3D Formulas with Multiplying and Dividing Fractions

A common formula in geometry is the volume of a cone $V=\frac{1}{3}\pi r^{2}h$ where r is the radius and h is the height. A common approximation is $\pi \approx \frac{22}{7}$. Suppose we want to find the volume of a cone with a radius of 5 in and a height of 21 inches. As you can see, to do this type of problem will require us to multiply fractions. Let’s review multiplying and dividing fractions.

Remember, to multiply two fractions, simply multiply the numerators and multiply the denominators and simplify your answer.

For example look at $\frac{1}{5}\times \frac{2}{3}$. Multiplying the numerators and denominators gives us the following: $\frac{1}{5}\times \frac{2}{3}=\frac{1\times 2}{5\times 3}=\frac{2}{15}$ Notice the answer is already in simplest form since there is no positive whole-number factor greater than 1 that divides both 2 and 15.

Let’s look at another example. $\frac{10}{27}\times \frac{9}{5}$. Multiplying the numerators and denominators gives us

$\frac{10}{27}\times \frac{9}{5}=\frac{10\times 9}{27\times 5}=\frac{90}{135}$. This is not in simplest form since 9 and 5 both go into the numerator and denominator. So let’s simplify the answer. $\frac{90÷5}{135÷5}=\frac{18}{27}=\frac{18÷9}{27÷9}=\frac{2}{3}$.

An easier way to do the problem would have been to cancel the common factors first. Here is how to do the problem by cancelling common factors.

$$\frac{10}{27}\times \frac{9}{5}=\frac{10\times 9}{27\times 5}=\frac{5^1\times 2\times 9^1}{3\times 9\times 5}=\frac{2}{3}$$

The volume formula above had a square in it, but how do we square or cube a fraction? Remember that positive integer exponents mean repeated multiplication. Look at the following example:

$${\left(\frac{3}{5}\right)}^2=\frac{3}{5}\times \frac{3}{5}=\frac{9}{25}$$

Try the following multiplication problems with your instructor and simplify your answers.

A common application of multiplying fractions is when we need to find a fraction of a total. To find a fraction of a total, we simply multiply the fraction times the total and simplify.

For example suppose we want to find three-fourths of $3000. All we need to do is multiply the fraction 3/4 times 3000. Remember 3000 can be written as the fraction 3000/1. Multiplying the fractions, we get the following:

$$\frac{3}{4}\times 3000=\frac{3}{4}\times \frac{3000}{1}=\frac{9000}{4}=\text{\$}2250$$

So three-fourths of $3000 is $2,250.

Try the following example with your instructor.

What about dividing fractions? To divide by a nonzero fraction, we multiply by its reciprocal. If you remember, the reciprocal is flipping the fraction. So to divide fractions we flip the nonzero divisor (second fraction) and multiply. Look at the example.

$\frac{9}{10}÷\frac{3}{8}=\frac{9}{10}\times \frac{8}{3}=\frac{72}{30}$. This is not in simplest form since the numerator and denominator have a common factor of 6. Dividing both the numerator and denominator by 6 gives us the answer in simplest form. $\frac{72÷6}{30÷6}=\frac{12}{5}=2⁤\frac{2}{5}$.

Try a couple of division problems with your instructor.

Practice Problems Section 3D

Multiply or divide the following fractions and put your answer in simplest form. If your answer is an improper fraction, convert that to a mixed number also.

1. $$\frac{1}{5}\times \frac{3}{16}$$

   Check answer

   $$\frac{3}{80}$$

2. $$\frac{2}{7}\times \frac{4}{9}$$

3. $$\frac{5}{13}\times \frac{1}{6}$$

   Check answer

   $$\frac{5}{78}$$

4. $$\frac{11}{15}÷\frac{2}{9}$$

5. $$\frac{8}{15}÷\frac{7}{11}$$

   Check answer

   $$\frac{88}{105}$$

6. $$\frac{3}{5}÷14$$

7. $$\frac{4}{25}\times \frac{15}{16}$$

   Check answer

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

8. $$\frac{9}{22}\times \frac{6}{7}$$

9. $$\frac{3}{28}\times \frac{7}{12}$$

   Check answer

   $$\frac{1}{16}$$

10. $$\frac{3}{4}\times \frac{8}{9}$$

11. $$\frac{3}{5}\times \frac{10}{9}$$

    Check answer

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

12. $$\frac{2}{3}\times \frac{3}{5}$$

13. $$\frac{4}{5}÷\frac{8}{15}$$

    Check answer

    $$\frac{3}{2}=1⁤\frac{1}{2}$$

14. $$\frac{10}{21}÷\frac{5}{7}$$

15. $$\frac{5}{6}÷10$$

    Check answer

    $$\frac{1}{12}$$

16. $$\frac{2}{5}\times \frac{7}{9}$$

17. $$\frac{8}{21}\times \frac{7}{4}$$

    Check answer

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

18. $$\frac{1}{7}\times \frac{5}{6}$$

19. $$\frac{6}{17}÷\frac{4}{51}$$

    Check answer

    $$\frac{9}{2}=4⁤\frac{1}{2}$$

20. $$\frac{7}{13}÷\frac{21}{26}$$

21. $$27÷\frac{9}{10}$$

    Check answer

    30

22. $$\frac{3}{4}\times \frac{20}{27}$$

23. $$\frac{7}{16}\times \frac{8}{9}$$

    Check answer

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

24. $$\frac{2}{9}\times \frac{3}{16}$$

25. $$\frac{7}{18}÷\frac{14}{9}$$

    Check answer

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

26. $$\frac{11}{25}÷\frac{22}{125}$$

27. $$\frac{12}{25}÷6$$

    Check answer

    $$\frac{2}{25}$$

28. $$\frac{6}{49}\times \frac{7}{9}$$

29. $$\frac{16}{27}\times \frac{18}{8}$$

    Check answer

    $$\frac{4}{3}=1⁤\frac{1}{3}$$

30. $$\frac{2}{21}\times \frac{14}{8}$$

31. $$\frac{13}{17}÷\frac{39}{85}$$

    Check answer

    $$\frac{5}{3}=1⁤\frac{2}{3}$$

32. $$\frac{15}{28}÷30$$

33. $$18÷\frac{27}{40}$$

    Check answer

    $$\frac{80}{3}=26⁤\frac{2}{3}$$

The following problems involve taking a fraction of a total. Remember, just multiply the fraction times the total and simplify.

34. Maria owed a total of $12,000 in student loans. She has paid two-thirds of the loan. How much has she paid? How much is left?

35. For this problem, suppose that 9/10 of the children in a small town have had their vaccinations. If there are a total of 23,400 children in the town, how many of them are estimated to have had their vaccinations? How many are estimated not to have had their vaccinations?

    Check answer

    21,060 children are estimated to have had their vaccinations; 2,340 children are estimated not to have had their vaccinations.

36. Ricky is a boy scout and is selling popcorn to raise money. It has been his experience that about $\frac{2}{5}$ of the people he talks to will buy popcorn. If Ricky talks to 140 people on Saturday, how many does he expect to buy popcorn?

37. Trevor wants to purchase a car for $24,600, but the car company wants one-twelfth of the money up front. How much money will Trevor need to pay up front?

    Check answer

    $2,050

For the cone problems, use the formula for the volume of a cone $V=\frac{1}{3}\pi r^{2}h$ and the approximation $\pi \approx \frac{22}{7}$ to find the volumes. For the circumference problem, use the formula given in that question with the same approximation for π. Don’t forget to do exponents before multiplication. Simplify your answers.

38. Find the volume of a cone with a radius of 7 in and a height of 12 in.

39. Find the volume of a cone with a radius of 24 cm and a height of 14 cm.

    Check answer

    8,448 cu cm

40. Find the volume of a cone with a radius of 10 feet and a height of 21 feet.

41. The distance around a circle is called the circumference and is given by the formula $C=2\pi r$ where r is the radius. A track is in the shape of a circle with a radius of 98 feet. Find the distance around the circle. Use the approximation $\pi \approx \frac{22}{7}$.

    Check answer

    616 ft

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