**Middle School Number Sense Lesson 51: Multiplying A Fraction by its Numerator**

We have already covered two different situations where we multiply a fraction by a whole number (in

**MultFracWhole**and

**MultFracWhol2**). Today we will cover a third variety--where the whole number and the numerator are equal. This concept appeared

**13 times**last year, with a median placement at

**question # 51**.

**Number Dojo Level: 225**

If you'll remember from

**MultFracWhol2**, we solved for the numerator, then the denominator, and then the whole number. We will go in the same order, but there is a slightly easier calculation for the numerator.

**How to Solve:**

- For the
**numerator**: Subtract the denominator**(D)**from the numerator**(N)**. Square this difference.*(Don't write this down yet!)* - For the
**denominator**: use the denominator.**Reduce this fraction if possible**, and write the fraction down. - For the
**whole number**: Add the difference from Step 1 to the original whole number**(W)**, and write this down.

**Example 1: 6 x 6/7 = ___**

- Subtract
**N - D**and square the difference. 6 - 7 =**-1**, squared is**1**. This will be our numerator. **D**is 7, so our fraction is**1/7**, which doesn't reduce.- Add
**(N - D)**to**W**. -1 + 6 =**5**. The answer is**5 1/7**.

**Example 2: 13 x 13/10 = ___**

- Subtract
**N - D**and square the difference. 13 - 10 =**3**, squared is**9**. This will be our numerator. **D**is 10, so our fraction is**9/10**, which doesn't reduce.- Add
**(****N - D)**to**W**. 3 + 13 =**16**. The answer is**16 9/10**.

**Example 3: 5 x 5/8 = ___**

- Subtract
**N - D**and square the difference. 5 - 8 =**-3**, squared is**9**. This will be our numerator. **D**is 8, so our fraction is**9/8**, which reduces to**1 1/8**. Write the**1/8**and regroup the**1**.- Add
**(N - D)**to**W**. -3 + 5 =**2**, plus the regrouped 1 is**3**. The answer is**3 1/8**.

**Example 4: 19 x 19/14 = ___**

- Subtract
**N - D**and square the difference. 19 - 14 =**5**, squared is**25**. This will be our numerator. **D**is 14, so our fraction is**25/14**, which reduces to**1 11/14**. Write the**11/14**and regroup the**1**.- Add
**(N - D)**to**W**. 5 + 19 =**24**, plus the regrouped 1 is**25**. The answer is**25 11/14**.

**Example 5: 13/21 x 13 = ___**

- Subtract
**N - D**and square the difference. 13 - 21 =**-8**, squared is**64**. This will be our numerator. **D**is 21, so our fraction is**64/21**, which reduces to**3 1/21**. Write the**1/21**and regroup the**3**.- Add
**(N - D)**to**W**. -8 + 13 =**5**, plus the regrouped 3 is**8**. The answer is**8 1/21**.

**Example 6: 18 x 18/14 = ___**

- Subtract
**N - D**and square the difference. 18 - 14 =**4**, squared is**16**. This will be our numerator. **D**is 14, so our fraction is**16/14**, which reduces to 1 2/14 =**1 1/7**. Write the**1/7**and regroup the**1**.- Add
**(N - D)**to**W**. 4 + 18 = 22, plus the regrouped 1 is**23**. The answer is**23 1/7**.

**Here's a free worksheet to help you practice Multa*a/b:**

multaxa_b.pdf |

**Up Next for Middle School: MultMixSameF**