**Middle School Number Sense Lesson 9: Multiplying Fractions by Whole Numbers**

On middle school tests this year, these questions appeared a whopping

**16 times**--10 of those at question # 9. The other 6 showed up around question # 42 or later. This concept comes in a few varieties; today I will cover the first three.

**Number Dojo Level: 34**

The simplest way to multiply a whole number by a fraction is to divide the whole number by the fraction's denominator and then multiply it by the numerator. I regularly emphasize to my students:

**"Divide before you multiply!"**It is much easier to work with smaller numbers than with larger ones.

For example, if I get a question like:

**6/7 of 28 is ___**, it is much easier for me to first divide 28 by 7 (to get 4) and then multiply that 4 by 6 to get

**24**. It is NOT as easy to multiply 28 by 6 (168) and then divide that by 7 (24). (You probably know that when speaking mathematics, we translate "of" to "times" and "is" to "equals.")

Another way these questions could be presented is

**3/8 x 96 = ___**. Again, it is much easier to divide 96 by 8 (12) and then multiply that by 3 to get

**36**. It is NOT very easy to multiply 96 by 3 (288) and then divide that by 8 (36).

The third variation we'll look at today is when the resulting product is not a whole number, such as in:

**15 x 4/7 = ___**. In this case, when we divide 15 by 7, we get 15/7 or 2 1/7. Multiplying that by 4 gives us

**8 4/7**. I will admit that in some of these questions, it may seem a little easier to multiply first and then divide. But I still prefer to multiply 2 1/7 by 4 than to divide 60 by 7 and then represent that as a mixed number.

**Example 1: 1/3 of 11**

- (Divide before you multiply). Divide 11 by 3 to get 11/3.
- Multiply 11/3 by 1 to get...still
**11/3**. (For some reason, my students tend to get tripped up on these simplest of MultFracWhole questions. If the test doesn't specify "mixed number" or "improper fraction," either is an acceptable answer.)

**Example 2: 5/6 of 42**

- (Divide before you multiply). 42 ÷ 6 = 7.
- 7 x 5 =
**35**.

**Example 3: 3/4 x 56**

- (Divide first). 56 ÷ 4 = 14.
- 14 x 3 =
**42**.

**Example 4: 22 x 5/9 = ___ (mixed number)**

- (Try dividing first). 22 ÷ 9 = 2 4/9.
- 2 4/9 x 5 = 10 20/9 =
**12 2/9**.

**Example 4 (again): 22 x 5/9 = ___ (mixed number)**

- (Try multiplying first). 22 x 5 = 110.
- 110 ÷ 9 =
**12 2/9**. This may be the easier method for you; just practice a few to see for yourself.

**Here's a free worksheet to help you practice MultFracWhole:**

multfracwhole.pdf |

**Up Next for Middle School: EstAddSub**