3G Adding and Subtracting Fractions

Lizzy has $\frac{1}{4}$pound of sugar left in one bag and $\frac{2}{3}$ pound of sugar in another bag. How much total sugar does she have? Questions like this are why we need to know how to add and subtract fractions.

Let’s review. Remember the denominator of a fraction is how many parts one whole is cut up into and the numerator is the number of parts you have. So if a pizza is cut up into 10 pieces and we now have 3 pieces left, then we have 3/10 of the pizza left. So suppose two medium pizzas were each cut into 10 pieces. We have 3 pieces left in the first pizza and 2 pieces left from the second pizza. How much pizza do we have left total?

$$\frac{3}{10}+\frac{2}{10}$$

First of all notice that the pieces are the same size. That is critical. We would not want to have 1 giant piece of pizza and 1 tiny small piece of pizza and say we have 2 pieces. To add the pieces we need to have the same size piece. In fractions, that means the fractions need to have the same denominator. This problem does. How many total pieces do we have? 3+2=5.

$$\frac{3}{10}+\frac{2}{10}=\frac{3+2}{10}=\frac{5}{10}=\frac{1}{2}$$

The first key is that we do not add the denominators. The denominator is how many equal pieces one whole is cut up into. We did not cut the pizza into 20 pieces; it was cut up into 10 pieces. So the denominator stays the same. We add the numerators and keep the denominator the same. At the end we put the answer in simplest form (1/2).

What do we do if the pizzas are not cut into the same number of pieces? Suppose one pizza was cut into 10 pieces and the other was cut into 4 pieces. We cannot add the pieces because they are not the same size. To make them the same size we can take the 10 piece pizza and cut all of the pieces in half (20 total pieces). We can take the 4 piece pizza and cut all those pieces into 5 pieces each. (That also gives 20 pieces). This is the concept behind getting a common denominator. Let’s look at an example. Let’s suppose we have 3/10 of the pizza left, but now we want to eat ¼ of the pizza. How much will be left?

$\frac{3}{10}-\frac{1}{4}$. We first find the common denominator. This is the Least Common Multiple of the denominators. So we need the smallest number that 4 and 10 divide evenly into. We see from the last example that this is 20. Now we convert both fractions into equal fractions with 20 as the denominator.

$$\frac{3\times 2}{10\times 2}-\frac{1\times 5}{4\times 5}=\frac{6}{20}-\frac{5}{20}$$

Now that they have the same denominator (each piece is 1/20 of the pizza), we can subtract. Remember we keep the denominator the same.

$\frac{6}{20}-\frac{5}{20}=\frac{1}{20}$. There will be 1 piece left.

Try a couple examples with your instructor.

Practice Problems Section 3G

For #1-24, perform the indicated operation and write your answers in simplest form. If an answer is improper, convert it into a mixed number.

1. $$\frac{5}{8}+\frac{3}{8}$$

   Check answer

   1

2. $$\frac{5}{14}+\frac{1}{14}$$

3. $$\frac{5}{18}+\frac{7}{18}$$

   Check answer

   $$\frac{2}{3}$$

4. $$\frac{7}{12}-\frac{1}{12}$$

5. $$\frac{5}{6}-\frac{1}{6}$$

   Check answer

   $$\frac{2}{3}$$

6. $$\frac{17}{30}-\frac{7}{30}$$

7. $$\frac{1}{3}+\frac{4}{5}$$

   Check answer

   $$1⁤\frac{2}{15}$$

8. $$\frac{1}{2}+\frac{3}{11}$$

9. $$\frac{5}{8}+\frac{2}{7}$$

   Check answer

   $$\frac{51}{56}$$

10. $$\frac{9}{10}-\frac{5}{12}$$

11. $$\frac{11}{12}-\frac{1}{3}$$

    Check answer

    $$\frac{7}{12}$$

12. $$\frac{13}{20}-\frac{2}{15}$$

13. $$\frac{2}{11}+\frac{3}{7}$$

    Check answer

    $$\frac{47}{77}$$

14. $$\frac{5}{6}+\frac{2}{3}$$

15. $$\frac{3}{5}+\frac{2}{9}$$

    Check answer

    $$\frac{37}{45}$$

16. $$\frac{7}{18}-\frac{1}{4}$$

17. $$\frac{5}{9}-\frac{1}{6}$$

    Check answer

    $$\frac{7}{18}$$

18. $$\frac{7}{10}-\frac{3}{14}$$

19. $$\frac{5}{8}+\frac{3}{5}$$

    Check answer

    $$1⁤\frac{9}{40}$$

20. $$\frac{5}{12}+\frac{4}{9}$$

21. $$\frac{5}{14}+\frac{3}{10}$$

    Check answer

    $$\frac{23}{35}$$

22. $$\frac{10}{21}-\frac{5}{12}$$

23. $$\frac{9}{11}+\frac{1}{4}$$

    Check answer

    $$1⁤\frac{3}{44}$$

24. $$\frac{9}{14}-\frac{1}{16}$$

25. $$\frac{5}{8}+\frac{2}{5}$$

    Check answer

    $$1⁤\frac{1}{40}$$

26. $$\frac{1}{9}+\frac{3}{4}$$

27. $$\frac{6}{35}+\frac{2}{14}$$

    Check answer

    $$\frac{11}{35}$$

28. $$\frac{5}{12}-\frac{1}{21}$$

29. $$\frac{7}{24}-\frac{5}{36}$$

    Check answer

    $$\frac{11}{72}$$

30. $$\frac{17}{22}-\frac{4}{33}$$

31. $$\frac{8}{25}+\frac{3}{10}$$

    Check answer

    $$\frac{31}{50}$$

32. $$\frac{9}{13}+\frac{1}{6}$$

33. $$\frac{4}{9}-\frac{5}{72}$$

    Check answer

    $$\frac{3}{8}$$

34. $$\frac{15}{16}-\frac{3}{32}$$

35. $$\frac{17}{19}-\frac{1}{2}$$

    Check answer

    $$\frac{15}{38}$$

36. $$\frac{17}{18}-\frac{14}{27}$$

37. $$\frac{13}{80}+\frac{33}{50}-\frac{3}{40}$$

    Check answer

    $$\frac{299}{400}$$

38. $$\frac{99}{100}-\frac{4}{25}-\frac{3}{10}$$

39. $$\frac{64}{77}+\frac{2}{11}-\frac{3}{7}$$

    Check answer

    $$\frac{45}{77}$$

40. $$\frac{19}{21}-\frac{7}{12}+\frac{2}{3}$$

41. $$\frac{2}{15}+\frac{11}{24}-\frac{1}{18}$$

    Check answer

    $$\frac{193}{360}$$

42. $$\frac{7}{8}-\frac{3}{16}-\frac{1}{24}$$

43. Are you ready to help Lizzy? Remember she has $\frac{1}{4}$ pound of sugar left in one bag and $\frac{2}{3}$ pound of sugar in another bag. How much total sugar does she have?

    Check answer

    $\frac{11}{12}$ pound sugar

44. Jim ran 3/4 of a mile on Monday but only 1/5 of a mile on Tuesday. How much farther did he run on Monday than Tuesday?

45. Three boards are nailed together when making a cabinet. The first board has a width of $\frac{3}{8}$ inch, the second board has a width of $\frac{1}{2}$ inch and the third board has a width of $\frac{1}{4}$ inch. What is the total width of the three boards nailed together? Write your answer as a mixed number.

    Check answer

    $1⁤\frac{1}{8}$ inch

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