}$$

41. $$\frac{15}{69}$$

    Check answer

    $$\frac{5}{23}$$

42. $$\frac{48}{144}$$

43. $$\frac{33}{220}$$

    Check answer

    $$\frac{3}{20}$$

44. $$\frac{75}{240}$$

45. $$\frac{28}{112}$$

    Check answer

    $$\frac{1}{4}$$

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